Wholemovement Blog

Reseed

2011-12-27T14:05:00.000-08:00

This month is a carry through from thelast three months continuing to explore the same foldsidentified in the Oct blog Circumference/TetrahedronNet and continued into Dec., Seed. The following pictures are reformations and joining multiples units to explore combining the newly proportioned units with thosefrom the regular equilateral triangle grid. This unit is from the same grid but revealing different proportional folds in relationship to the circumference.

Below are three views of 8 circles a combination of 4 of each grid folding using variations in last months entry.

Two views of the same model, eight paper plates.

Below: Three views of the model above with four more units added to the ends. Notice the formal difference. If this were an organism the change in form might result in a change of function, an adaption that might allow survival in a given environment. This is how I sometimes look at many of the forms and systems that I model. You do not see the movement inherent in the reformations and joining possibilities by looking at static images in fixed positions.

Twelve Paper plate circles in a tetrahedron pattern.

Below are two views of another model using 8 circles using the same folds for each set of four and reformed to different configurations

Eight circles in a tetrahedron patten.

Below is a model using the same proportional changes expanding to a differently formed centered. The grid is alternately colored areas using a black marker before reconfiguring and joining forms. The bluish color is a bleeding from one of the markers when coated with glue size.

Eight folded circles in an octahedron/tetrahedron pattern.

The end of each unit is colored differently to the same creases.

Below shows one of a the models from last month where four more units have been added.

It is always interesting to expand a system by adding on to what was thought to be finished.

Model from last month.


By adding four of the same reconfigured units a cubic pattern of two intersecting tetrahedra becomes obvious.

Below Another reconfiguration changing the angles and areas to generate
more options for combining and joining.

Two sides of the same reformation.

Two units joined together point to point.

Pushed together on surfaces will form a right or left handed unit.

Eight circles in four sets of two each joined end to end.

The same four sets joined point to point. The asymmetry of each

set does not disrupt the organizing symmetry of the tetrahedron.

Another view of the tetrahedron arrangement

Below: This model has been growing over the last year, a bit at a time, not sure where it is going. Here another layer of nine reconfigured circles have been added.

This is an outgrowth of an octahedron pattern using 32 paper plates.

Below Another variation in reforming four units and joining

around an octahedron center in a tetrahedron pattern.

This brings conclusion to the last couple of months of ongoing exploration into the development of another proportional relationship coming from the folds generated by where the circumference crosses the net of the tetrahedron. This coming year will no doubt reveal more connections and more models. For now I am off to The Joint Mathematics conference in Boston to start the new year.

Best wishes to all in 2012 and that this New Year will bring meaningful progress to each of us and more importantly for our global community. This is not unlike the importance in reforming individual circles that take on greater meaning and increased value by joining with other circles of different formations.

Seed

2011-12-11T04:59:00.001-08:00

Knowing the nature of a seed gives us information about how to facilitate and nurture healthy growth of that seed, from which we can derive benefit. It is important to know where a seed comes from and what to do with it. So it is with the circle. Without knowing the origin of the circle we have little idea about what it is and are left with conditions of appearance and symbolic meaning traditionally handed to us. Knowing origin and nature of the circle helps expand understanding about the circle and increases possibilities for application.

When thinking about origin go as far as possible to the largest context, back to a comprehensive singularity; no parts, primal unity, complete, undifferentiated, inclusive, absolute, God before evolving creation, undisclosed with infinite potential, not just one thing but everything unrealized. Wholeness in circle form reveals Wholemovement. By folding the circle one is decompressing the compression of a spherical seed giving demonstration to countless possibilities of diverse forms and information.

A part of what keeps me folding the same creases over and over again is that this grid of creases continues to reveal new reconfigurations and associations. I do not know what a circle is and find it difficult to accept a definition that describes the tool we use to draw a picture of it. It is equally difficult to accept the circle as being nothing just because we draw it that way and it appears empty. I need to see what is revealed through movement that is impossible to imagine by static construction. I am always surprised and intrigued by reforming the circle and the connections that are generated without loss to the circle.

Imagination is not required to reform the circle into extraordinary forms nor is prior knowledge necessary to reveal information. Folding circles is neither a process of construction or imagination, it is about looking at what is revealed within the Wholemovement of this spherical seed from which comes roots, a trunk, branches, and leaves to cradle life that then imagines and dreams of greater things.

With travel last month there was little time to explore or do much modeling and playing with the reconfigurations shown in last months post. Below are four models that are a continuation of that exploration.

Above) Two different arrangements. The top shows four units in a tetrahedron arrangement. The bottom shows a tetrahedron design of four units on an open icosahedron base breaking the symmetrical form.

Above) Two views of four reconfigurations added to a system developed in last months blog. There is an off-centered symmetry that expands the open complexity of the original system.

Below) Twelve reformed units from last month's blog are added to a preexisting model further expanding the complexity of a tetrahedral design into a more complex form in a truncated tetrahedron pattern.

Circumference/Tetrahedron Net

2011-10-27T11:19:00.000-07:00

Having folded thousands of circles I have again been surprised with what has escaped me over the years.

First fold a tetrahedron (www.wholemovement.com; How to fold the circle page.) Open it to the tetrahedron net.

Below left) Look at where the crease of the inner triangle cross the folded circumference. The creases have been drawn with a marker to more easily identify.

Above right) Connect the three points at the intersection between the circumference and creased triangle. Observe the relationship of this smaller triangle to the two larger ones. By tucking in the circumference going in the opposite direction reverses the orientation making a right or left hand positioned triangle.

Above) Fold a straight crease connecting the two points of intersection between circumference and larger triangle. This crease will line up with one of the folded triangle corners going to the opposite edge. Do this same creasing to all three sides of the inner triangle. Look at creases on both sides of the triangle to see the difference with circumference overlap.

Above) Two ways to reconfigure the circle using these creases. Let's start by exploring the one above on the right using multiples form a system.

Below) Four circles folded as above, reformed and joined in a tetrahedral star-like system.

Below left) By stringing six points around the middle with masking tape shows a zigzag like hexagon band.

Above right) A second taping of points shows a differently angled hexagon plane zigzaging around the center.

Below left) A third hexagon zigzag band is added.

Above right) The fourth zigzag hexagon band is taped revealing eight triangles and the six rhomboids. This shows the Vector Equilibrium pattern of twenty-four edges strung between twelve vertex points.

Below) Six shorter bisectors of the rhomboids are added using tape. This shows thirty edges that outline twenty triangles. By forming a tetrahedron arrangement of four tetrahedra the Vector Equilibrium pattern is revealed in the form of an irregular icosahedron.

The inherent relationship between the tetrahedron net and the folded-over circumference produces information to form a system of combined tetrahedron/vector equilibrium/icosahedron system. This in-between position makes reference to Buckminster Fuller's jitterbug, the collapsing of the VE into an icosahedron, octahedron to the tetrahedron by removing center triangulation. Here we see an unexpected relationship between the tetrahedron and the icosahedron in the folded circle tetrahedron net. Each of the four vertices of the tetrahedron have been extended to three vertices each making a total of twelve individual vertices, the Vector Equilibrium pattern.

Below) This open-centered arrangement is a variation of the four units above joined where the three points of each unit are connected, still keeping to the tetrahedron pattern.

Below) Using the open tetrahedron above four more reconfigured variations are added to the triangle end planes. Again each of the four end have been extended to three individual points, again showing twelve points of the VE pattern.

Below) Another example of an open-centered tetrahedron arrangement using another variation of four circles.

Below) Here the circumference has been folded to the outside rather than inside forming a different looking unit.

Above) Four of the above units are arranged in a tetrahedral pattern with open center, connected on the end points. This too can be extended to twelve vertex points.

Below) Here are three more variations that can be used in multiples or in combinations. There are many possibilities because of the shared congruences.

Above) Here creases of the circle/tetra net have been folded to both right hand and left hand positions showing a symmetry aligned to the tetrahedron net. The opened circle shows both back and front are relationships of the same side of the circle. It would be difficult to come by this information using traditional measuring and construction methods.

Below) Three variations in using right and left hand symmetry folds.

Below) Another variation using both folds. Joining two units becomes either a right or left hand twisting cylinder depending on which side they are attached.

Below) Four single units joined in a tetrahedron pattern forming an open star-like system. Again we see the VE pattern of twelve points.

Bellow) Four double units joined in a tetrahedron pattern.

Above) The two systems above combined to form a single system with slight variations in the units.

Above) Another variation in joining components.

Below) Again with the circumference folded to the outside formed to a tetrahedron pattern.

Above) Eight reconfigured units joined in an octahedron pattern which again give us twenty-four vertices of the VE in a very different form.

This are only a few of the many variations to be explored. Each system can be further developed forming greater complexity of form. Each reconfiguration shows different constraints of shaped planes and angle limitations, yet there are many possibilities in developing systems inherent in the primary patterns of the tetrahedron, octahedron, and icosahedron.

The Vector Equilibrium matrix comes directly from the closest packing order of spheres and is inherent in circle/sphere compression, otherwise there would be no sustainable order through all subsequent folding and joining circles. The tetrahedron is a four-sphere non-centered system and the VE is a thirteen-sphere centered system. The pattern of both can be seen through the rearrangement of four tetrahedra. Both centered and non-centered systems are the same packing of spherical order inherent in a single sphere that is revealed in consistent and principled folding of the circle.

I am always excited to observe things in the circle I have not seen before. I hope this will stimulate you to fold circles and begin to explore some of the extraordinary forms and relationships that are there to be discovered.

Icosahedron Seed

2011-09-24T19:46:00.000-07:00

One of the models from last months blog had enough interest to want to develop it further. I was curious about taking it in two different directions by adding-on different sets of elements that are consistent to the same seed form. At what point would each direction of growth stop, exhaust possibilities, or just break down, or would they?

Below are two views of the icosahedron from last month used as the seed.

Left shows the triangle face of one unit with folded over triangles ends folded 1/3 under to a trapezoid shape and joining the other three units at the middle third edge forming an opening.

Right shows a view of the opening where three units come together, much like an open pinwheel. There are four openings equally arranged around the icosahedron.

Below shows the two lines of development side by side; one on the left and one to the right using the same icosahedron seed. The first level of added-ons are different for each and will determine a specific direction for growth. The choices for what to add are determined by the seed properties and what looks like the more interesting from a number of possibles.

Left shows 12 tetrahedra reformed by folding in 2/3 of the tetrahedron leaving 1/3 bottom layer in a sloped prism configuration. A trapezoid side of each unit was attached to each triangle surrounding the center triangles of the four open tetrahedra that form the seed. They fit congruently to the trapezoids shaped by the creases.
Right shows 12 complete tetrahedra filling in the triangle corners of each opening, leaving 2/3 of each tetrahedron extending out and so changing the openings and telescoping out to larger triangle planes. The pinwheel openings are now regular open triangle planes.

Left shows adding a full tetrahedron to each of the four closed triangle plane.

Right shows a full triangle attached to each congruent open triangle plane.

Left shows the four previously attached tetrahedra where the vertex are opened as if to make room for something inside to come out.

Right shows four tetrahedron used above on left side, where the smaller triangle plane of the 2/3 truncated layer is folded in to accommodate fitting onto the previously added tetrahedra vertex. I decided to leave it this way for now because it shows 20 equilateral triangle planes facing outward reflecting the icosahedron seed. It is much like a seed breaking apart. When holding it in your hands is easy to recognize the interrelated patterns of the tetrahedron, the octahedron and the cube all inherent in this oddly expanded icosahedron.

Development on the left continues by adding some proportionally elongated tetrahedra to the open tetrahedra from above, as if they are coming up from inside. This leaves 12 remaining small open triangles, three for each of the original four pinwheel open ends.

Here the 12 openings have been covered with elongated tetrahedra folded to 1/3 proportion of the larger with the angle falling into alignment with the icosahedron stellations. The cube and the tetrahedron are apparent in this form, the octahedron is difficult to find. This uniquely irregular stellated arrangement comes from the symmetrical pattern of the regular icosahedron.

The icosahedron seed.

In both cases by adding units the inside of the open seed eventually closed making a “solid form.” This kind of expansion could continue by adding more units to surfaces. Eventually surface areas would diminish to where it would no longer be possible to add-on units of the same scale. As the scale of units diminish the add-ons would eventually become textural and possibly move into some kind of fractal like form. I may pick that up later and see if indeed that is what happens.

Why I pursue this kind of modeling is to actually see, to look at what otherwise would be difficult to imagine and mostly impossible to anticipate. There is much learned in the process of hands-on model making that can not be understood in any other way, and of course there is the enjoyment and satisfactions of having done something that holds personal interest and gives momentary resolve to ones curiosity before something else holds the attention.

Right Angle Triangle Icosahedron

2011-08-31T16:35:00.000-07:00

In thinking about the icosahedron and the various possible truncations using equilateral triangles in a tetrahedron patterned net I began to wonder about irregular triangles in the same icosahedron arrangement. Here are a few variations in exploring the same process using the tetrahedron folds to form a right angle tetrahedron; 3 right angle triangles and one equilateral triangle.

Using 4 right angle tetrahedra opened and arranged in the tetrahedron net pattern (one in the middle and 3 off of each side) reveals many interesting variations to the icosahedron. This tetrahedron net pattern for the icosahedron is in a pinwheel shaped net and uses 16 triangles rather than the traditional 20 triangles. The remaining 4 triangles are open planes of relationship due to off-setting the edge joining.

All the models use only 4 circles each in the same patterned arrangement, some using different combinations of the same folds, others using additional creases. This will give you an idea of how the same patterned net can generate a diversity of forms; some are made using a higher frequency folded circle.

Forming the right angle tetrahedron

See website for tetrahedron folding instructions; http://www.wholem ovement.com/index.php?option=com_content&view=article&id=51&Itemid=43 (2. Make a tetrahedron)

Having folded the tetrahedron, open it to the circle showing the triangle net; 9 creases; 3 diameters and 6 chords dividing the inscribed triangle into 4 equal areas. There are 3 sets of 3 points on the circumference, 2 points on each side of the 3 diameters.

The black lines make the creases visible.

Fold the 2 outside points in each set to center point and crease making 6 more creases.

This forms 2 more inscribed equilateral triangles.

Fold one of the inscribed triangles, either to the right or the left of the center triangle. Notice the center triangle is now off center as if it has been rotated where each division of each side in no longer evenly divided. The second line in from each corner is at right angle to the edge. When all 3 corner points are folded behind on this line perpendicular to the edge, 3 right angle corners are formed around the center equilateral triangle.

Bring the right angle corners together, same as folding a regular tetrahedron.

This forms a right angle tetrahedron.

Variations in icosahedron form

See website for folding instructions; (4. Make an icosahedron.) http://www.wholemovement.com/index.php?option=com_content&view=article&id=51&Itemid=43

Fold 4 of the right angle tetrahedra and open them flat and arrange to the net. Tape corresponding edges together as shown below

Here is the icosahedron, 20 faces, using right angle tetrahedra in the spiral net pattern. The proportions of triangles have been changed.

This time turn the net over and find corresponding folded over points less than the full edge length, match up and tape together. This will generate a modified form when assembled partial edge to partial edge.

Below are variations of the same tetrahedron patterned icosahedron net using more folds. Go to Blog Archives Feburary 26, 2011 "Unity Origami" for folding the 4-frequency diameter circle. From reconfiguring those 12 crease into the inscribed triangle and exploring the truncated possibilities from the same icosahedron net much can be discovered. If you get this far then fold the 8-frequency grid, 24 creases, (same blog entry as above) and explore further.

To the right are 2 units of the above variation (using 8-frequency grid) joined together forming a larger unit. The concave depression of each end can be pushed out to a convex configuration, one fitting nicely into the other. Multiples of this unit can be joined to form a benzene ring, an open sphere, the tetra/octa matrix and other interesting systems.

Other systems can be discovered by using multiples of other variations pictured above.

Another variation, tetrahedral in form, is also configured using 4 circles in a tetrahedron patterned icosahedron net.

In the photo below left, the circumference has been folded out rather than in as with those above. The patterned arrangement is consistent to both form changes.

There is no other experiential method of modeling that will demonstrate this kind of transformational process based on the tetrahedron pattern. Were each of the above to be traditionally constructed using individual polygons it would be a difficult job measuring, cutting, and joining, where some of them you would not even know to make, since polygons lack circumference.

The right angle tetrahedron movement of the circle is principle to the formed unit which is fundamental axial division of the octahedron. By using multiples of this unit there is much to be discovered. The icosahedron is only one of many directions to be explored. None of these formations would be possible if the tetrahedron was not principle, observable in the first fold and principle to all that follows. The circle would not be possible without spherical compression, itself a right angle movement.

Jury Duty and the Circle

2011-07-31T19:32:00.000-07:00

Last month I had jury duty for the first time. It was an extraordinary experience that gives me hope for the future that all people can join together in agreement. We were twelve randomly picked strangers that came together to make a decision. During the trial we were not to speak of the proceedings that was on our minds, so there was little else to talk about. We did not trade personal stories or chitchat. Social was not our purpose for being there, rather it was to find fairness in a conflicted situation.

On the fourth day we were given instructions to come up with a judgement and could not leave the room without agreement. A first straw poll showed ten agreed and two did not. It took about seven hours before the two differing views came to see a larger context allowing them to change perspective without making personal concessions. We all walked out of the jury room feeling we had all made the right decision. Then parted probably never again to see each other.

In that jury room was the influencing present of truth, beauty, and goodness; the absolute in a relative and finite situation. Millions of years of evolving civilization and each culture and individual have had to come to some understanding about these concepts that have so much to do with the choices and decisions we make.

We looked for truth in the facts we were given. Only within an enlarged perspective were we able to find meaning that allowed each of us to get beyond our personal interpretations and bias positions. All had to participate, expressing how each felt, so we could identify the social judgement we were assigned to make. That larger context towards greater value revealed what was fair, moral, and ethically right. The appropriate decision was reached by all participants for the millions we represented.

There was beauty in finding the harmony and balance, the rhythms and proportions of human interactions by acknowledging the greater context that embraces conflict. Beauty was not apart from the truth that gave meaning to the facts. By enlarging our perspective conflicting parts become co-ordinates revealing a perspective necessary to find agreement and beauty in that discovery.

Goodness is a term associated with volitional action suggesting personality of will; a consciousness of well-being towards others. In society it is ethics; individually it is moral direction towards appropriate action. Not only was there good in identifying the beauty in truth that underlies conflict, but in so doing we felt that we did the right thing in finding fairness in what had been given us to judge.

There was a joining of moral responsibility and social ethics through the triunity of truth, beauty and goodness, serving as unspoken yet practical guiding for what needed to be done. These Divine values are infinitely reflected in so many ways to move our lives towards something of a finer nature than what we are used to.

Compressing the sphere to a circle disc reveals a triune surface showing the triangulated nature of pattern, principle to all movement. The first fold of the circle, a 1:2 ratio, shows balance in the dual nature of forming the tetrahedron. Two more folds proportionally generate three equally spaced diameters increasing triangular relationships that branch out from there. Truth, beauty and goodness are self-evident qualities in the folded circle/sphere unity as principle root for all subsequent folding. There is greater meaning and purpose when thinking of unity as boundless potential rather than the restrictions of a defined circle unit. I could not help but to think how these qualities are abundant in our lives even when not noticed and over shadowed by conflict of one unit against another. The only resolve is to up-step the context towards unity. The circle functions as both part (simple to do) and Whole (difficult to recognize.)

What is so important...?

2011-06-19T12:08:00.000-07:00

Often I have been asked, “what is so important about the circle?” To ask what is so important about anything means we have lost some capacity to be curious by failing to observe the larger context of what is in question. Things often become static without interest and familiarity appears as generalizations. This question says more about our lack of attention and a need to separate and abstract things from context than indicating anything about the circle.

Each circle gives demonstration to the comprehensive nature of the Whole. All circles are multiple reproductions of the Whole as individualized parts. The physical circle is the only form that demonstrates the concept of Wholeness expanding our understanding about parts. The Whole is self–organizing where multiple parts are generated through the act of organizing. All potential of possibilities are realized through three folded triangular grids that are formed through information revealed in folding. The self-organization of the circle indicates endless potential in the forming process through the reconfiguration of and joining multiples in various combinations where there is no end to the number of parts that can be generated. There is no conflict about the circle as Whole and circle as part. This is obvious with all the ways the circle can be reformed into triangles, squares, pentagons, hexagons and all manner of combinations. We can no longer afford to ignore the implications of folding the circle as it relates directly to geometry and the more abstract concepts of mathematics found in the organizing principles of pattern formation.

Folding

The alignment in folding the circle in half proportionally generates three, four and five folded diameters from which three fundamental triangular grids are folded. Through the symmetry of forming, transforming, reforming and joining multiple circles, the forms and systems change, the grids are constant and the circle remains consistently Whole.

Mathematically we know three diameter lengths are always short of equaling the circumference. As we extend and diminish the length of the diameter the circumference increases and decreases revealing concentric circles The limitation of chords is that they are all contained within the parallel scaling of the circle, infinitely out and in. This includes all tangent lines and circles. The diameter is circle alignment, a symmetry revealing division without fragmentation or separation. The circle contains all expansion, contraction, converging and diverging of individually separated and isolated parts.

Sphere/circle unity

Spherical compression into a circle requires revision in thinking about the dimensionality of the circle and the importance of spherical origin. Through right angle compression nothing of the sphere is lost or added to, only the form has been changed; the transformation to circle remains Whole. No matter how we use it or what ideas we have about the circle it will faithfully reflect back the limitations of our own thinking. Reconfiguring the circle in any way tells us about the properties and limitations of the form of individual expression, not about the nature of the circle. This is true, for there is no logic or demonstration to prove the circle otherwise than Whole.

Experiential, hands-on folding is necessary to move beyond the concept of bringing separated things together to make a "unity." Unity is not made, it is realized. Everything is already together even as we perceive what is individually formed. The concept of unity, of God and creation is discussed in all religions and philosophies. There is no physical demonstration of what those ideas mean until they become personalized. The idea of unity as a structural, principled and sustaining reality can be demonstrated by the self-referencing and reconfiguring of every fold in the circle. Yet these ideas have no meaning until personally realized in some experiential way. This means the circle as Whole is not understandable until you become personally engaged and spent time folding. What other physical models do we have that demonstrate the transformations of unity with such far-reaching implications?

The circle is the only form that remains Whole through endless forming and reforming of surface. It demonstrates the interconnecting matrix that generates all classifications of polygons and polyhedra and many other unclassified formations. If these things were not in the sphere in the first place we could not make them, even with a circle. These forms are there as pattern before any folding is done. We do not create diameters or symmetry, they are revealed by aligning and creasing the circle; we simply identify what is now visible and give them names. This brings up a profound question about the circle, taking us back to those early childhood questions about existence, where we came from and why are we here, and for what purpose?

We explain using polygons to “prove” using construction methods of reason. The circle has no reason only an identifiable logic of self-organizing structural pattern revealed through a self-referencing Whole; first observable in sphere/circle compression, consistently realized in all folding thereafter. Reason proves facts where logic combines knowledge with revealed information through experience and insight. Facts of information, reconfigurations in the folds and in combining circles, have no meaning in themselves; the larger reality is in understanding the comprehensive nature of the circle. That is what I find important about the circle.

Whole-to-parts

Life is not a bottom up or top down process. We have formalized education as primarily bottom up, parts-to-whole, teaching only parts and expecting unity to somehow be understood. Adding facts together does not make a Whole. The reality of our lives reveals both bottom up experience and top down revelation that provides direction and moral insight, a decidedly Whole-to-parts perspective. The word Whole is capitalize referring to inclusive; distinguished from the idea of separated parts joined to make a whole. Parts in number will always fall short of unity.

Is the circle a force of nature, a form in nature, or an invention of human nature; did it evolve through practical usage to become a symbolic representation for the mathematical concept of zero? Or possibly the circle is a contextual creation of relationships that is the only possibility for infinite time/space expression of sustaining pattern? In line with many religious traditions the sphere/circle compression demonstrates an individualized tri-unity association disengagement from a trinitized monotheistic unity, which we have no other word for but God. Possibly all are true and relative within the Whole, as observed modeled in the form of sphere/circle compression and circle/sphere decompression by folding.

This detail of the model below shows individual units each reformed and joined in one of countless possible designs. Each circle is a uniquely formed reproduction of the Whole within a scaled aggregation of paper plate circles. Every circle is part of a larger system that is itself partial to complexities of larger design, yet all endless expressions of the circle Whole

Imagine this model of a limited number of multiple circles, combined with all the other possibilities of reforming and arranging endless multiple circles as the expression of one circle in all time/space instantaneously revealing all possible potentials at once. There is purpose in being given the advantage to experience through time, to spread out over space, to explore and discover with frustration, delight and with wonderment all the unimaginable possibilities and unseen potentials of far distant fulfillment.

Movement

2011-05-26T14:14:00.000-07:00

Having over the last month revisited some of the transforming systems I have explored by folding and joining circles has me rethinking movement. You can see videos of a few of the torus systems at; http://www.facebook.com/wholemovement?sk=app_2392950137 There will be more videos coming about variations on the torus and other kinds of moving and transforming systems inherent in the folds of the circle. Until I get a better understanding of moving videos around, here are a few thoughts.

We are all familiar with any number of geometrical transforming toys, many that appeared in the market after the rush of the Rubik's cube in late 70's, followed by a glut of “transformer toys,” and of course now the simulation of transformation through computer imaging. There is a long history of folded paper transforming and movement systems; the hexaflexagon and origami cranes most commonly known to many. We all have curiosity about and a fascination with movement. What is it to move, to be moved, to change and be changed? What happens as forms change is intimately connected to our ideas about time and space and the ordering of relationships that reveal interconnections between local and distance systems.

We can trace movement through layers of infinitesimally smaller organized subsystems of individualized moving parts; beyond the atom to even smaller imagined parts and systems, about as successfully as we understand movement at speeds beyond our capacity to comprehend. Movement appears relative, so we assume a binary position of movement and non-movement, giving ourselves a standard to measure the space between things, which is only scaffolding to get from the mechanical to the transcendental.

As a child looking at trees moving, I wondered where the wind comes from, who started it; what was that first push? what is it that is moving? I still wonder about movement and where things come from. Most explanations are imagined stories, elaborations in science about how things happen, not where it comes from or why. This moves into a larger context, into religion which has been around as long as we have been aware of forces existing outside of ourselves. Are we not suppose to ask these questions? There is no dogma in questions or in faith; only in the formalize answers and the fixed stories we tell to others, expecting them to believe our story when they have their own.

I am intrigued that we have used the circle as a static symbol for both everything and nothing. Today we agree on mostly nothing, zero, yet desiring everything. We really don't understand either. Together they suggest total integration in evolutionary understanding and in revelation through movement. A circle is a complete, inclusive, and self-referenced concentricity indicating there is no inner or outer boundary, only the perception of an infinite change in scale. The origin for the circle is the sphere, the only dimensional form that can be called Whole, demonstrating unity, and that looks the same whether moving or not. Going to origin reveals more about the nature of something and purpose, than only looking at the function or the thing itself. Only in origin will we find the necessity and understanding for change.

Compressing the sphere changes its form in a single direction of symmetry perpendicular to the expanding circle plane; similar to how galaxies are formed. Three circles are revealed in transforming spherical unity to a triunity. There is no separation only differentiation of surface and redistribution of volume. Nothing is added or taken away, the circle/sphere is Whole. Between the two circle planes is a circle ring, the dynamic agent of differentiation. Triunity of the circle is structural pattern revealed through precessional movement and is principle for all subsequent realization of potential formation; thus is the prologue for another creation story:

Wholeness through movement causes division becoming duality in triangulation with each part consistent to the movement and totally inner-dependent to the Whole. The individual nature of part to Whole regulates the interactions between all formed and unformed parts on all scale, in all time, with purpose.

This still does not answer my question about the first push of wind, it only shows that movement is the first action that happens out of what we imagine to not have movement. From a supposed spherical non-movement into compression is followed by the the folded line of division into the triunity of the circle. That first crease of movement can not be understood by describing a diameter; the information of understanding is in the conscious experience of moving.

Folding the circle decompresses spherical information. Every fold is spherical in nature. Folding is not to just demonstrate what you can do with the circle, it is about what you can discover by observing movement and what information is revealed in circle unity. Folding is not about the skill, inventiveness or creativity expression of the folder, rather it is in understanding the nature of what the circle can do and what is revealed that is instructive for increasing capacity in the folder. Only through observation of movement do we know what we are doing. Circle unity does not change, but movement changes the form, so it is not about the form, rather to understand the movement. The circle is both context and the content; it is the folders perception and understanding that changes.

As we observe the initial sphere/circle transformation, information is generated that upon reflection and consistent development of systematic symmetrical folding leads to an equilateral triangular grid of creases. This triangle grid is the primal matrix for moving the circle; for forming, reforming, transforming, and in-forming the circle/sphere. Reconfiguring this grid is the means to forming the possibilities inherent in the circle. Endless subsystems of interrelated movement and joining in multiples gives expression to what otherwise remains only potential.

The folded grid allows us to experience the transforming unity revealed from within the circle. This is only part of the story. It does not answer my question about the first push of wind, but it does move the story forward.

Transforming Systems

2011-04-20T13:07:00.000-07:00

Folding the circle in half is a transformation. The entire folding process is transformational. Without adding or taking anything away the form of the circle changes without changing the nature of the circle. Creases are the result of the self-referenced and self-organizing sequential folding, but the circle does not move by itself. You must be an active participant.

That first fold is a right angle movement forming a perpendicular chord half way between any two points on the circumference. This is the pattern for all subsequent movement because it happens first. In the following months we will look at various transforming systems by reconfiguring the circle in different ways and joining in multiples to this right angle pattern.

Open Torus Ring

You will need four paper plate circles, four bobby pins and some 3/4" masking tape. Folding the circle in half, then folding three diameters, reconfiguring and hinge joining all four circles together into a circle makes a torus ring. Eight tetrahedra are formed and joined at right angles to each other allowing the ring to move rotationally through the open center.

See the following site for instructions; http://www.wholemovement.com /index.php?option=com_content&view=article&id=51&Itemid=43

Above) After folding it in half and then thirds, open it to the circle and see three diameters.

Below left) Refold it to the cone shape and fold the top curved edge on one side over between the two end points and crease. Turn it over and do the same thing folding over the top curved flap on the opposite side. The four curved edges between the two folded over ends remain unfolded.

Above right) The open circle shows two opposite sectors having straight edges. Refold the curved flaps to the inside (as shown) in the opposite direction of the original fold.

Below) Fold the diameter, the one parallel to and between the straight edges, to itself and use a bobby pin to hold it. This forms two open tetrahedra joined by a common edge. The length of the joined diameter, two radii become a singe edge of joining, is at right angle to the two straight folded over edges.

Do the same folding and joining diameter using the other three circles.

Attach two of the above reconfigured circles together taping with a hinge joint.

To make a hinge joint bring two circle units together attaching on the straight edges with flaps folded over. Rotate one unit to one side where the adjoining surfaces are touching. Tape along the joined edges. Fold all the way to the other side keeping edges together and tape along the opposite side of the edges. This way both sides of edges are taped together making a strong connection with maximum rotational movement between the two units.

Below) Join the other two circles in the same way making two sets of two.

Join the two sets of two in the same way as before by making a hinge joint on each end. Bring straight edges together and taping on both sides of adjoining edges. You have to roll the ring to tape the last pair of joining edges.

The movement of this torus can be seen in the video (1:26) at:

http://www.facebook.com/wholemovement#!/wholemovement?sk=app_2392950137

To make a variation to the above tetratorus use eight closed tetrahedra you will need to fold eight individual tetrahedra. The instructions how to do this are on my site; http://www.wholemovement.com/index.php?option=com_content&view=article&id=51&Itemid=43
Put the tetrahedra together in a circle, edge to opposite edge using hinge joining. You now have a version of the open torus ring made with closed tetrahedra forms.

Elongated Torus Ring

The first of the three videos (1:02) at the site above shows an elongated torus ring which is the next system we will make.

This time use 8 paper plates, again fold three diameters in each. Then we will fold the end point of each diameter to the opposite end point of the same diameter and crease.

In doing this to all three diameters will generate three more diameters making six equally spaced diameters for each circle.

Below) Using the creases fold the circle in half and then into quarters. You will see the folded quarter circle is divided into three equal sections

Bring the two edges together and tape along the edge forming an elongated tetrahedron with a equilateral triangle open end.

Above) Fold and tape another tetrahedra the same way. Bring the two tetrahedra together as shown and hinge tape the taped edges together with the short open ends of the tetrahedra in opposite directions. Make sure to rotate the units as you tape on each side. this will give you the greatest movement.

Fold and tape all eight circles the same way, making four sets of two tetrahedra each.

The two open ends of each set of two will be hinged on the curved edge opposite the hinge joining on each set (the length of the shorter taped edges will be at right angle to the length of the longer taped edges.) Even thought the connecting edges are slightly curved and not straight, they can be taped and it will be strong with tape on both sides. This makes a right angle pattern of movement between the two sets.

When tapping the hinge of the two adjoining tetrahedra make sure the surfaces are face to face.

Rotate hinge to the opposite open face to open face and tape.

Make two sets of four each. Join the two sets of four together using hinge joining on both ends completing the torus ring circle.

Explore and enjoy the movement.

The third video http://www.facebook.com/wholemovement#!/wholemovement?sk=app_2392950137 is a variation using a more extended tetrahedra for the individual tetrahedron unit. We will explore more variations on the torus ring next month.

Properties of the Circle

2011-04-06T19:57:00.000-07:00

It is important to understand the origin and the properties of what we are working with. In this case the 3-D circle needs to be differentiated from the circle image we draw. The circle, the subject of the image, has spatial properties that are unique from all other 3-D forms. We need to observed the differences in properties between 2-D and 3-D if there is to be any clarity and understanding about each.

If we do not know the properties of what we are working with we do not know what it is or what to do with it beyond arbitrarily imposing our will that frequently ends up violating the nature of what we are working with, often being counter to expected results. Lack of understanding properties has proven over time to cause problems and confusion. Properties of the circle set the foundation for all subsequent folding (see previous blogs) and what else you will do with it.

This picture shows a circle and the image of a circle.

The circle originates through spherical compression. Both circle and sphere demonstrate a dual function as individual unit and unity simultaneously. The non-differentiation of spherical surface is transformed by a right angle movement to the direction of an expanding circle that reveals a triunity of three circles

The image shows one circle where as the 3-D circle shows three circles, one on each side and a circle ring. There are two edges where the three planes meet. There is an inside volume and an outside space. Three planes, two edges, two spaces; (3+2+2=7.)

The circle is a triunity of three interdependent circles that can not be separated one without the other. In order to conceptually take them apart unity is destroyed, being left with three abstracted, isolated, and imaginary units. The association of three anythings is a structural pattern and reflects unity. All number of units will never equal unity, for unity is always singular. Units is always plural and infinite in number. Three is the first active number and seven is the most possible associations of three possible.

A, B, C

One set of three (ABC)

Three sets of each individually (A) (B) (C)

Three combination of sets of two each (AB) (AC) (BC)

Drawing a diameter divides the image in two halves. When folding the circle in half it changes the properties where instead of two semicircles on one plane there are six semi-circle planes; six half circles. While this make no rational sense using a 2-D model, it is observationally logical to the circle, which remains whole, retaining unity even as folded.

There is no conflict between folding and drawing circles; they are two very different systems; one is an image/idea of the other. Knowing the difference in properties helps clarify some confusion and greatly expands our understanding of the circle. It introduces a new area of dynamic exploration that in no way denies the theoretical or 2-D mathematics that has been developed. There are well over a hundred relationships, functions, and math concepts in this one fold of the circle into six halves. This is not to suggest one is better than the other, but rather to understand the difference and benefits of both folding and drawing circles and the connections between them. We know the value of drawing circles but there is no precedent for folding the circle and that means we have no experience or understanding about it. Only through the direct experience of folding will we understand the value and benefits inherent in the circle.

Besides the information and the beautiful objects that are revealed by folding, it is fun, interesting and engaing. We have a prejudice of not wanting to have too much fun learning something we have already decided should be difficult; if it is serious we must work at it. We are at our most open to learning when we are having fun and engaged in what holds our interest and simulates curiosity. Long ago we decided that mathematically the circle as image is a symbol for nothing, a place holder to later be replaced by something of value. That has pretty much been the end of the story for quite a long time. It is now time to look at the value of the circle that gives contextual meaning to the other information.

The information and reformation possibilities by folding circles demonstrates we can no longer afford to disregard circle/sphere unity. Because we have not done it before is no reason to continue to ignore it. I am writing these blogs in an attempt to give some understanding about the importance of the circle and that it might possibly simulate you to want to fold the circle and to find out for yourself.

For more of what can be done with folding circles see previous blogs and go to my website;

http://www.wholemovement.com

Unity Origami

2011-02-26T19:57:00.000-08:00

Last month I posted a few pictures of folded circles on Wholemovement Facebook page. Some of you may have wondered how the complexity was achieved from paper plates circles. I thought it might be of interest to some of you to see the process of folding the triangle grid matrix that generates the forms used in assembling those and many other complex systems I have recently been exploring.

This process is similar to unit origami in that you are joining similar paper units to form complex systems. The similarity stops there. Because we are using circles and the circle demonstrates unity beyond all other shapes and forms, we then might call folding paper circles “unity” origami. This is not the unity of adding things together, it is the singular unity wherein all things are already together. It is then only a matter of giving individual expression through reformation of the circle and various arrangements in combination. Squares, triangles, and all other reformations of the circle are never less than the Whole circle. They are all formed from the same folded grid, simply reformed differently in multiples and joined. Unity origami does not carry the limitations and restriction that come along with folding square units. The circle demonstrates spherical unity revealed through patterned movements of reformation.

Steps in folding an equilateral triangle grid matrix.

First fold the circle in half by touching any two points on the circumference together. This first fold tells us this is a simply process of touching points and creasing to bring the circle into alignment.

Fold one end point of the half folded circle to a position half way between the two edges as you change the length by moving along the circumference arc. Do not try to measure; use your eyes to see the proportions. It may be easier to look for equal angles keeping the circumference even.
Do not crease yet.

Fold unfolded part under and line up the end points; one fold over and one fold under like a “Z”. When the points are touching the edges will be even. After turning over and checking both sides to see that everything is even, give it a good crease.

Open to the circle; there are three diameters dividing the circle in six equal sections, seven points. There are three choices to continue folding point to point. We will take one; the other two will also lead to the same complete 8-frequency grid.

Fold end point of one diameter to the opposite end point and crease. Do that the same with all three diameters; end to opposite end. This generates three more diameters that now divides the circle into twelve equal sections.

Fold one end point to the center point and crease.

Open the circle. The radius of the folded diameter is now divided in half and with the fold between radii of two diameters forms an isosceles triangle. Fold one end point of the first fold (the isosceles triangle) to the center similar to before and crease.

Continue around the circle touching every end point of each isosceles triangle to the center and creasing.

When you are back to where you started there will be a folded hexagon star of two intersecting inscribed triangles and three bisecting diameters. The three diameters form six star points with three bisecting diameters half way between each star point.

Each diameter now has two new points of intersections not there before. Each starpoint diameter is divided into four equal sections. I call this a 4-frequency diameter grid. The hexagon diameters are divided equally, the bisecting diameters are not equal in division. Don't look at this image, look at the circle you have folded.

Start with the one diameter; two end points, the center point and two new points that divide the diameter into four equal segments.

Fold end point to the furthest new point on that diameter and crease.

Fold the same end point to the closest new point on the same diameter and crease.

These are now three parallel folds that divide one radius into four equal sections. Do this to each diameter, both ends (six times total.) If you go from one to the other in sequence you will know when you are finished.

You have folded a grid of three diameters where each is divided into eight equal sections.

This I call an 8-frequency diameter circle grid. The three bisecting diameters are now part of the grid of three sets of seven parallel lines each. It is like an octave in music. All the notes you need are there to form endless arrangements between any combination using endless possibilities of intervals. The other two choices with the first three diameters that we did not take are now folded into this 8-frequency grid

This octave can be further divided by following the same process of folding the six end points to the new points of intersection on each individual diameter dividing each division again in half again. It goes from the first fold 1, 2, 4, 8, 16, 32 and so on, until you reach the size limitation of the circle. If you want to take it to a higher frequency start with a larger circle.

The following pictures show the development from the first, a 1-frequency grid to a 32-frequency grid. The higher the frequency the greater complexity can be generated from one circle. It is the 8-frequency octave that is fundamental to scaling out and scaling in, all to the pattern of three.

The pattern does not change while the possibilities in reconfiguring a single circle build in complexity with the increase in frequency. Kids in workshops sometimes fold a 32-frequency grid after showing them only the 8-frequency. They did not know what to do with it when done because of the amount of information. You will get the most out of a higher frequency grid if you have worked with increasing levels of frequencies first. Some students find it a challenge and engaging to just fold and explore the possibilities of reconfiguring the regulated movements of the circle to a higher frequency grid of creases.

You can see this process is straight forward always generating information to continue folding. Each of these individual frequency levels are different directions to explore and will reveal very different reconfigurations of the circle and when you start combining them the possibilities are endless. This is not different that any other kind of frequency modulation except you are doing it with a circle.

The more you understand the tetrahedron as basic to all pattern formation and that the other four regular polyhedra are pattern arrangements for reconfiguring and joining the more you will get from these frequency gird levels. Go to the five "Center Off-Center" blogs for more information, or my website, http://www.wholemovement.com/ for some basic instruction for folding the tetrahedron, octahedron and the icosahedron. There is more folding information on the Wholemovement face book page. http://www.facebook.com/wholemovement

Here is another useful folding for all frequency levels. It is folding an inscribed hexagon, I call an infolded hexagon since we are folding and not drawing it; unless you figure we are drawing it out from the circle by creasing. The hexagon comes directly from the infolded equilateral triangle.

Here is a 16-frequency folded grid circle.
Fold the circumference in forming the equilateral triangle.

Each side is folded under the previous side locking one into the other.

Refold on each diameter folding the equilateral triangle in half along the three perpendicular bisectors, the diameters. Do each one individually.

Open the triangle and put the circumference folds on the outside making three small vesicas. This is a another way to see the difference between three diameter lengths and the distance around the circumference. This proportional relationship is pi.

Open to the circle and fold in the circumference informing a hexagon shape. This is a more familiar representation of pi. There are now many more possibilities for using the circumference in exploring all levels of the triangle grid matrix.

For your first time in folding the grid do not be over concerned if all lines are not exactly parallel, some will be slightly off, but with attention to touching points they will all be close enough. Remember when points are touching the lines will be where they need to be, and subsequent points will be in alignment. Error compounds when you are joining individual units together, not in unity of the circle where error tends to be absorbed.

This is not a difficult process and it will become clear as you begin fold by fold. I would like to hear about and see what you come up with. Feel free to contact me if you have questions or need help with folding, and send pictures of what you have folded.

Enjoy the journey.

Pattern and Design

2011-01-31T19:29:00.000-08:00

Last month while playing with a few left over circle-folded reconfigurations, then picking up a discarded variation of the icosahedron (all open planes) I could see where together they might make an interesting object. For the delight of seeing what it will look like, the satisfaction of actually making it, and the joy of discovering what it will reveal, I spent some time folding more units redesigning them to fit this particular form of the icosahedron pattern.

This is the result of how those circles came together and some thoughts about the process.

There are fifty-one circles; twenty 9” paper plate circles and thirty-one 6” circle filters. They have been all creased to the same folded matrix and reconfigured differently as they are joined to form this patterned arrangement. This object has been coated with glue size which makes holding it an experience different than what you expect; very rigid and smoother than it looks.

This is not as regular as we would expect of the icosahedron. One vertex is open giving polarity to the system. The other vertices are open relationships between each triangle. Each triangle face is uniquely different. Where the triangles join are open locations of local centers for twelve pentagons. The edge channels defining the dodecahedron change in relationship to which triangle they appear and how they were reformed. The primary points of connection are the intersections of triangle and pentagon edges. There is consistency to the icosahedron pattern without the sameness of design. Each circle unit has it own unique characteristic, much like everything in particular that has not been generalized.

This object reveals the conditional pushes and pulls of the folding process in its forming, much like we would see in nature as individual systems grow to fulfill specific environments. The richness of the surface is in the irregularity of parts adhering to design criteria towards giving form to the pattern. After choosing the icosahedron pattern a series of design decisions followed where each unit is predicated on the developing organization and relationships already in place. The unit circle follows circle unity.

Each circle is reconfigured to a 3-6 symmetry, and collectively joined to reflect the 5-10 symmetry of the icosahedron. Each circle is a uniquely different aspect of unity; aligned to the process of forming pattern. Every small decision in folding was circumscribed by previous actions. There is nothing arbitrary, and yet it has none of the regularity and sameness of formulation that is so often seen in generic geometric models.

Having finished the above model I started to play with the icosahedron as an open form (16 solid triangles with 4 open planes.) The options are different because the forming of the pattern is different which changes the design possibilities. I wanted to keep to the same process going in a different direction. By using the same folded units in the above model, with reforming variations, they revealed different optional fits to the configuration of the open icosahedron. The option taken shows a tetrahedron arrangement extending beyond the icosahedron in a more open form. This combines both 3-6 and 5-10 symmetry, proportionally balanced in a way not obvious in the above model.

Two views of this model using 8 paper plate circles ; 4 open tetrahedra form the inner icosahedron and 4 circles form the individual tetrahedra vertex locations.

It is extraordinary to see over years of folding circles how many varied reconfiguration can come from reforming the same 3-6 folded triangle grid. When everything is folded from the same three-diameter grid everything is interrelated and inter-transformable in ways that are unique to folding circles. This means any configuration can be flattened to the circle and reformed into any of number of other units, recombined and joined into a variety of different symmetries and systems without adding any new creases. Once the grid matrix is folded into the circle there are an infinite number of unique possibilities for reforming and joining them. All this is possible because it is in the circle to begin with; all is revealed through keeping an eye to alignment in folding and reforming.

Here is another exploration using the open icosahedron form reconfigured from only 4 paper plate , folded to the same 3-6 grid. The circles are reformed so the four vertex locations of the tetrahedron extend into forming a centralized inside open icosahedron, revealing that the two symmetries are combined in the single tetrahedron/icosahedron pattern. The open icosahedron form is a variation of four open tetrahedra. This reflects back to the tetrahedron as primary structural pattern. The four remaining regular polyhedra are patterned formations in different symmetries of the tetrahedron opened and joined in multiples. This model can function as a unit in a variety of larger systems through small design variations of form changes.

We have gone from using 51 to 8 and now 4 circles. These models went through the same process, all folded to the same pattern, revealing the same symmetries arranged to different forms where each individual system requires a given number of circles to fulfill a uniquely designed expression.

These models can not be made using traditional methods without a preconception of design and a plan to instruct the assembly of each part which would be extremely labor intensive, time consuming, with a lot of frustration and to little purpose. These were made in process by following what developed from the specific forming of pattern down to individual designing of elemens, as revealed, each in turn, giving expression to that pattern. This can only happen with folding circles. What is in the circle is there for anyone that will take the time to fold, to play, and to pay with attention, observing the options and possibilities as they are revealed through the folding process. The difference of interest in each folder insures that every model will be a unique expression. These three models are different than the flat pictures we see of them, in the same way the circle is different than the image we see.

In further thinking about these objects and so many others I have made, they do not come from mathematical computations using “proven” formulas or technological means, or from preplanned designs. This is not technology as we understand it; it is a process of unity becoming individualized through divisional units revealed by folding and unfolding. Putting our need to control aside, and with mindful observation and a willingness to take direction from the information generated by systematically folding the circle we can then begin to discover what previously has been ignored and gone unnoticed. Frankly, with all the ways we use circles, I do not think we value or understand the inclusive nature of what the circle is.

The limitation in technology is that it takes choice away from individually being able to make changes, to conceive new connections and directions outside of that technology. Technology promises to relieve us from the hard work of living, to make our choices, to rob us of the suddenness of discovery and of being human. Technology is a tool; it is not a life process, even though we can fool ourselves into thinking otherwise. Life is where choice is the action, and consequences accumulate. Technology can look to many light years in the past, but it can not envision future evolution with higher values of human achievement. The vision of mind has no limitation. We are intimately involved in cosmic design and purpose and we need to better understand this spatial environment as well as the spirit context for our animal origin and human development.

Being overwhelmed with technological capabilities we are shortchanging our capacity towards envisioning our future. We have desensitized our minds to physiological, psychological, and emotional well being. This is all about alignment towards developing human consciousness to the greater mind and the intimate relationship we have with universe organization, structural pattern and unity that is undeniable. When there is no alignment in folding the circle there is no symmetry and everything is off balance (discussed in the last five posts.) There is no reason to suspect it is any different when making arbitrary choices that are not in alignment with the direction of life moving toward fulfillment. As I continue to play with and explore the circle I can not help but to think that the subtle aspects in human development are often what we have failed to understand about the greater context.

Center Off-Center #5

2010-12-21T12:43:00.000-08:00

Fold an off-center crease (below top left.) Line up the long part of the circumference with itself so the angle that is made on the off-center fold and new edge looks divided in half (below top right.) Turn over and fold the unfolded part to line up to the edge just formed dividing the folded circle into thirds; even up the edges and crease (bottom left.) The dark lines in the opened circle are the resulting creases (below bottom right.)

My brother asked what would happen if the fold was aligned to the smaller fold of the circumference when folding into thirds instead of the larger outside edge as pictured above. So the next step was to align the same right hand point to the smaller circumference edge dividing the new angle in half. Then turn over and line up all the straight edges and crease, again dividing the off-center folded circle into thirds. The lines (below) show a symmetry of folding the same end point to both the larger and smaller parts of the circumference, just as if we folded form both ends exactly the same to the large section. One point of crossing is both a right and left handed fold just by turning the circle over and doing the same thing. Orientation is an important and curious factor.

There are two diameters crossing at the center of the circle that intersect at two places with the first off-center fold. This forms two off-center points and five chords with two triangles on the first folded line where one point of one triangle is to the center of the circle.

Using information from the folds we can make parallel creases by accordion folding in all three directions forming an equilateral triangle grid (below left.) Using the triangle pointed to the center for position we can then fill in all triangles of the same orientation to see better how this triangle grid lines up with the circle (below right.)

As we see the grid does not line up with the circle.

We have previously established that folding the circle is about alignment, not about the center. Here we have a triangular grid centered to the circle without alignment. We can see the consistent symmetry of the grid and that a point of intersection is to the center of the circle but there is no true relationship of the dividing grid to the circumference. The first fold was arbitrarily off-center and still is.

Below are two more arbitrary off-center folds showing the same equilateral triangular grid to a different scale, depending on where the crease is off-center. A grid developed from an off-center fold will never be aligned to the circle.

Alignment comes from the concentric nature of the circle not from the triangulation or regularity of a grid. For this reason no polygon or polyhedron can be whole and will never reveal as much information as the circle to itself. Adherence to alignment between the furthest out and furthest in boundary of the circle reveals an order that far exceeds all other relationships since the movement in both directions is for all practical purposes infinite. When we start out misaligned it is sometimes difficult to discern when the reference is less than a circle. The accuracy of alignment with that first move within the established boundary has everything to do with determining subsequent development.

Let's see how it works folding the 4-8 symmetry. The process of folding is the same, the proportions are different.

The 4-8 symmetry shows one diameter and a change in proportions of triangles (above left.) There is no long or short circumference in the division, the folding is the same from both ends of first fold. Again there is information to accordion fold the right angle triangle grid matrix. Another diameter can be located by lining points of intersection perpendicular to the center crease which will place the center of the circle, thought it is not formed to this grid level. As before the grid is out of alignment with the circle boundary.

Folding the off-center crease to a 5-10 symmetry shows five chords where two are diameters. Again folding on both sides of the off-center line (below top left) show again differently proportioned triangles. By folding the triangle grid we see a very different division of creases (below top right.) By coloring in the triangles of the same orientation shows an out of alignment to the circle boundary.

We have seen folding the circle in half reveals alignment (previous post; Sept 20, #2 and Oct 19, #3.) Folding the half folded circle into thirds consistently forms three equally spaced diameters. This happens with the same consistency but with different proportions that correspond to the 4-8 and 5-10 symmetry. Let's look at the grid from folding the circle in half and how that is different from what we have just seen with developing the grid from the off-center folds.

Fold in half and then fold into thirds showing three diameters.

From this folded information we can fold the equilateral triangle grid similar to what we did previously, only in this case we are folding all possible combinations of touching points and then creasing. This reveals the triangle grid showing the enfolded hexagon star and three more diameters where there is exact alignment to the circle (below top.) The circle is equally divided into twelve sectors. There are three sets of three parallel creases and three diameters. Twelve lines in the grid reflect a pattern formation of three. This is not arbitrary, there is self-organizing and order that comes from alignment of the inner and outer boundary of the circle. This alignment is critical for the full functioning continuation of folding the circle (below bottom.)

This is not about the triangular matrix, or symmetry, it is about unity of the circle. There is an order of organization, balance, symmetry arrangement, and proportion when the circle is folded in half that only occurs with alignment. Any fold out of alignment and off-center will always reveal a consistent grid of triangles from which the center of the circle can be located. But there is only one way to align the circle to gain full benefit from the inclusive nature and potential of the circle and that is to fold the circle in half.

Folding the circle in half seems intuitive or at least a well conditioned first response.

This is the only fold that aligns the circle, a proportional ratio of 1:2. Sequential development reveals three possibilities of symmetries; 3:6, 4:8 and 5:10 (post Oct 19, #3.) The one Whole two parts ratio sets the structural pattern of three, a triunity that happens first with the compression of the sphere to a circle form, thus reflected in the first fold. Consistent developed from that first fold is a true expression of circle/sphere unity. This includes all limited expressions from off-center folding as we have seen.

The off-center folded grid, centered but not aligned, can be brought into alignment when you cut back the boundary of the circle (concentric nature of circle) to correspond to the primary points of intersection of the grid. Even with cutting back information is missing that is there when you start with alignment, but that information can now be generated by folding with the limited information you do have.

This experiential demonstration seems to have direct implications of how we might think about our off-centered and misaligned lives and how we might bring principled organization and balanced to the symmetrical and infinitely concentric proportional nature of life.

This picture is not unlike the disorder of the planet we are living on. It is how I find my couch at the end of the year, with not even a place to sit down. This coming year I plan to find ways to concentrically cut back towards aligning my personal off-centeredness (self-centered perspective) to the more inner and outward boundaries in this most extraordinary existence.

CenterOff-center #4

2010-11-23T14:57:00.000-08:00

Traditionally the circle has a center with concentric circles radiating out from that point defined by using a compass. Another way to understand the circle is that it has no center point, being itself center, where concentric circles go infinitely into and outfrom without boundary in either direction. One seems practical because we are familiar with it and the other is conceptual because we are not familiar with it.

Folding circles demonstrates both ideas. The unit circle has a discernible boundary. Through alignment of boundary a center point is generated and through more folding more center points are revealed. The circle is both unit and unity. It also shows that what is off-center can generate alignment to center. Only through boundary alignment can polyhedra be formed.

(Below left) concentric circles show the idea of a one point center; like one point perspective in drawing, it is a perceptual illusion that happens on a flat plane.

(Above right) through a principled process of folding the tetrahedron net, six center points are revealed (this is not counting the nine tangent points of the drawn circles.) The circle is the quintessential fractal pattern of self-similarity infinitely revealed through folding. Remember the circle is a compressed sphere and by folding the circle we are decompressing spherical information of proportions and relationships. The model is static while the process gives the idea of radiant energy spatially organized into a full spectrum interference pattern.

(Above left) a tetrahedron folded from a one center circle shows one of four sides with a center and the other three sides with partial rings of concentric circles. The 2-D image of a centered circle is not consistent in reforming the circle to 3-D formation.

(Above right) each of the six points of the tetrahedron net, generated by the circle, are a vertex or center point. There is an equally distributed, wrap around organization of surface design that reflects the order of spherical packing in this polyhedral form. Drawing in concentric rings to different circle organization forms the same polyhedral reconfiguration revealing the differences between one center and multiple centers.

(Below left) a tetrahedron arrangement of four tetrahedra made from centered circles all facing in the same direction show the concentric rings are on parallel planes without a common spatial center. Again the 2-D design does not reflect a 3-D spherical arrangement.

(Above right) the arrangement of tetrahedra is the same except the four centers are facing inward. The intrigue is that the emphasis now is on the open octahedron. The arrangement of faces show the concentric circles in axial symmetry with surface continuity. Projecting the concentric rings of each surface design as spherical shells will generate interference patterns with different wave configurations.

(Below left) using multiple centered circles and reforming to the same arrangement as above shows ten spherical centers indicating concentric spherical shells as if the tetrahedron were sliced through spherical packing. The edge division depends on the intervals between center points and here shows sixteen equal segments. These ten spheres reflect the four points and six relationships between them that is observed in the alignment of the fist fold in the circle.

(Below right) one more tetrahedron and two octahedra drawn with the same concentric surface design have been added to show how filling in tetrahedra and octahedra further reveals the closest packed order of spheres.

All observable spherical systems seem to develop from a local center within galaxies that themselves seem to be centered within universes all centered and filled with countless moving centers that are within what can only be called a cosmic matrix; Whole. The center is everywhere and is manifest locally on all scales. Unbounded unity contains everything down passed the smallest single unit. Alignment of any size circle will demonstrate a similar process of interrelationships of centers to a spherical matrix.

Concentric alignment into and out from itself demonstrates something of what we might call “true” center. All other centers are off-center, just as we find with less than half folding (Center Off-Center #1.) Boundary movement revolves around the central pattern forming generations of multiple off-center systems. Intention with purpose generates the movement that brings dynamic forces in time and space together towards alignment. To gain accuracy of alignment, in this case folding, happens in time with experience.

Folding the circle is a practical demonstration of moving from off-center to center, from planet to cosmic, from one to the many, from unit to unity. The value of unit lies within the full context of unity. We create confusion by calling the smallest visible circle a center point reinforced by the idea that concentric movement is away from this non-moving location. Within the idea of centered circle and circle as center are levels of truth that are demonstrated through the dynamics of folding circles.

In thinking about this I made a five-fold system using concentric circles with the off-centered folding and joined them in an icosadodecahedron pattern. We know from previous post, there is boundary distortion with the off-center folding. (Center Off Center #2)

(Above) three symmetry views of the off-center folding with the concentric circles colored in where the black ring is the furthest out boundary of a complete circle that is common to all six off-centered folded circles.

(Above) folding twenty elongated tetrahedra that fit into the tetrahedron openings.T he surface of each tetrahedron is individually designed to the same creased lines necessary for reformation.

(Above) four views of all twenty open spaces filled in with elongated tetrahedra leaving the pentagons open. It is beginning to take on a polyhedral look.

(Above) here all tetrahedra have been removed and twelve pentagon pyramids were folded to fill in the open pentagon spaces . Again the pentagon face of each pyramid were individually designed to the creases necessary for reformation. We can see a nucleus beginning to form within the fragmented boundary.

(Above) two views of evolving polyhedra where all pentagon and triangle spaces have been filled in. The faces of the triangles and pentagons are at different levels from the spherical center; not quite in polyhedral form, but identifiable as the icosadodecahedron pattern with some of the distortion still visible.

(Below) filling in the pentagons up to the radial boundary of the tetrahedra fully completes the icosadodecahedron. A little of the boundary distortion still extends beyond the polygon faces. To extend all faces a bit further out would then totally absorb the distortion from the off-center folding. What started as misaligned folding of circles has become spherically aligned.

Here fifty eight circles are folded to a 5-10 symmetry, reformed and arranged in an icosadodecahedron pattern. The initial first six circles were off-center distortions. The rest were center folded bringing the entire system to a traditional spherically centered polyhedron. By uniquely designing each surface to the information of the folded creases there is a proportional consistency through out all surfaces within each individual unit that reflects unity in the visual complexity of this system. There are profound implications in the demonstration of this boundary aligning process There is beauty in the proportional consistency and harmonizing of individual relationships that reflects alignment and appropriate interaction between all parts within the spherical unit. Each part can be viewed in relationship to the circle as well as the interrelationships between all unit parts and the larger unit to all spherical pattern.

These four posts cover only one short line of exploration into the nature of the circle and what can be observed about the consistency of information that is generated and that it can be used to go further and deeper. Make your own models, have your own observations, and feel free to add your thoughts and ideas about what you find.

Center Off-Center #3

2010-10-19T07:40:00.000-07:00

This is the third post of exploring the alignment of concentric circles as spherical shells starting with folding the circle in half. Folding less than half of the circle causing misalignment that will locate an off-center position and eventually peripheral distortion. We will continue with the center off-center concentric circles and how they relate to polyhedra.

The touching points are not connected except in relationship to the circle as a dynamic system. Aligning two points on the circle boundary will provide consistency in continued development of folding and combining circles with a common spherical center. We see this in the symmetries of the three primary spherical systems: 3-6, 4-8, and 5-10 (below.)

Concentric circles are self-organizing and self-generating structural pattern. The circumference and center are simply the largest and smallest definable circle units. Movement into and out from each circle boundary is a right angle movement. This is reflected in movement by touching any two points forming a crease at right angle to the movement between points. Centering of the circle comes from this alignment of the circle to itself.

(Below left) are equally spaced concentric circles. The intervals are arbitrarily ½ " apart; consistency of intervals is important. This will give an idea of spherical shells as the circle is reformed.

(Below right) shows two arbitrarily placed points on the circumference showing concentric circles from each point. The kite shape is straight-line connecting of intervals between the circles as they intersect. This is the same as folding the circle in half. An interference pattern is inherent between two locations of concentric circles in a right angle relationship.

(Above) Combining the touching points and the center point of the circle show the interference pattern of three center locations. Three is structural, two by themselves are not. Alternate concentric rings are shaded to make them easier to see. The right angle intersection of the kite shape comes from the intervals inherently patterned to circle movement. Folding the circle in half shows alignment of the circle/sphere context.

(Below, top row left) concentric circles with intersection of three folded diameters.

(Below, top row right) Using the same 1/2” intervals and shading alternate rings shows the interference pattern with three diameters. Each of the six points is a local center point with concentric circles. Only two rings of each have been shaded.

(Top, bottom row left) six primary points of the tetrahedron net (a two-frequency triangle) where the center points have two concentric rings each. Expanding the circles to all fifteen points would have been too dense to clearly see the net.

(Above, bottom row right) this wave pattern shows the 13 primary points of intersection forming the hexagon star pattern. Each diameter is divided into four equal sections where each point is a center point. The level of complexity is determined by establishing design constraints, in this case two shaded rings around each local center point.

(Below) here three diameters have been divided into eight equal sections, a consistent development in folding the circle. There are twenty-four creases with two concentric rings around each of the nineteen primary points forming a limited interference pattern. The intervals and shading of rings is consistently 1/2" to keep it simple. Each small white triangle interval is a point of three intersecting creases that coincide with circle intersections.

The interference patterns of concentric circles reveal the polyhedral nets that are inherent in the dynamic ordering of circle division. This is where the polyhedra forms relate directly to the circles and to spherical packing.

(Below, top left) a tetrahedron unit folded from the tetrahedron net (above with six centers) indicates four spheres in closest packed order. In this case the edge length is an eight-frequency division of a two frequency tetrahedron. A single tetrahedron unit does not exist in spherical order, only as an organization of four spheres that have been truncated.

(Below, top right) shows a two-frequency polyhedral form of four tetrahedra units showing ten spheres with the open octahedron space. The edges shows a sixteen-frequency division.

(Above, bottom row) the octahedron relationship is a formed unit of two open and joined tetrahedra. It shows six vertex points, six tangent spheres in spherical packing. The concentric circles defined on each face wrap around showing the spherical pattern in a polyhedral form.

(Below left) by adding eight tetrahedra and an octahedron to the vector equilibrium sphere we see spherical packing by filling in intervals between the thirteen points with concentric rings. There is an interference pattern created between the concentric circles of the centered vector equilibrium sphere and the thirteen local-centered spheres in the closest packed order.

(Above right) is a polyhedral representation of spherical packing of the tetrahedron/octahedron matrix showing eight equal divisions of each unit edge. This association of polyhedra shows patterned spherical origin in the form of truncated spheres.

(Above left) is the octahedron opened with two points joined forming a dual pentagon cap arrangement of four triangle faces and one open triangle plane; five triangle planes around the vertex point, ten triangle planes. Seven spheres are tangent on the surface, but unlike the matrix above, there is interior spherical distortion.

(Above right) add two open tetrahedra to the octahedron net in a tetrahedron pattern and bring edges together to form this icosahedron: sixteen triangle faces and four open triangle planes. This is a non-centered arrangement of twelve tangent circles on the surface planes, but again spherical distortion occurs on the inside. It is not a direct expression of spherical order.

(Below left) each circle forms 1/2 a tetrahedron unit using the 12 creases (shown above.) Each reformed circle reveals the square division of the tetrahedron.

(Above right) the two units joined on open square faces form the tetrahedron. The concentric circles surface design shows a higher frequency division of each edge. More creases generate more complex reformations, still keeping wrap around consistency of surface design.

(below left) this model of the off-center folding of the vector equilibrium from last month will be used to look at how the center off-center come together.

(below right) eight tetrahedra, from above, fill each of the open tetrahedron spaces. The tetrahedra come together at the center point showing a consistency of outward facing triangles that form the open square relationships. The completion of spherical packing has been drawn to indicate an interference pattern of interpenetrating spherical shells.

(Above bottom row) two views of octahedra filling in the six open square spaces This forms a large regular octahedron that begins to approach the off-center periphery. There is a lot of information about the interrelationships between spheres, circles and polyhedra in this system.

Continuing higher frequency development suggest the off-centered periphery will eventually be absorbed and become aligned to the circle center in a polyhedral form. Every location of local phenomena within circle/sphere unity is center, albeit local. It is then a matter of consistent higher frequency development towards bringing the center and off-center into alignment as one.

We will go more into that next month as this exploration continues.

Center Off-Center #2

2010-09-20T19:31:00.000-07:00

This month we continue the center off-center investigation. Can a local center be re-centered to spherical alignment? We have seen any point on the circle can be a center point by folding in half or less than half. Folding comes first before centering a location. This is not like drawing a picture of a circle where the compass first sets the center point. Concentrically scale and perspective determines which circle becomes center, usually the smallest in any system gets to be center.

Fold four circles with three diameters each (see last month’s entry.) Do the same with four less than half folded circles. Before joining the circles into the vector equilibrium arrangement, draw out concentric circles from each point of intersection on all circles in both sets. (Below) The circles are drawn on both sides, on one side every other ring was filled in to keep track of different sides of the circles. Be consistent with the intervals. Spacing does not matter.

(Above) concentric circles in a less than half folded circle.

(Below) join each set of four circles using bobby pins to hold them together. Look at the similarities and differences between the two systems.

The concentric circles in both are in alignment to a center location. The off-center folding shows gaps in the planes; parts of circles are missing. If we filled in the gaps, completing each circle, the spherical periphery would then be aligned to the local center in a concentric form. The local center would be realigned to the spherical center showing an increase in the size of the spherical form.

By filling in with parts the sphere would not be Whole. Nothing can be added or taken if it is Whole. This might seem a bit obtuse, but it is an important distinction to make between the Whole and parts that look whole. We don’t want to go around calling things Whole when in fact they are coherent parts in alignment. There is confusion enough figuring out about centers.

To give some perspective to all of this; …“Actuality exist centermost and expands threrefrom into peripheral infinity; potentialality comes inward from the infinite periphery and converges at the center of all things.”

There is a center of all things and then the multitudes of local centers that through peripheral or boundary alignment go to the same place. Knowing that concentric centering works for the 3-6 symmetry, the vector equilibrium. Will this also work the same for 4-8 and 5-10 symmetries? Will they all go to the center of all things?

The 4-8 symmetry starts with the same folding a circle in half. Fold the half circle into quarters by touching points and creasing. Then fold one point back to the opposite point; touch and crease. Turn over and do the same to the other side; point over to point and crease. This folds the circle into four diameters, eight equal sections. Fold four circles this way. The dark lines are creases.

Do this with four less than half folded circles in the same way. The proportional division of ½, ¼, ⅛ will determine the angles for off-center location the same as it does for the half folding. Do not measure, use your eyes, this is about seeing proportional relationships, not measuring.

Overlap two eighths, reforming the circle into a right angle tetrahedron with an open triangle face having three curved edges and three right angle straight edges. Use tape to hold the circle together. Join two on the straight edges going in opposite directions, forming two open right angle tetrahedra between them. Both curved and straight edges make right angle crossings. Bobby pin the circles together.

Now make another set of two units the same way. Join the two sets of two, symmetrically, with two closed planes completing the two open planes of the other with edges touching. Bobby pin together as before. This makes a spherical octahedron with eight equal open right triangle tetrahedra. Do the same folding and joining with the off-center folded set of circles.

(Left) an example of the half folding showing spherical form. (Right) the same joining using less than in half folding. Both show an octahedron centered pattern. One is spherical; the other is a distorted form. The difference is in peripheral alignment; going back to the first fold.

By filling in around the periphery, spherical potential then moves towards center that is now reflected in the outward form. This demonstrates the possibility of reforming ever expanding distortion of boundary properties to reflect spherical unity. The “converging at the center of all things” brings potential in line with a centered and balanced symmetry. The center is infinitely everywhere when there is alignment of concentricity.

In reforming the circle to a 5-10 symmetry the folding in half and folding less than half circle will be the same. The difference will be in folding five times to a different proportion and using six circles for each set.

1.) Fold circle in half. 2.) Fold one end point of the half circle to the point along the circumference showing a 1:2 division. What is left is one unit and what is folded over is the two units. When it looks correct, lightly crease. 3.) Fold first point back to point of the last fold, reversing the proportions to 2:1, leaving two units with one unit folded over. 4.) Then fold second end point of diameter to the edge of the previous fold. They should look equal with the fold edges down the middle. Lightly crease. 5.) Fold the two sections to the back and together. Everything should line up, if not, go back and make adjustments before giving a strong crease to all folds.

Open the circle to five diameters, ten equal sectors. As with the 3-6 folding, bring one diameter together joining opposite radii with a bobby pin. This forms two open pentagons with two open teterahedra intervals separating the two pentagons (below left). Three and five are odd numbers making these reformings different that with the 4-8 symmetry.

(Above right) Make another unit the same way and join them together. The sides of the pentagons of one unit will close the intervals between pentagons of the other unit.

Fold another circle as before and reform to a double pentagon. Then add that to the two already joined in the same way, completing five open tetrahedra around one of the open pentagons. This will be obvious when your see how the three reconfigured circles fit forming five pentagons around the sixth centered pentagon. As before use bobby pins to join the circles together.

Make two of these sets of three circles each. Join them the same way with edges of open pentagons closing the open sides of the tetrahedra intervals. Putting the two halves together forms an icosadodecahedron sphere. (Below left) This has twelve open pentagons and twenty open triangles.

(Above right) shows joining six less than half folded circles, reconfigured and joined in the same way showing a distorted boundary. They both have the same pattern center. The potential for distortion is endless; spherical alignment is one.

The demonstrations of the less than half folded circles are all folded about two to three inches off alignment of the circle. You can imagine that even a half a millimeter off will cause misalignment with distortion. This is not about the center or measuring, it is about alignment and symmetry. Symmetry is a quality of spherical formation. Alignment is what locates the center. Within the concentricity of the circle, which one is the center? When exactly is the periphery in alignment to the center? How close is close to be called accurate? Is anything less than spherical a loss of symmetry, or simply a distortion at the periphery of an always there center of everything?

Explore the symmetries and concentric circles of varied intervals. Next month we will continue to explore further the relationships between polyhedra, concentric circles, and the center off-center.

Another view of folding the vector equilibrium sphere can be seen at http://www.wholemovement.com

¹ The Urantia Book, Urantia Foundation, Chicago IL 1955, Paper115: section3, p.1262

Center Off-Center

2010-08-23T20:36:00.000-07:00

Accuracy in touching is seeing with the eyes, with the mind, coordinating with the body . To not see is to be off-center. This is not about measurement, it is about movement through space with purpose. Without awareness towards the Whole can we know the center? Can we know off-center? By starting with the circle as Whole, and through action of folding we will explore center and off-center locations.

Axiom for folding circle: Any two of an infinite number of points on the circumference of the circle when touched together and creased will equally divide the circle where the linear distance between points will move spherically at right angle to the diameter diminishing the distance between those points as they join on the circumference, where the linear distance between points becomes half the original distance.

This gives symmetry to the division of the circle

The corollary to this is where any one point on the circumference with any other point not on the circumference (or two points not on the circumference nor aligned to a diameter) when touched together and creased will generate a chord less than diameter where the right angle circular movement will diminish from one point to touching the other, and the line segment distance between the two points is reduced to half the original distance. There is no symmetry to this division of the circle.

The axiom is the only way to symmetrical fold a circle, assuming it to be the first act of folding. The corollary holds true for the circle and applies to folding any shape. Two circles will be used to demonstrate the same process for finding center and off-center locations. Points are touching or they are not; there is proportional harmony or disharmony to the circle. Center is always sustainable, what is not sustainable has no center. Inaccuracy in putting points together becomes arbitrary, random movement without attention to finite boundaries.

(left) axiom; (right) corollary. There is no right or wrong in observation

We see the difference in touching two points on the circumference and touching two points with only one point on the circumference. How does each develop as we apply the same folding process to both?

The next two folds are generated by folding the semicircle into thirds around the circumference. Fold one third in front and one third is folded to the back behind the middle third. It is not necessary to measure; use your eyes to know all points are touching by sliding the folds bringing them into alignment. Don't crease until all the edges are lined up and exactly even. Fold the off-center circle into thirds the same way; sliding circumference sections back and forth until the edges are even (points will not match up, you might look for division of central angle.) When edges look even, then crease. This folding process can be seen at the “How to Fold the Circle” page on my website: www.wholemovement.com

(Left) All chords are diameters. (Right) Only one chord is diameter The lines have been traced with a marker to clearly show the creases in each circle.

Both circles show three evenly spaced chords revealing a point of intersection, where each

is differently located to the circumference. If the points of intersection were a rotational axis, one would wobble and the other would show steady alignment to concentric movement.

To further develop folding in each circle in the same way, reform each by folding one of the chords to itself. The center is even with two opposite radii touching. With the off-center circle the division of chords is uneven, the end points will not meet. Use a bobby pin to hold the creases together. There are three folds, therefore three possibilities for folding a crease onto itself. Only with the off-center circle does it make a difference in the configuration. This joining a crease to itself is called a bowtie configuration.

Fold another circles as before, making another bowtie for each. Join together the two individual sets of each, straight edge to straight edge, using bobbie pins or other means to hold them together.

This makes obvious the difference between center and off-center forming and joining. Each of these sets of two circle each is one half of a spherical pattern. (In the off-center the circles are creased to one-quarter distance of a diameter of the circle, to hold to some proportional consistency. Otherwise they could be folded in anyway.)

Make a duplicate set of each ( two sets of two circles) center and off-center, and join them together respectively on straight edges. (Again described on my website in the How to Fold the Circle section). The off-center is now getting more difficult to work with and the center system goes easily together.

Each is shown in same orientation so the difference is apparent. The one diameter in each of the four off-center circles has been assembled to reveal the one composite circle for comparison to any of the four composite circles seen in the center joining. Other options in joining off-center circles do not show a composite circle. See below.

All center and off-center spherical systems show alternate open triangles and squares.

The off-center of each circle displaces the circumferences showing no regularity of a patterned spherical form. The form is deviant, while carrying an invariant pattern of three equally divided chords within a local center, not obvious in looking at the form.

Here one half of the centered sphere is joined to one half of the off-center sphere. This does not show balance, only that the pattern of developing three equally space chords for both is consistent and allows irregular joining of center and off-center..

There is only one of an infinite number of proportional symmetries in touching two points together in that first fold of the circle. Everything starts form this ratio of 1:2, one Whole to two parts. This is the first expression of symmetrical pattern. Anything short of that will eventually cease to generate. The off-centered asymmetrical forms, while they can be interesting and many, are limited and will not sustain generation through multiple joining.

Below are some examples of variations of off-centered spherical forms.

Below; the edges of each, center and off-center, are folded in-between the six points on each circumference before joining, this is more in keeping with the straight edge flat plane truncated look we are traditionally familiar with. The vector equilibrium, traditionally called the cuboctahedron, is pattern for both; clearly identifiable in center folding, not so easily to see in off-center folding.

Where is the center off-center? There is no outer boundary or inner boundary to concentric circles meaning there is no center to the circle; the circle is the center. The circle demonstrates the center is never out of center; there is only awareness about when inaccuracy slides into random movement and becomes misaligned. The mind functions as a balance, a connection, between seeing a finite center in physical form, and possible insight to perfection of infinite center regardless of form. This folding is a demonstration to seeing what works, therefore is sustainable, and what does not work and is unsustainable.

Over the next couple of months I will continue this center off-center exploration in folding and joining circles.

Math Word Problems

2010-08-03T04:45:00.000-07:00

Years ago when first reading word problems in my math book I felt I was reading about things on another planet. They did not make any sense to me. Why on earth would anyone try to figure out answers to the questions that book was asking. I entertained my self by drawing pictures in the margins, pictures that had a relationship to where I was sitting. The wording of those questions was as strange as the thinking I was supposed to do to get the answers. They always seemed to have a number of different answers even though we were told there was only one right answer. That just didn’t fit my young life.

Now I know there are questions that are not questions at all; where there are no right answers other than the answers that make sense. Traditionally they are called parables. Why didn’t we have parables in my math book? Why aren’t there parables in math books today? Too many right answers I guess.

Here are two contributions for a math book, if anyone cares to use them. They are certainly something one can draw pictures to, even if you can’t come up with the right answers.

Problem #1

The square when resting had a dream of triangles, pentagons, hexagons, trapezoids, octagons, and circles. Upon awaking he was troubled finding this dream disturbing.

The triangle when resting had a dream of squares, pentagons, hexagons, rhomboids, octagons, and circles. Upon awaking the triangle was also troubled finding this dream disturbing, but not as much as the square.

The circle when resting had a dream of triangles and squares, pentagons, hexagons, and all manner of polygons. Upon awakening the circle had a slight bitterness at the delightful recollection of this dream.

The sphere dreaming of all those things, upon awaking did not know it, and so continued dreaming on.

Problem #2

A man comes from a sphere; he does not remember. He grows up on a sphere, surrounded by endless spheres of all different sizes. He looked around and it appeared to him flat. He holds a ball in his hands; bounces it on the “flat” to entertain himself, then for profit, and possibly for the enjoyment of others.

Another man comes from a sphere; he does not remember. He holds an imaginary sphere in his hands; no one can see it. He compressed the sphere into a circle and then sticking his head into the circle his mind was consumed knowing spherical reality.

Beyond Circle is Circle

2010-07-07T05:02:00.000-07:00

Folding circles is not just about geometry, mathematics, education, patterns, information, transformation, beautiful forms, and interesting movement systems. These are simply what is generated by folding and joining circles. Beyond those benefits folding the circle provides a different way of thinking; to reevaluate what we think about the circle, and by extension everything else we think we know. There is no comfort here, growing pains are real. The choices we make are all we can claim to be our own.

With many thousands of years of accumulated knowledge all we can think to do is to destroy the planet to which we have been born, that sustains and nourishes us, and to destroy other life forms, and each other, by conscious design and willful acts of choosing selfishly to consume. From a cosmic perspective this is irrational, irresponsible and totally unacceptable human behavior. We consider ourselves the most intelligent creatures on the planet while collectively we appear to demonstrate the opposite. We have developed relatively high levels of technology without developing any moral responsibility in using it.

Every culture builds on experience of human needs in a local setting. We all access the same mind spirit filtered through local cultural habits. These habits of thinking distort and undermine individual experience. There is a great need to rethink those things we take for granted, that we accept without giving thought. Each culture defines life by its own boundaries for its own survival. Observable facts combined with multi-generational stories cause information fragmentation and global confusion with little understanding of where we are or for what purpose. Reading the genetic code is only a story about local events that stops just before now.

Changing how we think is similar to rearranging elements in an equation and keeping the two sides equal. Nothing changes except the arrangement of symbols. Many of us have e-mail but none of us have e-quality. E-mail is the easy way out of the hard issue of e-quality. If only one person is considered to be more, or less, than others, there is no equality. But equality is an imaginary issue; it only functions in virtual reality.

When holding a circle, it is the same for everyone, nothing more or less than a circle. No matter how many different reformations can be folded, the circle remains Whole. When ask to describe the circle, most people will talk about the image; it has a center, it is round, etc, etc. We are then confronted with reconciling what we have been taught and what we individually experience. So often we deny our experience to uphold the stories we have been told. The circle is our experience which far exceeds the simply stories we have been told about it.

We fight for cultural histories, stories that are in conflict with human values and moral development. We think our story is better just because we happened to be born locally, here or there. Well, we are all born locally, in the same way on the same planet in the same space that is so much larger now than it was when Buckminster Fuller called where we are “spaceship earth.” Years have passed and disorganized mutiny continues. We have not yet figured out who is in control or where we are going. Science does not know how to repair the ship. Too many religious maps get in the way of being able to agree on any one direction.

We seem to be lost in space, warring against each other for ownership without much thought of a cosmic navigator or moral mind that knows the territory. Maybe the fight is not so much about who gets control, but more about flight by those who do not want to participate as crew members to keep this “spaceship earth” in working order and in keeping to the course towards human potential and spiritual birthright.

Within the circle every movement is different; there is no conflict. Everything is interrelated and works together in unity beyond the circle/sphere shape.

Animal fear response is no excuse for separation. Fear prevents us from finding value in differences, from learning math, from approaching the unknown; the “X” factor. Fear disturbs comfort, restricts interaction, reduces meaning and fails to accommodate progressive change. It prevents us and gives us opportunity to realize our better selves. Fear inhibits curiosity, diminishes the wonderment about the diversity of life. Overcoming fear does not happen in the abstract, it is the reality of our lives, the choices we make. To overcome limitations stimulates the mind, enriches the heart, nourishes the soul, and delights the spirit.

We all come up against personal/collective origin. The biggest unknown is the idea of God. We understand the fact of our beginning, yet cannot explain it. We are a purposeful part of creation and there is no agreement as to why. The creator is always more; always embedded within creation. We all are created in the same way, nobody is special, but we are all a different and unique personality. What ever story we accept about origin gives meaning to our existence and has direct impact on the choices we make. The accumulation of facts weaves different cultural stories. It is insight to what happens between stories that tie the facts together making up the reality of our lives.

Holding a paper circle is to experience unity. The triunity of the circle demonstrates an elegant truth in the simple beauty of absolute symmetry, where goodness of movement is reflected in human striving towards these eternal values. These are qualities reflected in the movement of the spherical creation of “spaceship earth”. Busy with local self-interest and ownership we have paid little attention to where we are and where we are going.

Only unity can withstand endless differentiation of division and the appearance of separation. To take away is to pretend there is no unity. There is nothing to add to what is inclusively Whole. The circle is never less, it cannot be more. It should not be a surprise that within endless diversity of all things is unity, the foundational connective. God is the only concept of absolute unity; larger than human mind, beyond human imagination, and personal context for both. Anything less than unity is not; anything more than unity never was. So here we come up against origin and purpose.

There are no facts that give direction, no diagrams and numbers of explanation. Where in our math education do we find what makes us a better person, giving us the experience to change the way we relate to the world; to think differently? What in the abstractions and constructed generalizations of the stories is there to change our thinking towards higher ideals, to ennoble individual growth, to support personal relationships and advance human civilization? The pursuit of mathematics has much value, but very little that nourishes the human soul or elevates understanding through recognition of living truth, responding to beauty, and doing good; bringing triunity to a higher level of experiential reality. The only choice we really have is giving to faith in finding the one story that only we can hear.

Everybody folds the same circle, but creases a different diameter. In that first folding there is inherent value that provides opportunity to be mindful and to experience a different way to think about these things.

One Fold Circle outreach project.

2010-06-07T18:45:00.000-07:00

Years ago when starting to fold circles and working with students, the question came up; “Why don’t we fold circles?” There is no simple answer, but it was clear we would benefit in ways yet outside our thinking by actually folding circles rather than just draw pictures of them. Children fill many pages drawing circles as they imitate writing, and draw representations of many things. We get older and are trained to draw circles in writing and with numbers; as an analogy for going nowhere (around in circles) and in some cases it is everything, most often the circle means nothing, zero. We continue to use the circle to store data, but favor bar code and pixel transfer of information. We accept the utility of circles for multiple mechanical advantages, and marvel at the beauty and perfection of circle forms and movement in nature. We are mystified by the appearance and meaning of crop circles. All mathematics is rooted deeply in the circle.

We cut the static image into parts for mathematical constructions, distort circles for topographical demonstrate of surface equivalents; always using only the idea and image of the circle. We do not have experience or directly dialogue with the circle to understand the nature and unity of what it is. We use the circle for everything except information.

Every child should be folding circles as much as drawing pictures of them. Because we do not fold circles does not mean it has no value. This is simple an ignorant intellectual position of condition that favors folding squares and making circle images. In realizing there is no in-depth information about circles, or folding them, I took it on as my job to explore and understand the nature of folding circles and give demonstration to its value.

I found much that was unexpected about the circle. First, the circle is the only form that can demonstrate the concept of the Whole (which we chose to ignore), while simultaneously functioning as a part across all disciplines. Second, it stimulates and requires stretching our mind beyond the conditional limitations we have accepted. Third, the circle is inclusive to the physical demonstration and development of fundamental patterned information that is not possible with any other method of modeling. In short, folding circles requires physical engagement in coordination that stimulates mind function and engages the spirit of human potential. I do not see this in any other materials or process in such an accessible, direct, principled, and comprehensive way.

Folding circles is a process with far more educational value than any tool we have constructed. This sounds ridiculous to say in an age of extraordinary technological development with such a focus on increasing simulation education. I make this claim from twenty years of folding circles. We have lost sight of a larger perspective about purpose in favor of proving ourselves above all else, even above what we do not understand. We don’t spend much time deeply contemplating things we discover, we are more interested in trying to figure out how to use things for our own rewards.

From the beginning of this exploration I have envisioned folding circles as an integrated educational activity for all students. We have no idea of the difference this would have in the practical, purposeful, progressive, and meaningful effects of educating future generations.

A year ago I decided to count the number of mathematical functions and concepts in the first fold of the circle in half. This was to tie curriculum based information required in formal education directly to folding circles, which would help teachers in making connections. This is not to illustrate math concepts and learn vocabulary, but to observe and discover what is inherent in the circle. More mathematical concepts continued to appear and now there are close to 150 individual math functions, all in one place at the same time, where nothing is added or taken away. This also seems like a ridiculous statement given that we accept learning bits and pieces spread out over a lot of years.

Enable to see this information we must first fold the circle and observe what we are doing. Information is revealed in how we do things, not in the facts about what we have done. We train teachers to teach other peoples after-the-fact experience, thus robbing ourselves of our natural ability to observe and learn from our own experience. Folding circles provides a tool and supports a natural way of learning. This does not preclude guidance from those more knowledgeable in specific areas. The circle is not hierarchical nor favors those with more education. If you can fold it in half, the information is there for anyone that will spend the time.

People that teach teachers tend not to look beyond their own understanding of abstract prescription about math curriculum and how it “should” be taught. Even to achieve what is hoped for with greater benefit is not in their sights. The only way they will know there is an alternate approach to parts-to-whole education, is for someone to give them a Whole-to-parts process. Part of my job is to do that. Folding circles can not be rationally discarded just because we do not do it, or never seriously thought about doing it.

I don’t know of any other way to get this information into the hands of people that teach the teachers that teach our children, other than just giving it to them so they may prove to themselves the worth by their own experience. They don’t have to pay anything, only to read a bit, make one fold in the circle and consider the possible benefits. I have twenty years of doing something that nobody else has felt important enough to do that tells me this is one educational approach we have not yet tried. This project is an effort to pass on some of this information to people that could find ways to meaningfully integrate folding circles into primary level curriculum. It is also a way that others can become involved in this effort of seeding information that will, during some future generation, take root and grow to be as common as grass on which to stretch out and observe the universe in ways yet unseen.

For more about this project go to: http://www.kickstarter.com/projects/1619831705/one-fold-circle-outreach

Thank you for your interest in Wholemovement.

Unity and units

2010-05-03T11:55:00.000-07:00

The circle image is mathematically defined by a specific arrangement of infinite points all equidistant from the center point. The circle is not the arrangement, nor the imaginary points, or the line that forms the image. Nor is there a center to the circle if you think about the movement of concentric circles going in and going out. The circle is the center. Unity is not the uniformity of units. Unity is the context that generates endless units. Units come in all descriptions, material, sizes, and arrangements; even ideas become unitized by similarities as we minimize differences. We add and subtract, divide and multiply them, cut them into parts towards constructing some kind of imaginary unification. All of our knowledge is gained in unit segments. Everything is subject to unit measure and becomes uniformity. We are born, nourished, educated, measured, live by, and die as cultural units identified by name and number. The uniqueness, the interactions, relationships, the context, the values that gives meaning to life has been eliminated in order to generalize formulations of unit construction.

The circle is a one-line symbol. We have broken the line and straightened it. Circles have become a barcode of straight-lines. The longest unit-measure of a circle is the diameter. We assume this straight line times three plus infinitely smaller units find equality with the circumference. We give proof using a straight-line formula thinking we can reconcile the measure of a finite part with the infinitude of unity. It is confusion and irrational to think any number of parts could ever equal the circumference boundary.

It matters little, how many, how much, or in what arrangement we put units, we cannot construct unity. There can never be enough units to demonstrate Unity of the Whole. When something is considered a unit, it is taken from context, isolated, striped of individuality. Meaning has been generalized away leaving an abstract idea, a mathematical term, a comfortable generalization so we do not have to deal with the messiness of individual differences. There is no obligation to a unit. This takes a toll individually and in cultural progression towards any meaningful purpose of value or moral responsibility.

To reduce people, animals, environment, ideas, to units provides a way to rationalize greed and destructive self-centered behavior. For those that have seen the bumper stickers that goes something like; ‘he/she who dies with the most of what ever it is wins,’ can recognize the humor and the seriousness with which we embrace this kind of win/lose separation. This is all too evident by the condition of human culture on this planet. Every human being bought and sold, in every way possible, is a unit mark on somebody’s ledger. Disassociation and separation are cause for confusion, disharmony, and the fears that plague us all.

A unit is an idea about a part in separation from the Whole. Without equality in unique and individual differences there is no unity and we are left with sorted parts and constructed unification. The Whole is ordered to patterns of harmonies and rhythms of integrated and coordinated differences to infinite expression; not one part being better than another. To understand unity as our progressive destination we must first start with unity as our highest ideal, then with discerning mind and determined will, move towards that reality.

The Whole is singular and through movement creation becomes a plural expression of Unity. The Whole is both unity and unit without separation, without number. This brings us around to the circle, a symbol for everything and nothing. The circle is the only form to represent unity, but unity is not the circle. The circle reveals infinite parts in creased lines, areas, and points of intersection. The circle can be reformed to endless configurations. By defining isolated properties rather than interrelationships and interdependent functions between all parts within the Whole we have become confused about the nature of forms of expression of the circle and the circle, of the Whole and parts.

Our animal inheritance binds us to the particulate, abstracted by mind to individual units. Our moral and spiritual endowment motivates us towards unity. We have the capacity to make choices and influence the direction of change, individual and social, towards higher ideals of transcendent perfection. To make progressive choices we must be clear about differences that give value, without being caught in separation and the isolation of units, or in the static form of uniformity. With endless reconfigurations of the circle it is never less than Whole, regardless of the levels of reformation and multiple joining. Each circle remains whole, visibly functioning to reveal endless expression in part. In what ever Religion, and in all the ways we chose to describe and acknowledge this energetic, personally realized, creation of seamless existence, unity belongs to God. Without the Whole there are no parts, without the circle, no lines; without unity there are no units.

Folding a circle in half: Part 2. Principles

2010-04-01T07:06:00.000-07:00

Let’s get back to the first fold in the circle (Feb. entry) and talk about the principles behind the parts that determine the interrelationships between those parts and all subsequent folding with a circle. In deciding that the circle is a mathematical symbol representing nothing, and using fragments to draw 2-D constructions to prove abstract formula, we have failed to understand that the circle is both Whole and part. Understanding the circle is Whole allows us to observe what is principle. Principles happen first; they affect all parts and all folds, reconfigurations, and joining of multiple circles. If we do not know what is principle, what comes first, we do not really know what we are doing.

The sphere is Whole before all other forms. Compression transforms the sphere to circle. Spherical unity is principle to the nature of the circle; it comes first. It is triunity; a disk in space showing three circle planes. By adding the two edges there are five generalized parts. We can add the inside volume and the external space, seven individualized associations of unity. There are seven observable qualities that happen first in this act of compression and again in decompressing spherical information by folding the circle.

Stating with the WHOLE, there is MOVEMENT that creates DIVISION forming a DUALITY in TRIANGULATION, where there is a CONSISTANCY of all parts to the movement of the Whole, and each part is INNER-DEPENDENT to the Whole.

Every fold in the circle reflects these seven principle qualities. They apply to all aspects of our lives.

Notice the first five observations are about the mechanics. They reveal the manifest functionality of external order. The last two observations are relational. They give us the most trouble. We are not consciously consistent in developing progressive habits, and we do not like the idea of being dependent on anyone or anything. We have trouble connecting with the inner, the unseen intention of first cause, the absolute pattern that regulates all purposeful evolutionary formation. The degree of clarity about these two qualities has everything to do with how we relate and interact with each other, and the connections we make that give meaning and value to our lives. These ideas about principles are not just a philosophy illustrated by folding a circle. It comes from direct observation about what happens when a sphere is compressed and again when the circle is folded.

Cutting the circle into parts violates the principles and destroys unity causing disruption, and confusion; then breakdown occurs. We are left with separated pieces and must rely on the inconsistency of construction methods. Unity can not be constructed; it is a function found only with the Whole. Principles are not a function of parts, but of the Whole. We only have to look at what we have done to this planet and the condition of people’s lives to know there is destruction, lack of clarity, and little understanding of principles or purpose. Humanity lacks knowledge of unity, is confused about the Whole, and has become addicted to fragmented construction using bits and pieces for our own short-sighted pleasure. We are under the illusion that we create unity with parts. Yet we do make extraordinary bigger parts from smaller parts.

Understanding these principles help us to recognize unity, support the beauty of endless differences and to progressively benefit from such diverse expressions of inner goodness. We lack responsibility to the interconnections between all people for our lack of understanding our dependency on the source of all being. To better understand what is inclusively principled can help clarify and reconcile the confusion between widely differing forms of social and religions cultures and our individual experience.

The Wholemovement approach to geometry and pattern development differs from traditional understanding based in a mixture of false assumptions, inconsistencies, and an over abundance of self interest. Ask mathematicians what the principles of mathematics are and keep tract of all the different answers you get. We do not have an understanding of what is principle beyond what we think is more important than what somebody else thinks. Folding circles offers a demonstration of principles that are inclusively dynamic to all pattern development of formation, in all fields of study. Knowing what is principle helps to increase our capacity to a clearer understanding about consistency of appropriate behavior about our place in the universe. The sphere/circle reformation and folding is the only principled, experiential, hands-on activity that demonstrates anything about the idea of a comprehensive and inclusive Whole.

Origin and meaning of Wholemovement.

2010-03-03T07:30:00.000-08:00

Over twenty years ago, from curiosity, I went to my first math educators’ conference to see what people do there. I ended up looking at lot of geometry books to see how the education industry defines geometry. Surprisingly over half the books gave no definition. The others gave ‘earth measure’ and ‘measuring things of the earth.’ Geo means earth and metry meaning measure. So this is why measurement is important to teach; because the word says so. That made no sense to me. Measurement is simply a way of keeping track from one location to another. Geometry had to be more than that.

I thought about how the word is broken down. Geo means earth; the earth is spherical, and the sphere is fundamental to everything we know about this universe. The sphere is Whole beyond all other know forms. Metry is simply to measure and measuring is about movement from one time and space location to another. That is when it made sense; geometry comprehensively and inclusively means Wholemovement. The movement of the Whole; complete, self-referencing, revealing through movement everything that is, is not, or yet to be; everything we know and do not know, and all unimaginable potential. The movement of the Whole generates endless parts. No amount of parts will ever realize the Wholemovement. Now I had a context from which to better understand geometry, mathematics, and all other patterns of order and organization of the things in our universe.

Geometry is about relationships, dynamic interactions between everything in time and space. It is not as presented in static, isolated, and abstract images. The concept of Wholemovement was so much more interesting, made more sense, without denying anything of what is in all the books. Wholemovement gave a new meaning to geometry and mathematics that I was unable to find in all the overwhelming amount of fragmented and abstracted information in a discipline defined by its own definition.

We talk about unit in math and geometry; isolated facts and functions we are suppose to learn and the isolated unit sections in which we are to learn them. Units are a function of a linear concept from what appears simple to complex. Nowhere do we talk about unity except maybe as a collection of single units brought together to make a bigger unit, usually call sets. Whole numbers are no more whole than whole math, whole language, the whole family, the whole planet, or the whole universe, all in an unimaginably large cosmic environment. All are systems of relative size and complexity of parts, bigger and smaller, endlessly forming systems, moving, changing, transforming, all interconnected and inner-dependent to the movement of the Whole. No part or combinations of parts, no system can ever equal or reveal the Whole. Without the absolute of the Whole there is little meaning in the relative relationships between parts. Unity belongs to the Whole; unit is separation.

When we consider the Whole we are looking at origin to all possible locations, relationships and interactions. To know the origin of anything is to better know what you are working with. To know origin is to know what happens first, and what happens first is principle to all that follows. There is no meaning without context and the larger context gives the larger meaning. If we do not start from the Whole we will not get there by constructing, there are an infinite number of parts. Without context we are lost in a confusion of constructing what will eventually fall back onto itself. Starting with movement of the Whole, everything is all in the same place; simply spread out over time and throughout space. Thus, through observation and reflection we learn about this extraordinary existence of being.

Learning to act appropriately within the largest context is the most practical thing we can do to survive locally. We all have in common spherical origin. Individually we come from a spherical egg cell on a spherical planet, in a universe of spherical objects, moving in ways too large or small to see. None of this we understand, it just happens and we create informational stories about it. Our experience is not constructing life one unit at a time; it is more comprehensive and dynamic than that. Wholemovement is much closer to our individual experience of the inner-relatedness and outer interconnections that we experience and observe about all life. All of this gives meaning and value beyond just that of local concern.

When using the word Whole as inclusive I capitalize the “W”, otherwise when using whole referring to a big part or what seems coherent, a lower case “w”is used. Whole used to mean health, unity of parts, and unity without parts. It has recently come to be used as a modifier to give greater meaning to what ever we think is important and want to emphasize. The word Whole has been corrupted and currently holds little meaning.

The word geometry reflects a parts-to-whole process of endless parts where the Whole is never reached. This has become the direction of formal education in this country. On the other hand Wholemovement is a Whole-to-parts process of modeling that has allowed me to find a way to make sense of the all the bits and pieces of fragmented information that keep us on the unit level of separation. Understanding that the word geometry comprehensively means Wholemovement can help us move towards health and unity.

Folding a circle in half: Part 1

2010-02-15T15:32:00.000-08:00

The questions in the first post were around the advantages of folding circles. Since we do not fold circles, we fold squares and draw pictures of circles, maybe we should do a little folding for some first hand experience. Then maybe the questions will make more sense and we can have a discussion around a shared experience.

Hold a circle in you hand, you will recognize it, but you have not seen it yet. What questions do you ask? What are its properties? How do you describe it; not what you have learned about the image, but the circle that is in your hands. This is about what you do not know, not what you know. Most students start out without know anything, so they say what it reminds them of rather than what they see. A few things that can be said are that it is Whole, it shows unity, it is dynamic, it moves in space, has an edge, a circle band that connects two circle planes, three circles total. It has volume. Given this information, how do we know what to do with it? Without getting caught up in parts, we can say the Whole circle moves. With that information we move the circle to itself; touching every where. In other words, we fold it in half. Then we crease it leaving an expression of that fold in the creased line.

Open the circle. The question is; what do we have that was not there before we folded it, what has been generated? We are looking for what we have not been trained to see. Most people will say a line; even math teachers will say a line in the middle rather than calling it diameter. When asking how we know the line is in the middle, rarely do I hear the edges and areas are congruent; mostly I hear because it is in half, or they are equal, or are the same. This is a good time to start introducing new words to add clarity to describing our observations, or reconnecting to what we only know in a math context. We can talk about curved and straight lines, points, areas, volume. What else? We have folded a ratio of 1:2.

How many half circles are there, two, four? The circle showing three circles folded dividing three circles into six semicircles. Only the concept and a 2-D construction shows two halves, the physical properties of 3-D are different. There are now many things to observe and talk about. All parts are multi-functional………….

Did you notice what you did when you were folding it? Describe what you did. You will discover what I have observed most people do when folding the circle in half. We all put two imaginary points together, look to line up the edges, and then we creased it.

We now have more information; two specific unseen point locations we touched together. There are two points, one line, and two semicircle areas and two imaginary points used to fold it; we just don’t know where they are yet. Seven parts folded from three circles.

Take a new circle and mark two points anywhere on the circumference: it dose not matter where. Now we see the two points, touch them together and crease. You don’t have to even it up; if the points are accurately touching the circle will be exactly in half. From this we can explore the idea that any two points on the circumference when touched together will fold a crease half way between, at right angle to the distance between the two points, and we see this folding process is about touching points. The first things that happens is principle to all else that follows. What are the principal qualities that you see in this one fold?

We have all folded the circle in half showing different diameters. How do we know that? No two people ever pick the same exact two points. The proof is in the relationship of points to diameter. Look at the properties; the similarities and differences of the parts that are now visible. How many of what we have are there? What are specific relationships between parts? There is a lot to observe. When you think you have seen everything, then draw lines connecting all four dots. This shows the distances that are already exist, this gives the unseen relationships shapes and visibility. Now look again.

We have six relationships between four points in space. Four points in space is a minimum description of a tetrahedron with six edges. Movement is always in two directions; folding in both directions is a 360°spherical pattern of movement. The diameter functions as an axis. That means there are two tetrahedra, one inverse to the other; an inside and outside, or a positive and a negative. Everyone has a differently proportioned kite shape, thus different proportioned tetrahedron. If by chance someone folds two points furthest apart, the kite shape will be a square. What are the differences between a kite and a square?

This is plenty to get you started and demonstrates how a little curiosity and attention, when catching our interest, reveals a lot of information. You should be able to recognize many of the over one hundred and twenty mathematical functions and relationships in this one fold. Guidance with clarifying questions will allow your students to discover many of these functions through their own observations about what they have done. It makes no sense to tell students what somebody else has discovered when they have the capacity to discover for themselves through their own experience. If they do that, it will always be theirs. A process for discovering things has begun, and quite possibly they will discover things other people have missed.

Fold the circle in half and let us know what you have discovered.

Think about it

2010-01-28T07:57:00.000-08:00

If there are over one hundred and twenty mathematical functions and relationships to be observed in one fold of the circle in half, why don’t we fold circles in schools to teach mathematics?
If math can be discovered by anyone observing what is generated by folding the circle why aren’t our children folding circles along with drawing pictures of circles.
If the symbol of the circle is nothing (zero) and the circle is everything (Whole) wouldn’t it make sense to start with everything rather than nothing?
If the circle is Whole, a compressed sphere, why do we continue to teach mathematics in a fragmented fashion using bits and pieces where we cut the sphere apart to show a circle?
If the Pythagorean Theorem can be discovered in one fold of the circle why do we teach it as an abstraction to be proved by constructing it?
If the first fold of the circles reveals the principles of pattern development, fundamental to mathematics and systems formation, why don’t we teach the principles of mathematics? They are the same principles for everything else.
If the circle allows us to model using straight and curved edges why do we use only the straight edges of polygons?
If folding the circle one time reveals the Fibonacci progression of numbers, why don’t we let young children discover this progression?
If the circle is origin to all polygons, why do we teach children mostly about polygons?
If folding the circumference of a circle reveals proportions, ratios, and relationships not possible in folding the perimeters of polygons, why don’t we fold circles?
If the circle is comprehensive why do we only give students parts and tell them what to look for rather than allowing them to tell us what they see making their own connections within the folded circle?
If the circle in space is dynamic and generates information why do we only drawing static pictures of it?
If the square is limited to four sides why don’t we fold circles that are without limitation of sides?
Why do we think simplicity is one thing in isolation rather than unity as a comprehensive Whole of all things revealed sequentially?
If the meaning of anything is defined by the context why do we take things out of context and teach isolation, limited to only a few connections?
If all information about triangles is revealed in three folds of the circle why do we want to continue to teach about triangles piece by piece?
If folding a circle in half generates six semi-circles and constructing a drawing with a diameter yields only two semi-circles, why do we keep constructing circles showing only two?
If everything in the circle is in the context of every thing else why do we want to take it apart as if there is no context?
Why do we put emphasis on measurement when understanding is in the ratios and proportions of the self-referencing, self-generating, and self-distributive Whole?
If three is structural and seven is the most possible combinations of associations of three, why is this not taught as basic properties of numbers?
If all parts folded in the circle are multifunctional and interconnected, why don’t we teach students about the unity and interrelatedness of all parts.
If all polygons are demonstrable by folding circles, why do we limit ourselves with constructing polygons piece by piece?
If one fold in the circle forms a dual tetrahedra pattern, and nine creases make the “solid” form, why aren’t students folding tetrahedra rather than constructing with templates made by someone else?
If all regular polyhedra can be formed by folding tetrahedra, opening and joining in multiples, why don’t we do that?
If the circle is both Whole and parts, not demonstrable by any other shape or form, wouldn’t it be to our advantage to fold circles?
If every fold in the circle is a spherical pattern of movement revealing a straight line perpendicular to and half way between the points, why do we draw straight lines to show the distance between points?
All fundamentals of geometry and mathematics are generated by folding the circle; why do we require students to construct this information when it is inherent in the circle and is there to be observed by anyone?
If the circle is Whole, inherently containing everything fundamental about geometry and mathematics, pattern and pattern formation, why don’t we fold circles?

We do not fold circles because we do not believe it is Whole, even though we call it whole. We only draw pictures of circles and fold squares. Neither of those activities will tell us anything about the nature of the circle and what can be generated by folding circles. If they did we would already be folding circles.