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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.
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.
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.
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.
In thinking about the icosahedron and the various possible truncations using equilateral triangles in a tetrahedron patterned net I began to wonder about irregular triangles in the same icosahedron arrangement. Here are a few variations in exploring the same process using the tetrahedron folds to form a right angle tetrahedron; 3 right angle triangles and one equilateral triangle.
Using 4 right angle tetrahedra opened and arranged in the tetrahedron net pattern (one in the middle and 3 off of each side) reveals many interesting variations to the icosahedron. This tetrahedron net pattern for the icosahedron is in a pinwheel shaped net and uses 16 triangles rather than the traditional 20 triangles. The remaining 4 triangles are open planes of relationship due to off-setting the edge joining.
All the models use only 4 circles each in the same patterned arrangement, some using different combinations of the same folds, others using additional creases. This will give you an idea of how the same patterned net can generate a diversity of forms; some are made using a higher frequency folded circle.
Forming the right angle tetrahedron
See website for tetrahedron folding instructions; http://www.wholemovement.com/index.php?option=com_content&;view=article&id=51&Itemid=43 ( 2. Make a tetrahedron.)
Below is the tetrahedron net, the folds from which the right angle tetrahedron is formed. There are nine creases; 3 diameters and 6 chords dividing the inscribed triangle into four equilateral triangles.
The black lines are the creases traced for better viability
There are two points on the circumference on each side of the diameters, three sets of three.
Below) Fold each point individually to the center and crease. This makes 6 more creases. Having folded the tetrahedron, open it to the circle showing the triangle net.
This will form two more inscribed congruent equilateral triangles.
Fold one of the 3 inscribed triangles, either to the right or the left of the center triangle. Notice the center triangle is now off center as if it has been rotated where each division of each side in no longer evenly divided.
The second line in from each corner is at right angle to the edge. When all three corner points are folded behind on this line, 3 right angle corners are formed around the center equilateral triangle.
Bring the right angle corners together in the same way as folding a regular tetrahedron. This forms a right angle tetrahedron. The folded over right angle corners will fit one into the other to hold it closed
Variation in forming the icosahedron net
Fold 4 of the right angle tetrahedra units and open them to the net and tape them together as shown. Face side up using right angle tetrahedron in icosahedron net.
Turned over with bottom side up.
We can see the proportional difference between using the equilateral tetrahedra (left) and the right angle tetrahedron (right) to form the icosahedron net. The net pattern for the icosahedron is the same, each showing twenty triangle faces.
Here is a more open and irregular form of the icosahedron by changing the proportional attachments of the right angle tetrahedron in the icosahedron net.
Below are a few more variations using the patterned icosahedron net. These are made using tetrahedral units folded from the 4-frequency diameter net. Go to “Unity Origami” blog entry from Saturday, February 26, 2011 for folding the 4-frequency diameter circle. By reconfiguring those 12 crease into the inscribed triangle and exploring truncated possibilities and using then in the icosahedron net, there are many variations to be discovered.
This variation above is formed using four 8-frequency diameter circles in the icosahedron net.
Here are 2 units of the above variation joined together forming a single unit. The concave depression of each end can be pushed out to the same configuration, thus one fitting nicely into the other. Multiples of this unit can be joined to form a benzene ring, an open sphere, the tetrahedron/octahedron matrix and a variety of other complex systems.
Another variation in the form of a tetrahedron reconfigured from the same four circles in a tetrahedron patterned icosahedron net.
There is no other form of experiential modeling that will demonstrate this kind of transformational process. Even though some of the models take the form of a tetrahedron and a variety of other truncated and open systems, they are all from the same icosahedron net in a tetrahedron pattern. Were we to make each model using individual polygons it would be a difficult construction job of measuring, cutting, and gluing, where some of them you would not know to be able to make.
Above shows a few directions to be explored using the right angle tetrahedron patterned to the icosahedron.
In the photo below we see the same pattern in different forms. The model on the left is formed using equilateral triangles where the one on the right is from triangles of a higher frequency folding with the circumference folded out rather than in.