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Three primary circle patterned grids; 3-6, 4-8, and 5-10 can be folded using a scrap of paper. They are inherent in any flat plane surface and can be folded without regard to boundary configuration. Last month’s blog, “Order Without Boundary” showed in the first fold, a ratio of 1:2 (a division of one unit in two parts) providing only three options in a regular division of the plane. The circle is primary from which other shapes can be folded, demonstrable by anyone willing to fold a paper circle in half, observe what is generated and use that information to continue folding.
Any point on a plane can generate one of three specific division of symmetry. This suggest every point on a plane is a center point for dividing the plane symmetrically. The circle itself is the center where symmetry becomes the means of encoding directives for division starting with the diameter. The circle is the center and a local centered systems can happen anywhere on the surface. This supports the idea of center as origin regardless of location or configuration.
All regular and irregular flat plane perimeters are inherent in the circle for the same reason the circle can be drawn on any plane. One center to one circle traditionally uses the compass to draw the circle giving importance to the radius as measure. Movement out, when going back stops at or goes through the center point. When moving into the center the scale changes to find other endless concentric circles. The center is both inside and outside by nature of the concentric alignment of the circumference.
Traditionally polygons are constructed using the circle and straight edge. Folding the circle produces the straight edge and yields the same information plus more. The triangle, square and pentagon are all discernible in the hexagon, primary in the circle. This arrangement of three straight line divisions from a single arbitrary fold is inherent to all shapes. The information is encoded to the angles of division around any location.
Years ago when tracing 2 and 3-D geometry constructions back to the circle I did not see that it was not about the shape of the circle, but about proportional movement that happens where straight line relationships hold consistent to a circle pattern. Information is encoded in the relationships of creases directive to what will and will not work. Folding is sequential coding that generates progressively complex systems, it is not construction.
Following are a few pictures expanding on the three primary polygons and the straight-line grids that exhibit the nature of a circular pattern.
Above left) A piece of paper folded to show the hexagon star with the 6 equally spaced bisectors. The star points are all equidistant from the center point making it an enfolded circular pattern. The lines forming the two large triangles do not go edge-to-edge; they are partially formed creases. For one sequence of folding see last months blog: http://wholemovement.com/blog/item/130-order-without-boundary
Above right) Each of the six partial creases forming the triangles has been fully creased across the paper, similar to every fold in the circle being a chord. This fully completes the division of the hexagon, six triangles around the center, each bisected in three directions with all lines extending to the perimeter.
Above left) The partial folds forming the hexagon have been folded to papers edge thereby expanding the matrix.
Above right) One sixth of the hexagon is folded in making a pentagon star. (6-1=5) The pentagon pyramid can also be formed with both concave and convex functions.
Above left) Folding in two sixths (6-2=4) leaves a square-based pyramid, half of an octahedron.
Above right) Three sixths or one half of the hexagon folded in forms two tetrahedra, a three sided regular pyramid with four triangles. One is inside of the other where both have an inside and an outside making a full count of four tetrahedra.
Above) The small equilateral triangles in one direction are colored to show the regularity of the triangle grid. The information is there to extend the grid over the entire plane.
This 3-6 folded grid gives options for reconfiguring the paper to the five, four, and three-fold symmetries.
The folding process is the same for the 4-8, and the 5-10 symmetries. With each reduction of symmetry the possibilities for complexity increases. Symmetry can be seen as the inherent gatekeeper for directional development.
Above left) The 5-10 folded grid shows the pentagon defined by partial folds as seen with the hexagon.
Above right) The completed five-fold grid is traced in black to show all lines folded to edge of shape.
Above) The completed grid is colored to show alternating areas.
Above left) The pentagon grid is reformed to a square-based pyramid. (5-1=4)
Above right) 5-2=3 shows a triangle-based pyramid, or tetrahedron. These are the same polyhedra formed by the hexagon grid, but they are differently proportioned because of the different in symmetry.
Above left) The square is defined by partial creases as observed with the hexagon and pentagon shapes. This is the same folding process using different angle directives.
Above right) Folding continues to fill in and out the square grid, in much the same way as the hexagon and pentagon.
Above) The square grid is colored showing the alternating triangle grid that is basic to a square division.
Above left) The 4-8 grid folded showing the right angle tetrahedron inside of a larger truncated tetrahedron.
Above right) Accordion pleating the tetrahedron is a function of parallel creases in the gird.
Below) The above folded paper is reconfigured into a cube with only six squares to the outside and the rest of the paper is folded to the inside.
Below left) Four right angle tetrahedra are folded to the same measured square grid in different places on different shapes of scrap paper.
Below right) The long edges of the tetrahedra are joined edge-to-edge in a tetrahedron pattern forming a cube. The excess paper coming from six of twelve diagonals to the cube show where the tetrahedra have been joined.
The coded order is that a 3-6-12 folded grid can be reconfigured to the pentagon, the square and the triangle. The 5-10 is limited to the square and triangle. The 4-8 grid is limited to the triangle. This is not just a construction formula, but the inherent function of the folding process. The 3-6-12 folding being first, contains the geometry and mathematical relationships that are fundamental to further development and is important for math.
The place to start is with observations by the folder; how they fold and what is there that was not before the fold. Angles are the most obvious directive since that is primary starting from the first fold, a ratio of 1:2. The individual proportions determines the direction of symmetry.
Above) The differences in angle divisions from any point of symmetry on any plane is determined by the proportional decision of congruent angles. Here the divisions of angles is revealed in the 3-6-12 grid. There is much to add, subtract, multiply and divide once we can determine the unit angles, and consider each proportionally within the entire plane. Even without knowing arithmetic or numbers, the proportional relationships of units to units are easy to see when they are all in the same place sharing functions.
Above) The square grid shows a different angle division. The central 90° angle is shared by both the 3-6-12 and 4-8 grid.
Above) The pentagon arrangement shows different angles that have different proportional properties from the other two. It shows 90° angles, but not centered as with the two above. This makes the proportions and ratios unique to the pentagon. There are different symmetries dividing the space around each point of intersection depending where they are located in the grid.
The number pattern for the angle degrees for each symmetry is surprisingly consistent and reflected in numbers.
With the first fold there are three possible choices to continue generation of structural order. Direction is coded in angle units consistent to 9 as the last number of the first level of natural numbers. Folding 60° in 180 gets two division of 60° three times and again 30° that will reveal 90°. Folding 90° again in half get 45°(4+5=9). By folding the 180 (9) into 144° angle (9) we automatically get 36° (9) with half of 144 being 72 (9). There are no other options to the consistency of angle direction encoded to the circle. When we start with polygons we are limited by the specific angles of the shape.
In looking at the angles of each folding 9 is a common base seen in reducing the numbers to a single digit .
630° 36 9
450° 39 9
540° 45 9
It would follow that the summation of all these numbers reduced down to a single digit would be 9. They all started from a single 90° movement. Three squared, structure to itself, is the number 9, the complete sequence before transition to the next level out. There is a consistent development even in the numbers as they represent angle units.
There many math functions and spatial systems that comes from these three folded grids. The more math you know the greater depth of discovery, particular when it comes to guiding children towards becoming aware of their own observations. At the same time math is not necessary to explore the beauty and complexities inherent in reforming and joining folded scrapes of paper.
While this information is accessible using any scrap of paper it is most visible in folding the circle shape. Information generated in the circle is not possible with other shapes since only the circle has a circumference. Every fold is a chord; an acknowledgement of the complete unity/unit of shape. Information is coded into the circle and revealed through a consistent directive of sequential folding. It is important that each crease be a full chord, otherwise contextual information is missing. This is equally important using irregular paper.
The 3-6 and 4-8 share a 90°central angle. Starting with either the three or the four folding both end up with 12 divisions. (only the 3-6-12 is consistent to the ratio where the other two are not.)The 4-8 is inherent in 3-6-12 making the 3-6-12 accessible through the 4-8. This interconnection between symmetries expands our ability to see what otherwise is not possible when things are separated.
Below left) Folding the 4-8 into a 12 symmetry division, six diameters, reveals two hexagons one enfolded to the other. The difference is not so much visual as procedural at this point.
Above right) Folding the hexagon star shows the internal triangular grid. The creases are outlined to better see the grid.
Below left) Starting with the 4-8 folding in a square and incorporating the 3-6-12 directives reveals the hexagon in the square intentionally not in alignment with the perimeter, showing the hexagon center can be anywhere in the square. Following different directives of the same pattern reveals many possibilities for combining symmetries, freeing us from having to rely on constructing polygons.
Above right) The creases are lined in black to see one possible combination between the square and hexagon sharing 12 fold symmetry.
Above) Alternate areas have been colored in to show the pentagon star is inherent to the combing of the 3-6 and 4-8. There are a great variety of intersections and shapes for reforming the square where the center of the hexagon can appear at any location in the square area on any scale. First there are pattern directives, then forming of branches that generate information with which to design. Symmetry seems to be the constant where discernable or not.
Above) The same folding of the hexagon using the regularity of the square shape as directive for the folding. This is more inline with traditional symmetry of polygon alignment.
That first fold shows coded information is activated by proportional folding which determines different symmetries. I call this information geocoding for lack of something more descriptive. It happens without construction or formulas. We do not think about geometric shapes as being coding devises for structural development. The folded ratio of 1:2 is structural; two quantities without separation is a set of three. The number 3 is structure coded into the first fold and with consistent development assures structural regeneration. Each of 3, 4, and 5 are further coded to branch into different matrices. Three, four, and five add up to twelve; another form of 1 and 2 or set of 3. Being first this primary code is core to all folding that follows revealing the geometry observed in nature and imbedded in the abstracted symbols we used mathematically. Geometry is as much a coding devise for spatial development of complex systems as it is a tool for proving abstract mathematical concepts.
We do not think of basic shapes as coding devises that generate complex geometries because of a traditional position about 2-D construction with emphasis on product. Geocoded information is important to the development of technology just as it is in the geometry which cannot be separated from the instructive information inherent to shapes that make up the forms and systems of the physical and biological world.
While I get excited about the product potential in folding circles, it is the information that it generates and the consistency of development that holds my attention. So here again, as twenty-five years ago, I look at the circle and ask, what it is? Why does it reveal what other shapes can not, yet contains information for all other shapes? How does it easily generate much that we have learned to construct? Why don’t children fold circles? Why isn’t folding circles part of our curriculum? Have we missed some crucial aspect of the circle by only drawing pictures of it? Has Euclid’s influence limited us? Is it because the idea of the circle is a symbol for nothing, where zero prevents us from seeing anything there? Maybe because of how we have defined the circle? I guess it all has something to do with why we continue to ignore folding circles.