Thursday, 01 March 2012 19:40

## Folding January Into February

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As January moves into February I continue using my business cards to explore more of the primary tetrahedron configurations (refer to last months entry.)  Following is some of what emerged.

Trying to be more systematic using the tetrahedron with 1/3rd of the circumference folded out and putting four circles together in a tetrahedron pattern I found more possible variations than I was willing to make or keep track of.  Rather I took to exploring what caught my attention.

Below. Here are a first few examples before realizing this would take longer than I wanted to give to  accounting for all the possibilities of joining four tetrahedra with variations of circumferences.

This is one helix arrangement of eight sets of four tetrahedron from the possibilities suggested above. Each individual sets has the potential to form some manner of helix formation.

Below Five sets of another arrangement are joined in a helix of a different configuration. Each of the sets above can be joined in multiples ways to reveal a wide range of helix formations.

Two more sets of four open tetrahedra; beautiful little things.

There are too many variations and subtle possibilities. I decided to again go back to the beginning.

We are only familiar with the tetrahedron as solid T, the second T1 has 1/3 circumference out, T2 has 2/3 circumference out, T3 has the entire circumference folded out. Each folding out forms a tetrahedral pocket. With the T90 and T60 the circumference is out but folded flat to 90 degrees and 60 degrees to the tetrahedron edge. Over twenty years of exploring these individually I felt it is time to line them up and look at the options of a tetrahedron when folded from a circle. There are many other interesting ways to reconfigure the tetrahedron, these are limited to the circumference folded in combinations in to out.

Below  In the mean time I played with expanding the look of this sphere (a carry over from last months entry using eight T3 units above.)

There are many ways to fill in a sphere without changing the pattern while altering the form and design. Here there are tetrahedral spaces and two sizes of octahedral spaces that open up many growth possibilities. How forms grow and develop is always fascinating.

Above Two views of another variation to the tetrahedron related to T3 joining four units in a tetrahedron pattern revealing the triangle in a hexagon and at the same time we can see from another view the cubic pattern as a function of the tetrahedron.

Below Two models formed using T1 units.

Above left Three tetrahedra are placed with one corner into the open pocket of the others forming a triangle set with an open tetrahedron. Four triangle sets have been joined into a tetrahedron pattern.

Above right  Each T1 unit is placed into the pocket of the preceding unit in the same orientation forming a line that begins to make a helix. With eighteen units it comes back onto itself where no more of the same size can be added. If each unit either got successively larger or smaller it would become a spiral growth.

Below The tetrahedron patterned set from above left is a configuration that fits within the space of another sphere from last month entry,  below right.  They have been formed separately, yet one is the seed for the other as it fits exactly the interior space.
By traditionally cutting off the circumference to make polygons we eliminated possibilities to see many of these kinds of  interrelated expanding relationships.

Another view of the open tetrahedron patterned sphere from above.

Below  24 solid tetrahedra fill the open tetrahedron spaces in the sphere from above.   Following is a continuation of expanding the sphere by adding congruent tetrahedral units. They could have been filled with T1 or T2 units as well that would have changed the direction and look of forming.

Twenty-four more T units were added on to the ones previously added.

Twenty-four units from above have been replaced by T2  units, as if the tetrahedrons from before have opened, much as buds open to flower.

Here we see a continuation of the flowering by adding more units. Growth can continue with each new tetrahedron that adds three more open tetrahedron pockets and the possibility of three congruent triangles as places for development to occur.  By sequentially reducing the scale and continuing the growth it would take on a fractal development.

I took another directions by opening the solid tetrahedron with circumference out, below left. It is one of a number of configurations of the tetrahedron that I did not consider to be primary to the circumference being folded in or out as listed above.

On the right shows two of these joined with the open tretrahedra at right angles to each other forming a solid tetrahedron with 720 degrees of circumference on the outside.

Above. Two units of two are joined one fitting into the other in a tetrahedron arrangement. There are other ways they can be joined since all the triangles are congruent.

I wanted this seed to arbitrarily grow to see what would develop. The only rule was to add T3 units attaching on the bottoms to open triangles where it would comfortably fit without distortion.

Above are various stages of adding only T3 units in random placement adhering to directives set by the configuration of the seed. There is no apparent symmetry to the developing mass, but it became clear there were repeating relationships between groups of units that was not reflected in the random placement of units.  Individual clusters began to appear repeating the same options of attachment throughout no matter where they appeared. Randomly added units were limited to a given number of options by any cluster configurations. Had the seed been a different configuration the growth would have shown different clustering limitations.

There were other options to pack units even closer had I decided to use T, T1 and T2 units, but in this case I limited myself to only T3 to get a better idea of what was going on. There are two aspects of dependency;  each part to the Whole that determines the relationships between parts that determines the design of the system. Parts can be added and subtracted, understanding that each added part, or group of parts, are different divisional reconfigurations in multiples of the same unit circle assuring unity throughout.