### Archive

- June 2015 (1)
- May 2015 (1)
- March 2015 (1)
- January 2015 (1)
- December 2014 (1)
- July 2014 (1)
- June 2014 (1)
- April 2014 (1)
- March 2014 (1)
- January 2014 (1)
- August 2013 (1)
- July 2013 (1)
- June 2013 (1)
- May 2013 (1)
- April 2013 (1)
- March 2013 (1)
- February 2013 (2)
- December 2012 (1)
- November 2012 (2)
- October 2012 (1)
- September 2012 (1)
- August 2012 (1)
- July 2012 (2)
- June 2012 (1)
- May 2012 (1)
- April 2012 (2)
- March 2012 (1)
- February 2012 (1)
- September 2011 (2)
- July 2011 (2)
- June 2011 (1)
- May 2011 (1)
- April 2011 (2)
- February 2011 (1)
- December 2010 (1)
- November 2010 (1)

## One Pattern Many Forms

Folding any piece of paper by touching two points together reveals a tetrahedron pattern. All circles inherently carry the properties of this pattern in that first fold, which can then be reformed in countless ways. The tetrahedron is principle because it is first in alignment.

The following pictures show a variety of many possibly variations of tetrahedra formed to an equilateral triangle grid using only four 9” paper plate circles each. They are all folded the same way, reconfigured differently, and joined to the same pattern in forming a wide range of designs.

The forming of the regular “solid” tetrahedron formed from one circle using the same folds can be seen at; http://wholemovement.com/how-to-fold-circles. Higher frequency folding of the circle grid can be seen at; http://wholemovement.com/blog/item/97-unity-origami.

**Below top)** shows the “empty” circle representation we all carry in our minds.

**Above bottom left)** shows the 8-frequency grid (much like an octave in music) inherent to the circle, though not seen until folded. Moving from right to left we see each folded stage within the context of the inherent non-changing grid. This grid can be taken to much higher frequencies through the same ordered process of touching points and folding.

**Above****)** a traditional tetrahedron solid from four circles with the circumference folded to the inside. The four vertex points are the only connections to the folded-in circumference of each circle.

The properties of the tetrahedron are 4 points in space defining 6 edge relationships revealing 4 triangle planes. The 4 points and 6 realtionships between points is sufficient to fully define a tetrahedron; a number 10 (one circle one diameter, the first fold.)

**Above****)** the same tetrahedron with the circumference folded to the outside.

**Above****)** the same tetrahedron where one sector of each circle is folded out leaving the remaining circumference folded in.

The following pictures are a selection of tetrahedra done over a few years using 4 circles folded to the same 8-frequency triangular grid and reforming the creases, joining in different ways to a single pattern. My intention in showing so many pictures of the same pattern is to get across the idea that forming is ongoing and there is more to the circle than we imagine.

**Below)** exploring reconfiguration of 4 circles in a variety of inside and outside combinations.

Bobbie pins and tape hold them together. Others are held together using glue.

**Above 3**) the 8-frequency folded grid in three closely related variations.

**Below)** 4 circles joined in a tetrahedron pattern forming an icosahedron. Some surprisingly aren’t in recognizable tetrahedron form or symmetry, yet are the same pattern from same grid.

**Above right)** holes punched in each corner allowing it to be twisty-tied together.

**Below)** random selection of variations in forming tetrahedron pattern.

**Above 2)** two variations in star designs from tetrahedron pattern. The three projections on each of the four sides are reconfigured from the same grid. There are slight differences in reformation showing variations.

**Above) **another type of star with slight variations in configuration.

** Above right) **the model to the inside is folded using four 22" diameter circles and the one on the outside of the same design is made from four 9" paper plates.

The tetrahedron is not the only possibilities in joining triangles nor do these even come close to the number of reformations possible from this one of three fundamental triangle grids inherent in the circle. This gives you some idea of the tremendous breadth of design possibilities by folding and joining circles and indicates extraordinary possibilities in higher frequency grid folding, reconfiguring, and joining multiples. Any of these objects can well serve as a single unit for developing greater complex of fractal systems in different symmetries.

Imagine all of the above models folded from the same 4 circles where each of the four circles is a multiple of one circle folded to one grid. This is all a transformational process of a single circle without the limitation of sequencing locations through time.

## Counting 2-D and 3-D

What came up this month is numbers negotiate between 2-D and 3-D.

Euclid wrote, a point has no part. The point is whole. All parts inherently carry the whole. The point and circle are forms of unity scaled up and down beyond our limited perception of concentricity. Unity is singular; one unit is always in plural. Both inherently carry unlimited endless potential in generations of parts.

To start, DRAW a point. Draw a single regular curved line ending at the starting point. Make a second point approximately centered in the circle. Starting from the second point draw a single curved line, the same size of the first, around the first point ending at the second point. You will have two intersecting circles from two points showing four points.

Above) the two circles where the circumference is part inside to the other, both sharing the same measure. They do not have to be exact, you can draw free-hand for the same results.

Above) to FOLD a circle (mark any 2 points on the circumference, touch them and crease) two more points are generated on the circumference (end points of diameter) making four points, the same number of points as the drawing above. Two points on one circle show four points. To the right all four points have been connected with straight lines showing the 6 relationships between them (10 parts.)

Below) continue drawing on the circles by connecting all four points with straight lines, each starting and ending on the circumferences. All connections will be chords showing 6 diameters (realtionships.) There are 10 equilateral triangles, 5 pointing up and 5 in opposite direction. There are other equilateral triangles of different sizes within the two intersecting hexagons along with many other unmarked shapes and relationships between the points.

Above right) by shading one triangle in opposite orientation on the two ends reveals a rotation-slide symmetry.

Above left) a drawing of two circles with 4 triangles shaded to show a reflective symmetry on the vertical axis of both ends.

Above right) are two images of the icosahedron. One shows the solid and the other the placement of 4 triangles equally around. The properties of the drawing and the folded icosahedron are the same but with different symmetries; one is 3-6 and the other a 3-5 symmetry.

2-D as compression of 3-D opens unseen connections. 10 points are counted on the viewed side and 2 more points on the underside of the drawing, 12 points in all. There are 10 individual equilateral triangles showing and 10 underneath making 20 triangles planes. There are 19 lines (edges) on one side and 11 on the underside making 30 edges.

In the drawing are encoded the properties for the icosahedron; 12 points, 20 planes and 30 edges. When using four circles folded into a tetrahedron, opened and joinied in a tetrahedron net, then reformed to an icosahedron will show four open triangle faces (see http://wholemovement.com/how-to-fold-circles) The four shaded triangles in the drawing corresponds to the open triangles of the folded icosahedron triangles.

Below left) fold two circles to a 4-frequency grid (blog http://wholemovement.com/blog/itemlist/date/2011/2?catid=140) and join them to the arrangement of two intersecting circles. This shows two large triangles reformed and joined to show a symmetrically opening on each side through the inside space. This is one of many possible combinations by joining these two folded circles.

Above right) is a 2-D grid of two intersecting circles with shaded openings seen in the folded circles to the left.

This reveals full information in the first construction by Euclid in Proposition No.1 (proving an equilateral triangle.)

Above) folded four-frequency diameter circle. The infolded hexagon is optional in the folding.

Below) two views of the configuration from above with 16 triangle surfaces (front and back) stellated with 16 tetrahedra made from 8 circles. The 4 open triangle planes are left open.

Below) two more reconfigurations from the same circle grid are added to one end. All the triangle planes are congruent giving many options for attachments developing a variety of systems, this being only one possibility.

Below) different views as another unit is added to the opposite end extending the system.

Below) two views of an open spherical unit from four circles that is added.

Above) another two circles make an open icosahedron (previously described) that is joined to the original body of the stellated system. Because of the congruency of triangle planes from a single folded grid, reconfigurations can be attached in many different places creating a wide variety of complex systems.

Below) are different views and repositioning between two segments.

Above) there is a folded hinged unit in the system allows for some articulation in changing positions between two sections.

Above) are five basic reformations of the 4-frequency folded grid that were used to reform the units used for developing this system coming from drawing two intersecting circles. This just happens to be where my interest at the time took me. There are many hundreds of directions to explore with this kind of developing process. Possibilities are different for each folder since no two people will respond in the same way to make the same choices or reformations, yet all coming from the same folded grid.

By understanding how drawings are abstract representations that hold compressed information from 3-D configurations we can use counting to help make connections between 2-D and 3-D about what otherwise goes unnoticed, where each is held in separation from the other. Reality is not flat, nothing is separated; it is spatial and we flatten it to make a simple 2-D conceptually organized system, using numbers to give location and meaning. Numbers represent groupings where every fold is circular movement that changes the numbers. To get the most from a circle we must fold it, from numbers we must count. To get the most from nature we must pass through the experience necessary for understanding. Looking at the picture does not count as the experience

## Where do numbers come from?

The short answer is where everything else comes from.

For a longer explanation we have to start with the concept of origin. The most comprehensive, expansive/compressive concept possible must be in the form of a sphere; it is undifferentiated and concentric without scale. The sphere is unity, it is whole without qualification, without inner or outer boundary, in all direction. Unity contains units, where each unit inherently contains unity.

Two arbitrary diametrically opposite point locations on a unit sphere of any size, when compressed towards each other creates tension inward and movement outward at 90 degrees to the direction of movement retaining spherical volume. A sphere to circle transformation reveals differentiation without adding or taking anything away. This can be demonstrated by rolling a ball of clay and squishing it flat using your hand.

In this spherical transformation there appear three congruent circle planes; top, bottom, and the circle ring connecting the two. When considering individual parts there are two edges connecting the three planes making five individual congruent circle parts. There are seven properties to the disc when considering the internal volume and the space surrounding the circle unit. The concept of a unified whole suggest no surrounding space, no taking apart or separation, only expanding potential into and out from any location. Nothing is added or removed through compression, only a flattening spherical displacement showing form differentiation.

The sphere to compressed disc from which an image is then drawn loses the dynamics of spherical force to the 2-D symbol, a static stand-in to which we assign meaning. Images that appear on paper or screens are always a representation of something else. A holographic image projected into space does not embody energy uniqueness, the dynamics resides with the original. A reproduction will never be the original no matter what means of sophisticated technologies are used.

Circle/sphere origin is pattern to countless multiple copies from which are formed expressions of all kinds; it is a process of divisional revelation. The circle disc is a dynamic self-referencing system of spherical unity with infinite symmetry. Numbers have not appeared, only multiplicity of structural pattern in-formed with differences.

Working backward from the image towards spherical origin gives another perspective. Draw a circle and cut it away from the paper making a 3-D circle disc. It becomes dynamic that can be rolled and spun revealing countless rotational axis. Curving the circle around and touching any two points on the circumference will form a curved open conical plane. When the two points touching are diametrically opposite each other the curved plane becomes an open cylinder; a special cases cone.

Regardless of which two points are used when touching, crease the circle flat. Alignment occurs dividing the circles equally into halves. Rotating on the crease in both directions reveals a spherical pattern of movement reflecting origin. Any fold less than half (two points on circumference not touching) will lack alignment to origin.

Alignment shows duality in unity of three (1+2=3.) These numbered parts do no happen separate from the circle. The crease is a line of symmetry, a diameter, an axis of rotation; a multifunctional straight line with two end points on the circumference; a tri-unity system of three individually identifiable parts, four points, six relationships revealing a tetrahedron pattern of movement, and much more if you include all five circles. This all happens simultaneously. Numbers are used to spatially separate partial events.

This folded straight line crease is the symbol of our first number (1). Number one is the first mark of self-referencing movement of unity. From one line of symmetry we surmise an infinite number of symmetries and countless relationships in combinations and arrangements. All possible combinations of number functions are coded into the relationship of one straight line and one circle (the first of an infinite number of folds.)

Without unity of the circle the idea of one isolated from context is non existent. Yet separation is used to develop abstract relationships by adding, then subtracting, multiplying, and dividing to get a larger and smaller segmented number scale. We teach a unitized system of separated numbers in sequential ordering from simple to complex. We have assumed this is how it is.

Addition, subtraction, multiplication and division carries the assumption of building from nothing, zero (O) to one unit towards developing complex system of calculated functions using multiple units. We accept units coming from nothing with little understanding about how and why or there being no logic.

In retracing the process from sphere to circle then first fold we see an opposite sequence of functions to what we are taught. First is circle/sphere unity, wholeness through compression shows origin and with first movement is a folded division that generates a multiplication of individualized units that can be subtracted from and added to each other. Unity has not been destroyed, there is no separation.

Division, multiplication, subtraction and addition are observed when starting with unity where a process of ongoing interrelated complexity is revealed all within the circle. Through observation and thinking about what we see about what we do when folding the circle, numbers are then symbols signifying inherent differences that are acted upon in relationship to each other.

Nothing exist in separation; only in context. One is an abstraction. Two in separation does not exist. Only in context does relationship of two ones become three. Starting with unity the one line of division is not separate from the act of dividing in two parts. In context there is no separate between one and two, they are three; structural pattern. Traditionally we abstract one and two from three using the circle as a symbol for nothing from which numbers mysteriously emerge.

There is no conflict; one illustrates the abstraction of 2-D and the other reflects 3 dimensional reality. Both systems are correct. One is part of the other. Both taught early at the same time would advance understanding of geometry and mathematics with benefit to other disciplines. By introducing unity into a unit based system would expand thinking towards a more comprehensive way of understanding the nature of where we are, who we are and maybe get closer to why we are.

Looking at a larger picture is simply a bigger unit. The syntheses of many units of all sizes is not unity. There is no summation that gains the Whole. When you start with the whole it is all there, mostly unrealized and through a principled process of generation potential is endlessly revealed within unity.

There is no ultimate boundary, no conditions of confinement to circle unity. The circle demonstrates, order, structural relationship, and principles that are foundational to all subsequent evolving of information. This does not happen with polygons or polyhedra, no amount of numbers can account for it.

We can say we invented numbers, or maybe discovered them, possibly they are inherent in how the mind keeps track, or it is an abstraction of what occurs before number recognition later constructed as language, maybe numbers were given through higher insight, or by creatures from off of this planet coming from other worlds. Nobody now was around for the first couple of numbers so it is pretty much guess work about what it is that fits the accepted version of the number story. It is demonstrated that numbers have a context. How do we substantiate they came from nothing?

Where do your numbers come from?

## Practical Application of Folding Circles

Inherent in the circle are structural forms and proportional relationships of many design possibilities for a wide diversity of products for the human landscape along with the obvious connections to a variety of geometric and biological forms. Often people come up to me and suggest I should make this or that idea that comes to mind from them looking at models of forms and systems generated by folded circles. My response is always “you thought of it so it is yours to do.” I do not develop specific product, that would detract from my job exploring the circle, a transformational process showing what is there that we do not know, to leave some record of the comprehensive nature of folding circles, the only shape that can inclusively demonstrates unity. The development of product I leave to others to explore.

A week ago Wyman Williams sent me a video of a wind generator he made from two tetrahedra. He calls it a TetraGen, a concept he developed from having attended a folding circle workshop about eight years ago and buying a book about the folding. I want to share his work so you can see folding circles does not have to be fancy or complicated to reveal potential for a practical, efficient, and economical solution to a serious global problem. He is developing a wind-powered generator simply and cheaply.

I will let Wyman tell you his own story:

“One night in late 2012 is was using your book (Folding Circle Tetrahedra) as a guide to exploring folding the tetrahedron with circumference outside. It occurred to me that if the resulting material outside the tetrahedron were attached to three faces and stacked on another tetrahedron with the material arranged to catch wind like the one above, it should rotate around a shaft through to center. I folded and taped the tetrahedra and attached them to a shaft made from a coat hanger. It worked quite well in the wind.

I then built a more durable model out of corrugated sign board and aligned it in the same arrangement as the paper plate model. I attached it to a threaded rod and inserted it into two lawn mower wheels attached to the top and bottom of the base. It worked, but took a much higher wind speed to turn it than I had anticipated.

I bought a toy propeller wind generator kit and attached my improved arrangement of the tetrahedra made from poster paper instead of the propeller. Instead of matching the faces of the stacked tetrahedra, I splayed them into a hexagonal arrangement. You can see in the video it really works well.

I then restructured my outdoor model in the hexagonal arrangement. It now begins spinning at very low wind speed. This structure has been spinning outside for over five months and is still in good shape”. - Wyman Williams -

This generator design could not be stronger, lighter, or more economical due to the structural nature of the tetrahedron by using the entire circle. Without the circumference the tetrahedron would remain a regular solid polyhedron as traditionally defined. The circle reveals the tetrahedron to be a functional relationship of the first fold of the circle in half; four points moving in space showing six relationships in a dynamic, structural pattern that is principle to all subsequent reforming and joining circles. The dual nature of that first tetrahedron is beautifully shown as Wyman forms two tetrahedra, relocating them around one of the seven major axis, finding advantage for the circumference folded to the outside.

Aside from the traditional geometry and mathematics generated in folding and joining circles there is a practical side to the process, as we have just seen. 3-D printing revealing new uses for reproduction and prototyping complexities, with an ever enlarging choice of materials, applications and design possibilities. It is important to retain a hands-on designing experience of the imagination that is not tied to computer design tools and programs, but that is structural and self-organizing by nature. The circle is purely transformational since nothing is added or taken away, as we do with all other designed forms of production. Folding requires the observation, imagination of the human mind, courage to explore, and the spirit of discovering unknowns that come from direct hands-on doing. With some of the extraordinary complex forms and beautifully individual expressions coming from traditional origami, there is nothing that will reveal what can be folded from within the circle. Only the circle has a circumference, which can be diminished to any number of sided polygon and folded to any polyhedral form. All reformations comes from the same folded grid matrix making it a truly unique transformational process where all folds come from a single source first movement. There are three possibilities of consistent folding in half, the ratio 1:2 that generates only three option of proportional grids; 3:6, 4:8 and 5:10 (these are discussed in previous blogs.)

There is much to be discovered by folding and joining circles that cannot be anticipated by drawing pictures or constructing preconceived ideas based on traditional adding parts or of taking away excess. I hope this makes the point that there is tremendous design potential in what is generated from folding circles that is not accessible using traditional ways of designing and modeling. While products that are derived from folding circles are an important part, they are not to take precedence over the experience of working with and observing the inner-connections between all parts in an inclusive and principled transformational process. The circle is the only form to demonstrate division generating a multiplicity of endlessly diverse units and systems starting with unity, where parts can then be added and subtracted.

Wyman has discovered the circle to be instrumental in stimulating new approaches to old ideas. I am interested to see what other people have developed by folding circles and what kinds of products might emerge. If you have pursued your own direction in folding circles I would love to hear from you and add you to a list of people exploring folding circles in directions I have not taken.

## Yellow Circles

This continues my exploration into three diameters in the circle; three lines the same length equally spaced as they bisect each other. This goes back to a blog two months ago, Circles From Scrap, starting with the Off-Centered Circle series of five blogs from August through Decembers in 2010.

Let's start with a quick look at an example of transforming a geometry model to a more realized expression of specific biological design. These transformations bring to my mind questions about how we separate art and math by isolating one from another.

*Above)* is the geometry of folding and curving seventeen circles modeling a spiral form held together with bobby pins. It is pretty straight forward in organizing reconfigured circles. I like using bobby pins, tape and other means of holding circles together as part of the forming process, easily removable, and they become informational about the nature of what is being joined. There is no need to hid what we think gets in the way of a “clean” geometric presentation.

*Below)* the same model developed from liking the bobby pins and adding more as they began to take on a hair like quality. It seemed like an appropriate development of the material already being used.

*Below)* other views of what is now indication of a possible life form that may have inhabited the shell like spiral model, were that what it was, or might be. Without underlying geometry there is no art, yet art envisions geometry.

Interdependency of parts gives meaning and value associated with the organization of forms in a process within a context of change. The moment of maximum value is maybe when no more bobby pins can be added, no more glue applied.

*Above)* another paper plate model that has gone through similar metamorphosis using bobby pins and clear tubing explores a specific growth formed to an imagined environment.

*Above)* again another model of three circles using bobby pins and twisty ties to complete an idea for biologically dressing of geometry. The mind clarifies the geometry and the heart gives expression to the art of it.

*Below)* four circle 6” diameter can lids folded three times and joined forms a Vector Equilibrium using clips and half covered with a black rubber coating. Sometimes I use other materials to finding the harmony that connects them giving form to pattern in specific and unique designs.

The three diameters do not need the circle boundary but will always reference the circle pattern of point relationships. Reference to blogs;

http://wholemovement.com/blog/item/121-circles-from-scrap

http://wholemovement.com/blog/item/92-center-off-center

Three lines bisecting each other, equal in length and angles is the structural nature of the circle. It is principle to alignment of all systematic organization of subsequent information revealed directly through folding, indirectly in 2-D methods of construction. Concentricity in reality is the self-centering without measure or construction, eliminating all scale reference, even to the point we call “center.

Euclid stated, there are not parts to the *point*. From that I conclude the point-circle/sphere is total unity, completely, infinitely self- referencing. Symmetry is first expressed by unity to itself, leaving a line of movement to a self-reflective three parts relationship. Unity is reflected in multiply closest packing of spheres with consistency measured in relationship between any two spheres. All geometry resides in unity. Euclid further defines a *line* by points with no parts, then the *circle* by the lines defined by points with no parts. Unity of no parts is reflected in the largest and smallest of parts. No number of parts can show unity; is then endlessly reflected in part of the whole contained. It is through this inherent organization of that which has no parts that there is any organization at all. Value is in relationship, meaning is what we bring it.

*Above) * four 8” diameter circles folded in half, then into thirds forming three axial divisions of rotation; the same creased pattern as the tin lids. Each circle is reconfigured to two tetrahedra with the four circles joined together on the creased lines. http://wholemovement.com/blog/item/92-center-off-center

http://wholemovement.com/how-to-fold-circles

*Above)* four 8 x 8”squares folded and joined in the pattern of the circles above. Alignment is only at the point of intersection. The form has no coherent perimeter yet the pattern of relationship is the same.

*Above)* comparing the same expanded folded grid in the circle, square, and rectangle. The grid position to the perimeter in each is very different. Only with the circle is there complete alignment and comprehensive self-organizing.

*Above) * folding a hexagon star using the six creases of the 8 x11 ½” rectangle has visual interest, but is limited.

By exploring other possibilities in combining shapes of different kinds folded to the same pattern there is much greater possibilities for variations that are visual interesting.

*Below)* a range of shapes will be used; rectangles, squares, torn circles to approximation, hexagons, torn star-like configurations, all from the same folding with three equally spaced and bisected creases.

*Above)* combining two squares and two torn circles with bobby pins holding them together.

*Below)* examples of combinations of the above configurations forming a few possibilities.

*Below) * the center point of intersection have been torn out, some holes punch through for surface interest. The variations show a continual decreasing the planes by tearing parts away, increasing the length of the perimeter of each circle. The form continually changes to an unchanging pattern.

The fractal quality is arbitrary to both the inside and outside; the edges of attachment remain unbroken. The structural pattern is the triangulated relationships of the six axis holding this object together. If the bobby pins joining them were removed the model would collapse. As long as part of the planes connect the axis, it will stand. The three diameters seem to function well with any shape of paper as long as there is some connection between the six axis, thus suggesting endless design possibilities in the ways of forming modifications without disrupting the structural pattern.

The pattern is the structural nature of three and the form is the circle/sphere coordination showing a 3-4-6 symmetry. The design is what we do with modifying parts (above) without changing the underlying symmetry.

*Above Top)* two views of the Vector Equilibrium sphere with the creased grid matrix of triangles and squares from six folds each in progression of the 1:2 ratio of the first fold. There are two sets of three diameters, each set functions differently. (The creases have been traced dark to see them clearly.*)*

*Above Bottom) * 4 circles are joined on the star points rather than the bisecting diameters, as above, showing hexagons and octagons. The spherical form has not changed but the grid pattern shows different designs with two different ways to attach the circles that are unobserved using only 3 creases.

*Above) * each half of the sphere is the same grid combining the two different forms of grid expression. This breaks the symmetry on a level not seen in the form without the folded grid.

This folded grid indicates the pattern of structural relationships and stability that is observed when removing parts of the plane surface above.

*Above Left)* once the circle is folded into the triangular sector, then fold each point to the others (folding in half in three directions) will locate a midpoint in each of the 6 sectors resulting in the grid matrix seen in the spheres above. There is more about this folding at: http://wholemovement.com/blog/item/121-circles-from-scrap.

*Above Right)* when the circle is reconfigured to a quarter sector both grid functions appear sharing a 30 degree sector.

*Above Left)* sectors cut from circle. Cutting destroys unity of the circle leaving it to function as parts; each part showing a center point of unity.

*Above Right)* four parts rearranged forming a second net for a tetrahedron.

Cutting the sectors from the circle makes it is easier to fold them individually rather than with many layers of folding but we are now working with unit parts and not with unity of the circle. But then other things become apparent. There are three ways to reconfigure using the three creases to forming dual tetrahedra, two are right and left handed relationships to the curved edge.

*Below)* here are four sections folded to one of the three possibilities showing two open tetrahedra each.

*Above)* two sets of two tetrahedra are formed by connecting them in pairs on similar edges. There are four different ways these sets can be joined.

*Above Left) *two sets are joined in the same manner as the spherical Vector Equilibrium, open triangles and squares. Given the number of choices in building sets, this is only one of many possible arrangements.

*Above Right) *all open tetrahedral and octahedral spaces are regular. Four regular white tetrahedra fit into the tetrahedron openings. Two are to the longest radial measure and two are to the shortest measure forming the inner and outer regular VE boundary of the irregular yellow.

Okay, that’s enough yellow circles.

## Have You Folded Your Tetrahedron Today?

My brother Ernie passed on this month; not anticipated but expected. Seeing him live through fifteen years with Amyotrophic Lateral Sclerosis (ALS) and his support in my exploration in folding circles brought us into a friendship we did not share as kids or in growing up.

He was a gregarious sort that took energy from many people only to increase and enhance in giving back. He used those years of a slow progressing disease experience to help many people with ALS whose lives would be much shorter. He enjoyed giving to all kinds of people in diverse ways. It took most of our adult lives to begin to appreciate and love each other for our individual differences and life choices.

Over those fifteen years I would stay with him while going to math conferences and doing workshops in the area. We did conferences together. He would move through the aisles of the exhibitor’s hall in his wheelchair asking attendees, “Have you folded your tetrahedron today?” He would playfully engage them to fold as he instructed, showing them with not fully controllable fingers. Having been with Ernie people would excitedly come up to the booth holding their folded tetrahedron to see what else they can do and buy a book about folding circles. In watching Ernie, thinking his good old boy approach was cornball, yet it worked to get people interested and coming to the booth. Chalk one up for Ernie. We sold a fair number of books over the years while he sold people on folding circle and enjoying life. This worked because he was a genuine and generous man that approached people with an open heart; enjoying people every bit as much as I enjoy exploring folding circles. People saw how he delighted in showing them, how easy it was even with large hands that year-by-year became less responsive to his control. He impressed them; a big man in an equally large motorized wheel chair with attached table loaded with paper plates, folding sticks, and tape, engaging them as they passed by. He openly, honestly, directly approached people for what he could give to them, receiving much in return. He loved doing this even as it strained him, pushing him against the goodness of his decision to live a diseased life fully; and we enjoyed our time together.

Sitting around the house playing with new ideas about the circle he would sometimes ask if I had thought of doing what I was from the other direction, or maybe to count differently. I would explain to my students to fold every other point to the center and crease. This can be confusing for some people. He says fold point 1,3,5 to the center and crease. Clear to everyone. From the start he put the process of choice into the hands of the folders. I now often use this in instructing students; everybody understands.

Something he said revealed another way to look at folding the circle in less-than-half that opened deeper levels of exploration. We folded the thick paper circle underneath a take out pizza dinner; we began to eat lots of pizza dinners. Some of his ideas sounded obvious at first but I learned to listen and value them. In his own way he appreciated the depth of my interest in folding circles and at the same time I developed a great respect for his courage, practicality, openness, joy in living, his humor, and the choice he made to live his life fully even as the ALS was slowly closing down his body. We were both proud to be each other’s brother. I carry him with me and I with him where he goes.

Ernie was a man with many activities and interest. The disease did not lessen the value of his life or limit giving to others as he continued cheerfully to serve in various capacities through times when it was difficult for him to care for himself. His self-sufficiency was grounded in his relationships with others. After his passing many people came together sharing personal stories enlarging everyone’s understanding of this life well spent. This helped fill in those years we were unable to find a friendship that later found us.

Ernie was a living demonstration of the value of relationships, not in an abstracted, generalized, or conceptual way as I observe in folding circles, but in the everyday encounters he had with people. Whether between points, lines, areas as they shift and change in the circle or the living relationships expressed with many people that come into and out of our lives; the same unity, principles, order and consistency of individual potential is revealed and realized in the human spirit. Ernie showed me a living geometry in his choice to live life as best he could with his capacity for love and service, despite his disabling body.

Below are a few pictures of fold circles done with Ernie.

Four 14” pizza circles

Ernie liked transformational torus rings and movement systems using paper plates.

A few objects around his house.

Four 25” diameter circles upped the scale.

We made soccer balls from party plates

and a number of off-center foldings.

An 8-frequency collapsible tetrahedron

and a spherical cubeoctahedron column served as conference displays.

## Circles From Scrap

We hear concerns by people about thinking outside of the box. Yet nobody wants to leave the comfort of the square plane. A box is constructed with 6 squares. Give up our obsession with the square and there is no box.

There is no excuse not to fold circles. Pick up any scrap of paper, wrinkled, stained, any shape or size. Fold and tear it and you have a circle good enough to model the 2 and 3-D geometry found in any math book plus much that is not in books. It is all there in any scrap of paper.

1. Not to be put off by step-by-step instructions (words are often difficult to interpret) below are videos showing how to fold and tear a circle from any shape of paper and a few first steps. There is no need to be overly concerned about accuracy in tearing circles, just pay attention to what you do as you are doing it. I use a throw away catalogue, it has lots of thin pages, easy to tear and easy to fold.

**The following videos have no sound, this is a visual experience.**

It is important to accurately fold the half-folded paper into thirds; this is done by proportionally adjusting edges until they are even, this insures the folds are angled equally. Measure and mark a point the same distance up from the corner point on each edge insuring the edges will be the same length. Accuracy in tearing the arc between points does not matter as long as you start at right angle to each side and tear towards the middle. When tearing use your fingernail to accurately position where the tear will start.

*Above left)* here is the vector equilibrium sphere made with four torn circles. The thirteen points are all the same distance apart. The paper images add to the seeming disorder of an organized association of points.

*Above right)* is the same model with the edges folded over between points to show the regularity in geometric form we are familiar with. The circles are held together with bobby pins.

A tip making for when using multiple circles. Make an accurate traingular folding, mark this as a template. After folding another paper in half, line up the folded template with the edge on the first fold from the approximate center out. Bring the first fold to the edge of the template and crease knowing it is accurate. Remove template, fold over, line up edges and crease. Using an accurate template eliminates proportional adjustment and saving time in folding.

2. This video shows folding a tetrahedron folded from a ragged-edged circle without losing accuracy to the polyhedron form. Weather the circumference is ragged or not has little to do with the regularity of relationship between points.

3. This video starts with the torn circle folded to the triangle and expands the folding to a 4-frequency diameter grid of twelve creases by making three more folds. All twelve creases are full chords as a function of the relationship between points as the layers are folded.

Make a comparison of this folding to folding each chord individually as shown in a past blog Unity Origami

The two approaches reveal different information about the same grid. With heavier paper it is best to fold and crease each chord individually because of inaccuracies created by the thickness of folded layers. There is a consistency in direction when folded one at a time. With the folding in the video each cord is a combination of forward and backward directions of folds. Since every fold is an axis moving in both directions, it makes little difference except for thickness and amount of folding.

*Above)* two different sectional divisions have been colored to show the importantce of diameters as coordinate aspect of the 4-frquency grid. The left triangle relies on one point on the circumference to divide the radial bisector in two equal sections, where on the right the diameter is the edge showing the same division. There is an unequal division of the center bisector on the right showing reciprocal division between edge and diagonal. This reflects the primary hexagon division. These two triangles can be individually separated, but together are the means of infinitely scaled division of circle unity to any unit measure. The difference between them is important to the interrelated dynamics of the grid and foundational to geometry.

*Below) *this exploration started with a yellow pad of paper, wanting a circle and having no scissors. All I needed were three equally spaced diameters and I could tear the circle from the paper. With six folds in a rectangular piece of paper this is what I ended up with, a fundamental in/out folding of the four-frequency grid.

Notice the hyperbolic dynamics talked about in recent blogs; In/Out Hyberbolic Surface

and Folding Hyperbolic Surfaces

Circle unity has no points, lines or separated areas. The circle unit can have these properties. A point is a scaled down circle. Euclid defined “A point is that which has no parts.” He is talking about unity then uses it as a part of construction. We start with a center by way of the compass. There is no center to unity, it just is. The circumference gives meaning to the point in a formed circle unit, not to unity. We do not make unity; we rearrange parts and work with proportional relationships between units within unity.

Tearing the edge of the circle from point to point is relative as is using regular curves or cutting straight lines. Edge boundaries are always a bit ragged. Each edge changes the form and relationship between parts but does not affect the consistence nature of structural pattern of arrangement. We have straight edges, regular circumferences, and now arbitrarily torn edges that gives us opportunity to see something different about the circle and giving some clearity about the idea of pattern formation.

*Above left)* both are a tetrahedron. One has the circumference folded to the outside and visible while the other has the circumference folded to the inside and invisible.

*Above right)* is a folded tetrhedron with the edge on the outside. There was little attenion to the configuration of how it ws torn. This is no less a terahedron but much more in-formation.

*Above) *you can see the beautiful symmetry in the folded tetrahedron net regardless of the forms it takes. These are the two tetrahedra fron above opened flat. This reflects a process we see in nature.

*Above)* six tetrahedra, four in a tetrahedron arrangement and one to the side in line with the others. The tetrahedron with the circumference folded out forms three open tetrahedra cavities. When tetrahedra are fitted into the cavities it will hold the tetrahedron in opposite position. This inside and outside is an interesting and efficient way to build a tetrahedron/octahedron matrix.

*Above left) *a two-frequency tetrahedron, four circles torn form scrap paper.

*Above right) *the same two-frequency tetrahedron with the circumference folded to the outside. The two are exactly the same arrangement of four tetrahedra with an open octahedron space.

*Above) * using the two-frequency tetrahedron from above and adding four more tetrahedra to the four open planes reveals a cubic pattern of eight tetrahedra in a spherical rhombidodecahedron form. This a function of the inside circumference being on the outside.

*Above)* the six vertexes of the rhombidodecahedron have been opened to the outer creases in each tetrahedron net. This introduces six squares and twenty-four open triangle faces that are tetrahedral and another twenty-four open triangles combined to form twelve smaller rhomboids that are octahedron cavities. The twelve rhomboids have opened to reveal a complex spherical division of tetrahedra and octahedron cavities.

As far as I know this is not a classified polyhedron because you cannot get it through truncation. This a transformational process by opening the six vertex points that open 24 edges to a very regular spherical division. This can not be derived by cutting away corners because there are no corners on a sphere. Circle unity is inherent to all possibilities of unit configurations.

I consistently find levels of information in folding circles that surprise and delight me that I do not observe elsewhere. To share some of this is why I write this blog each month.

Now you know paper circles are as cheap as bending down and picking up scraps of paper. By simple folding and tearing the circle away from the paper you can explore the beauty and mathematical relationships we never see in the paper trash we throw away, or in the books we are given to read. Everything I have done with paper plates can be done with torn circle scrapes. This is yet a beginning towards understanding the perfection of unity and the ragged nature of boundaries in division.

## Folding Patterned Blocks

On a number of occasions towards the end of a workshop a child will ask how to make a cube. Then I would have to figure out which one; how much time is left and can we do it without diverting the class from what they are doing.

I tell them yes, but not with the folding we are doing. I show them an easier way using the 4-8 folding rather than 3-6 we have been doing. A cube is made, others see it; very quickly half the class is making cubes without distracting the other half.

Here are five cubes made from both the 4-8 grid, right angle triangles and the 3-6 equilateral triangle grid. They have been colored to the creases used for each folding.

We will explore folding cubes with both the 3-6 and 4-8 symmetries of the circle.

*Below left)* fold the circle in half, and in half again; four points on the circumference, five with the point of intersection from the two perpendicular bisectors. Alternate areas have been colored in to show areas of division.

*Below right)* the inscribed square has been creased and two lines of division parallel to the sides have been folded and alternate areas colored. There are five squares, one inscribed and four divisions of smaller squares.

*Above left)* Two circles of the net above are reformed and joined forming the cube. It is the same as joining two open tetrahedra to form an octahedron; (see http://wholemovement.com/how-to-fold-circles. Scroll down to #3.

*Above right)* is the same forming and joining only with more creases showing more areas colored in. This is pretty straightforward to what we understand about divisions of and construction of the square and cube.

*Below)* is another way to make the cube; it takes a little more folding but is richer in transformational and 2-D designing possibilities.

From folding three diameters we fold the 4-frequency diameter grid. http://wholemovement.com/blog/itemlist/date/2011/2?catid=140.

This grid divides the circle into 12 equal parts that show three differently positioned squares.

*Above)* the grid is shown with one of the squares colored to show that part of the grid that lies inside the square net. Two extra creases were made from the grid to show the axial division perpendicular to the sides. Two folded circles have reformed to a cube. The cubic folds are the same for the 3-6 and 4-8 symmetries, different in context making the divisions from the 3-6 gird uniquely different.

*Above)* the flat circle shows two radii that become the diagonal and edge of the square in different formations. Folding under one-quarter of the circle a right angle tetrahedron is formed. Starting with the folded square the same occurs. They are interchangeable to different scales and forming of the cube.

*Above left*) the circle is formed to a right-angle tetrahedron with perpendicular bisecting diagonals.

*Above right)* the tetrahedron edges are pushed in becoming the diagonals of the squares formed by the three edges of the tetrahedron. There is a reciprocal function between edges and diagonals. The right angle tetrahedron is one-quarter of forming a cube and the three squares are one half of a formed cube.

* *

*Below)* is another design of division from the 3-6 folded symmetry.

*Below left)* the 4-frequency grid with one folded square. The areas have been alternately filled in to make a more interesting proportional surface design. With these folds there are many possibilities for designing the surface.

*Above right) *two circles joined forming a complete cube.

*Below)* the circumference is folded to the back to show only the square. With out the folded circle it would be difficult to come up with this proportional design.

*Above)* all three creased squares colored to show the three square-compound. By adding creases for the three right angle axis to each square. It generates 24 division in the circle.

*Above)* the circle with circumference folded behind showing only one square. Two squares of the same design are reformed and joined. There is the option of putting six squares together into a cube.

In both the 3-6 and 4-8 folded grids the primary points of intersection represents centers and tangent points of a circle matrix. Curved lines are used to redesign straight creases. These show only two of many possibilities.

*Above left)* the 3-6 triangle grid has been colored using the curves that resemble the closest packing of spheres in a traditional arabesque design.

*Above right)* the 4-8 square grid has been designed the same way.

*Above)* The same design can be “centered” differently in how it lies within the circle. This goes for any grid

*Below)* right and left shows the cube formed from each of the above designs.

*Below)* primary folding possibilities of one circle folded to a 4-8 symmetry where the edge of the inscribed square is divided into 8 equal segments. this is different than starting by folding the diagonal into eight equal divisions.

*Above middle right) *is one half a cube.

*Above bottom right)* shows ¾ of a cube formed from this square division.

A “full” cube can be formed from one circle with the 8-frequency division folded on the diameter/diagonal. Out of infinite possible diameters in the circle the square uses two. We can then see root function is the diameter of the circle which goes for the square as well.

*Above)* Two diagonals/diameters divide the circle into four equal parts. Each diameter is folded to 8 equal divisions. The circle has been reformed into a complete cube where the four part colored areas shows a volumetric division in surface design different than what we usually see.

*Above)* two cubes; one from four circles and one from a single circle are combined making a beautifully proportioned cubic system from five circles. Pattern blocks are proportional systems of relationships that have origin in the circle.

The square and cube while they can be constructed separately are relationships of triangles inherent to the circle. There are a lot of possibilities for making your own set of pattern blocks from circles.

*Below) *some three-quarters cubes and some partially formed right-angle tetrahedra make up another kind of proportional cubic system.

Let’s look at other possibilities by folding the circumference to the outside of the cube.

*Below left)* two perpendicular diameters and the square relationship are folded to the 4-8 folded symmetry where the alternate areas in the square are colored.

*Below right)* four circles folded to the right-angle tetrahedron with the circumference folded in, then joined forming a cube.

*Above)* one circle is reformed to the three squares (half cube) with the circumference folded to the outside.

*Above)* the cube from above has been reformed with circumferences on the outside. It reveals the spherical tetrahedron, one of the two tetrahedra forming the cube. The four circles are held together with bobby pins.

*Above)* two views of a different division of proportional folds with circumference on the outside. The cube is smaller with more circumference revealing a spherical truncated tetrahedron.

*Above)* a circle folded to the 3-6 symmetry with alternately areas of one square colored in.

*Above)* Two views of four circles, shown above, have been folded and joined with circumference on the outside. This time part of the second spherical tetrahedron is revealed as they intersect crossing on two opposite faces.

There is another way to form a cube solid is to fold the tetrahedron net; http://wholemovement.com/how-to-fold-circles.

*Below)* the tetrahedron net. Fold the two end points of the three creases that form the inner triangle to the center point and crease (one at a time.) This gives you two more congruent triangles, one on each side of the first folded triangle.

*Above right)* using the creases from either the right or left side of center inscribed triangle, fold the circumference behind forming the triangle. The center triangle will be off center to the just folded triangle. Fold over on second line in from each end point forming a right angle hexagon.

*(below left.)* bring triangle points together forming a right-angle tetrahedron as you would form a regular tetrahedron Tuck triangle flaps inside to hold the right angle tetrahedron together.

*Above right)* the tetrahedron is opened with flaps folded out forming right angles that shows three sides of three-quarters of an open cubic arrangement.

*Above)* two open units joined on surfaces and taped together.

*Above)* two sets of two, above, are joined. The backside cavity is half an octahedron.

*Above)* two sets of four units are joined with the eight open cubic units on the outside. The 24 external points reveal a truncated cube of eight octagon and eight triangle planes. The center is an octahedron axially divided by three intersecting square planes.

There are many variations of cubic division and arrangements, some with circumference folded in, some folded out, and some combined. There is much to be discovered. A cube can be made from one circle or with many, from the 3-6, 4-8, or 5-10 folded circles.

Sometimes students will make a transforming system by hinging four solid cubes together in a square arrangement.

A full cubic transformational system takes eight cubes, all hinged in the same right angle system reflecting the torus ring where the surface rotates through the center opening. You can see an example at:

https://www.facebook.com/photo.php?v=1870781780855&;set=vb.125203137500210&type=3&theater

The surfaces in this torus system were designed to show the closest packing of spheres as well as a few straight-line possibilities. There are other torus systems in the same album made with other forms.

Within the circle are the proportions for blocks having a variety of measures with lot of interesting surface design possibilities.

enjoy...

## In/Out Hyberbolic Surface

Picking up on last months entry I went a little further in exploring the hyperbolic surface since having observed for many years that it is inherently within the folding of the circle. First it was necessary to crumple some paper and to look to see what happens in a random act. There is no order to the surface plane when crumpling a piece of paper (in this case a paper plate circle.) The reforming of the surface is arbitrary, unpredictable, and sometimes with interesting results.

The surface is contained by the edge showing a chaotic wave action taking place in the plane as the interior surface moves perpendicular away from the flat plane forcing the edge to move in two opposite directions. There is a warping of mountains and valleys that move through the horizontal surface plane. Other than the random forming it is similar to dropping an object in still water. A tension and compression is created in the movement through the plan that reflects the dynamics in compressing the sphere to a circle disc. The outward tension gives balance to the inner compression.

*Below)* The hyperbolic surface is mathematically portrayed as a smooth curved saddle form in two directions, a distortion of the square shaped plane. It shows an underlying tetrahedron pattern inherent in opposing right-angled movement.

*Below) *Looking from a polygon perspective there are three basic modes of planer forming. Starting in the center there is the hexagon surrounded by hexagons (6,) a flat tiling plane. The pentagon surrounded by hexagons (5) shows openings between hexagons where a continuous surface must curve away from the plane in a spherical direction. The heptagon surrounded by hexagons (7) finds overlapping of planes that must buckle off the plane to keep a continuous surface. This demonstrates a hyperbolic plane.

*Below)* Start by folding three diameters into the circle; that is the hexagon pattern with seven points. There are six equal areas (6.) The circle can be reformed by folding in one radius a pentagon pattern of 5 (6-1), to the square pattern of 4 (6-2,) then to the 3 pattern (6-3 on right.) With each decrease in parameter there is an increase in height, a precessional movement off the circle plane. There are two ways to decrease the circle by half (6-3), one an in/out star-like configuration (left) and the other a tetrahedron (right.)

*Above)* Each increase in altitude shows a decrease in circumference and when all are in alignment on the flat circle of origin there are concentric rings that correlate to the eight divisions across the circle; a folded 8-frequency diameter circle ( http://wholemovement.com/blog/itemlist/date/2011/2?catid=140)

* *

*Below)* Each decrease is inscribed onto one circle, then accordion pleated in alternate fashion. This forms a hyperbolic curve to the surface going up and down in perpendicular direction to the plane. The saddle curve is seen in the reformed edges not the continuity of surface.

*Above)* Three points on the saddle surface form an inward curving triangle. The same three points also show a relationship off the surface to be a flat open triangle plane. By locating a fourth point anywhere on the surface, a tetrahedron pattern of six relationships and four triangle open planes are formed.

*Below left)* A paper plate circle with five inscribed circles accordion folded forms an extreme saddle configuration. Many people have done this; most notably Jose Albers with his students in the 1920s at the Bauhaus School in Germany and later in the 1940’s at Black Mountain College in North Carolina. There are more current variations by others slicing through the circle and cutting out the center. Here the circle remains whole keeping unity throughout, which I believe was the point of the exercise in the first place.

*Below right)* Two circles are reformed and joined, one into the other.

Concentric circles give order to the plane by determining where the surface will go. With crumpling the circle the same dynamics randomly occurs without any sense of order.

When three diameters are folded into the circle a similar ordering occurs as seen in the concentric circles. The 8–frequency triangular grid is a straight-line polyhedral reflection of concentric circles. The above illustration using three diameters shows the flat hexagon transformation from 6 into 5 the pentacap, to 4 the square based pyramid, and 3 the tetrahedron; all revealed in concentric circles and formed using straight lines all self-referrenced to the circle through proportional movement.

*Above) *Using the 8-frequency grid shows a simple folding of two views of reconfiguring the in/out of the circumference.

* Above)* On the left is* *arbitrary crumpling to the center where on the right are combined a few of the folds from the 8-frequency grid giving an interesting combination of a semi-controlled crumple.

*Below) *Two variations where the circumference is more controlled by the regular symmetry towards the interior of the circle using a 3-fold symmetry.

*Below)* This show basic opposite direction folding* *where the center is going in one direction and the circumference moving at right angle in the other, much like the saddle only in angular form. A tetrahedron interval is formed at the center by two solid faces and two open faces of triangles; four points and five edges of six relationships.

There is a slight fold variation between the left and right hand pictures.

*Above) *Two units on the left shown above have been joined bringing the open tetrahedra together forming a double saddle system with a solid tetrahedron center. Two views.

*Below) *A picture from lost month* * http://wholemovement.com/blog/itemlist/date/2012/11?catid=140 shows the same hyperbolic organization around the center tetrahedron as above, only this is formed using the 4-frequency folded grid and shows a single saddle-like system.

*Above)* Four circles were folded to a tetrahedron net of nine creases and crumpled into balls, flattened out to the circle and joined in a tetrahedron pattern. The surface is determined by the random crumpling of the circles and does not affect the patterned arrangement of four circles; the curmpled surface gives interest to the form.

*Above)* The four-circle sphere from above with four circles added to it, each a reformation of the tetrahedron net forming three pronges each.

*Above) *Two views of the above system with four more circles added that have been reconfigured from the 8-frequency diameter grid; one to each three pronged extension. Three are a slight variation to the other two and one is in the form of an icosahedrom.

*Above) *This is another direction in* *developing the above tetrahedron sphere. More crumbled circles combined with a creased triangle net have been added. The last four on all four sides have been folded but not crumpled. There are nine layers on each side making a thirty-six circled sphere displaying a tetrahedral design of four open triangles.

*Below)* Here are a few variations of individual units that can be made form the 8-frequce gird where the circle is reformed to a hyperbolic in/out pattern, in some cases looking totally different than what we associate with the mathematical saddle form. But then that is the advantage of folding circles, it will do the mathematically unexpected.

Following are a couple of systems developed from exploring multiple units from above. There are many possibilities for combining the spherical, the flat, and the hyperbolic systems by folding and joining circles that does not happen with other shapes.

*Above) *a system made from joining two of one of the reconfigurations, making another the same and joining the two sets of two together.

*Above)* Two views of an eight-unit circle using one of the units from above. From the side view you can see I decided not to complete the circle but keep it going in a helix form. I may continue to add by sequentially decreasing the diameter of each circle moving it into a conical helix or spiral form.

*Above)* Two views of two circles folded to a hyperbolic variation and joined in a tetrahedron arrangement.

*Above) *Another system with four circles folded to hyperbolic configurations and joined in a centered tetrahedral pattern. There are two views showing each end; one is an irregular variation of the other.

I always enjoy seeing the factual *beauty* in folding and reforming the circle, seeing the underlying structural *truth* about the nature of pattern as formations emerge, and to reflect on the *good* that comes from unity where all parts are multifunctional and interconnected allowing endless potential of expression throughout.

## Folding Hyperbolic Surfaces

A hyperbolic surface warps a 2-D plane perpendicular to that plane in two opposite directions forming a 3-D saddle like shape. This can be demonstrated by adding more than 360° to the circumference of a circle, popularized by Daina Taimina’s crocheting. Given that the circumference is fixed with the paper circle, and one can not add anything to it, folding reveals another approach to forming a hyperbolic surface. Compressing the flat plane perpendicular to the surface towards the center curves the circumference in the same way. This warping of the flat surface is inherent in the nature of folding circles. Rather than going into any explanation about hyperbolic surfaces or the mechanics of folding, let’s just observe for now what happens when we distort the circumference of the circle. We will continue next month going further into folding hyperbolic planes.

Starting with a 8-frequency-diameter grid and reforming by folding around the hexagon center forms a hyperbolic surface.

*Above left)* This is the reconfiguration I will use in multiples to explore various systems. To the right shows another variation that is more open. The curving of the circumference can vary depending on the design of folding the circle.

*Above)* A random warping of the circumference coming from a couple of different ways of folding the center into itself. Both of these used a 16-frequency diameter grid (each of the three diameters is folded into 16 equal divisions.)

*Below)* A more open distortion of the circumference using a simpler in/out folding of a 4-frequency grid..

*Below)* shows two of a number of ways these two units can be joined; on the left one is nestled into the other and on the right edges are joined.

*Below)* three views of another kind of surface-to-surface joining. Here four individual units are joined in a tetrahedron arrangement making a very open and rigid system. The open areas are the six edges of the tetrahedron's four vertex points.

*Below)* two tetrahedral systems are joined one into the other forming a double unit. They are joined on one of the four openings on each system. Joining on the other openings would expand the complexity of the system

*Below)* three systems joined together on curved edges. This retains the open flow through all units.

*Above)* two views of combining six circles using the two unit and the four unit systems.

*Below)* two other variations in rearranging ten circles using the same units. On the left there is an open flow between the open units and on the right the flow is closed off between connections.

Moving on to other directions in thinking about the hyperbolic surface as a natural outgrowth of folding circles I looked to a simpler model.

*Below) *an example of using the folded net of the tetrahedron in two circles where the circumferences form an interesting hyperbolic system, one fitting into the other, showing a continuous wavy surface around four of the six edges of the tetrahedron.

*Below)* By using four circles with three diameters each folding two parallel chords connecting the two triangle arrangements of six points, they can be joined to reveal a collapsible transformational system that will fold flat. Three open square planes function in the same way to collapse the system that is similar to the collapsible system of four circles forming two open square planes from last month’s blog.

The collapsing squares brings to mind the collapsibility of the Vector equilibrium using four circles with three diameters each and six open squares, which brings up the off centered systems (from previous Center/Off-Center blogs.)

From here I thought about the collapsing of the vector equilibrium and how that works with the off-center systems and the importance of the center location. Each VE is concentric to the center; the off center is not. Every two points have an implied relationship of a third part connector; without the line relationship they remains two isolated points. By using a sequence of tetrahedra folded to different layers of the triangle grid and connecting them to both centers they reveal a relationship between the two centers that is a uniquely different expression of two points connected by a straight line.

*Below)* There follows four views show what it looks like. You will see bobby pins holding the circles together that form the off-centered vector equilibrium’s end points. The connecting tetrahedra are glued together.

While making tetrahedra it seemed reasonable to finish another piece by adding four tetrahedra to it.

*Below)* three views of the complete model where the tetrahedra show an exploded 2-frequency tetrahedron with an elaborated octahedron center.

There are always other directions to be explored. While focused in a particular direction there are always divergent tangents being pursued at the same time that come out of the dialogue between bringing different forms together that otherwise are associations that would go unnoticed. Another one developing over the last few months is shown below. This model uses 12 folded and curved circles.

Next month I will go more into the "ins and outs" if you will, of folding the hyperbolic surface using the circle.