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Monday, 26 September 2011 12:40

Developing The Icosahedron

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One of the icosahedron variations from last months exploration interested me to develop it further. I was curious to see how two with many possibilities would develop. By introducing elements that are consistent to the icosahedron form (not arbitrary design elements) I wanted to see at what point would possibilities run out, and growth stops.

No individual systems is self-contained, they are all parts of larger systems in the consistency of pattern development. In development of two different directions added-on forms are different in configurations and locations where they are placed. The direction for both is determined by the information within the icosahedron form itself. My choice from all the options was that which looked more interesting to spend time pursuing.


di-1   di-2

Above are two views of the same icosahedron. Left shows the triangle face forward with folded trapezoids joining the other three units at one place on the edge. On the right shows the open space forward where three units come together. There are four of these openings equally spaced  reflecting the tetrahedron relationship.


Left below shows 12 forms of I/3 bottom layer of the tetrahedron division placed on each trapezoid surface.


di-3  di-4

Right below shows filling in three triangle open corners of each opening at the 1/3 top layer mark of the tetrahedra; there are 12 tetrahedra.


Top Left shows the addition of full tetrahedra attached to each closed triangle plane.

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

 di-5  di-13


 di-6   di-7

Left show the above attached tetrahedra opened as if to make room for an appearance from inside.

Right shows the 1/3 layer of the tetrahedron used above on left side development. The smaller triangle side is folded in to accommodate fitting onto the perviously added tetrahedra. I decided to leave it as is with 20 equilateral triangle planes facing outward reflecting the icosahedron of origin. When holding it is easy to recognize the tetrahedron, which positions the octahedron and reveals the cube pattern all inherent in the expansion of the icosahedron in a unique expanding development.


The Left photo above shows development continues with some elongated tetrahedra attached as if coming up through the open tetrahedra. Remaining are12 open small triangles from the original four openings.


Here the openings have been covered with elongated tetrahedra folded to the 1/3 proportion with the angle naturally occurring for alignment with the icosahedron stellations. The cube and the tetrahedron relationships are apparent in this form, the octahedron is there but difficult to find. This is a uniquely irregular stellated icosahedron.


In both cases we have by adding units eventually closing off the inside making them “solid.” Further expansion could be done by adding more units to surfaces, as if generated from the inside. Eventually the surface would diminish to where it would no longer be possible to add on in the original scale and what is left becomes textural.

Starting with two open tetrahedra forming the octahedron and by adding two more tetrahedra we form the icosahedron. The seed tetrahedra allows movement between the 3, the 4, and the 5 as it develops through various stages of growth in the icosahedron form. Any one of these stages can take different directions by the form of and how we arrange, or what constraints we set up for continued growth. 




di-11      di-12










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Bradford Hansen-Smith

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