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While in Boston the first week in January and with limited space at the Joint Mathematics Conference, I began to fold my business cards, they are round with three and a half inch diameter. I wanted to explore some multiple systems and did not have the space for using 9" paper plates. As it was I ran out of space anyway, and so have continued working small this month folding my business cards.
Below are my cards and the models I started with. Each helix is made from four circles folded to a tetrahedron. Two tetrahedra were opened and joined to formed three tetrahedra. Each unit is two circles making three tetrahedra. they can be closes or open in form. Four tetrahedra are reformed and joined to make a helix form of twelve tetrahedra. One shows closed units and the other open.
Below on the (left) is a closed 3-celled tetrahelix seed, it has no twisting and is folded from two open tetrahedra. On the (right) the same two open tetrahedra are reformed by opening edges and joining points forming a 4-celled tetrahelix having a directional twisting either right or left hand. There are four different combinations of attaching a 4-cell unit to another in the same direction.
Above right shows three different compactness of twisting in helix form. There is a 3-directional rotation possible with each attachment of the seed units causing the relationships between units to change. There is also the possibility of changing right and left hand twisting at these junctures. The bottom helix is the one we are familiar in that it holds the DNA double helix formation.
Below Putting three different helix forms (above) together in a single strand of three angles of twisting keeping the same directional twist gives an idea a the movement that might occur.
I soon ran out of space and decided to stop the helix exploration, instead focusing on just the nine creases of the tetrahedron. As periodically happens, things appear not having been noticed before. I am not as systematic as it might appear, going with what at the moment catches my attention.
Knowing two open tetrahedra also form an octahedron and by adding two more open tetrahedra to the octahedron net (completing the original tetrahedron pattern) we have made a net for the icosahedron, which I opted to explore further. (Folding of the tetrahedron and octahedron nets have both been covered in previous blogs.)
Below (left ) shows the icosahedron net. On the (right) it starts to immediately form the icosahedron open pentagons through the open squares and down to four open triangles.
This is a 3" diameter icosahedron made with four circles of the same diameter.
The four open triangle faces in a tetrahedron relationship suggested that maybe two icosahedra could be joined showing some kind of interwoven icosahedron type compound. Not having tried that before, they fit nicely one wrapping through the other.
This I had to do with paper plates on a larger scale.
The alternating dark and light triangles do not match up on the joined edges to continue the design. Yet without the colored in triangles the creases match up perfectly where the 3-6 symmetry is consistent to the 5-10 symmetry. This is not a math problem as much as a human problem, bringing us back to helix formations and DNA.
Next I wanted to see the sequential development of the open triangle, square, and pentagon planes. What traditionally is called truncation and stellation is a bit different in using circles in that nothing is added or taken away, it is revealed through movement and multiple circles.
Above shows development from the circle to the tetrahedron, two tetrahedra make an octahedron (here the octahedron is with three open triangle planes.) From expanding the octahedron net we have the icosahedron. The regular polyhedra from triangles; tetrahedron, octahedron, and icosahedron now shift revealing two semi-regular polyhedra, the snub cube and the snub dodecahedron. The three, four, five symmetries are open planes of relationships of circle-folded tetrahedra joined in multiples of 1, 2, 4, 8, and 20 circles. The triangles are structural, both open and closed faces; the square and pentagon are non-structural being relationships of triangles.
Below is a line up showing the stellated developmental stages of icosahedral growth. First the icosahedron, then stellated with four tetrahedra, one on each center triangle of the four tetrahedra that make up the net. They are equally spread around the icosahedron revealing a tetrahedron pattern with an open icosahedron center. The third expansion shows four more tetrahedra equally spaced (on open faces) closing the figure forming a cube pattern with an icosahedron center. The fourth figure is fully stellated showing the dodecahedron pattern with an icosahedron center.
The question came up; how many different ways are there to form the icosahedron net by joining only the edges? As it turns out there are four different combinations by attaching edges that form an icosahedron, only one is regular. Each form is chiral, right and left handed, making eight different configurations from this one icosahedron net.
Above shows only three of the four reformations from the net. The center icosahedron is regular where the one to the right and left, plus the one missing, are irregular deformations. They all have 20 triangle faces; 4 open and 16 closed planes, and 30 edge relationships between 12 vertex points.
Below are the four icosahedron stellated with a tetrahedron on center of the four circles used in the net. They are all tetrahedra in pattern; the regular icosahedron is on the (left.)
Below three of the four have been stellated on four open faces forming eight vertex points. The regular icosahedron is in the center, the others are deformed cubic pattern.
Below. Here all faces are stellated making them all dodecahedron in pattern, with only one being regular. It is not difficult to pick out the distortions. Here again only three are shown. The third distortion is missing because I was late to see that one.
Below is another way to close up the icosahedron net by joining edges. There are now only two open triangle planes making them 18-sided polyhedra of various configurations. Here are twelve different configuration of how many I do not know. Because of the chiral nature of the icosahedron each of these can be right or left hand. There are many possible combinations of multiple systems that can be derived from any number of these irregular 18-side polyhedra; there is much here to be explored.
Above shows a further folding down of the icosahedron net eliminating the open triangle plane altogher. Now they are 16-sided polyhedra, still consistent to edge joining. These are three of an unknown quantity which are for another time to discover.
Below are two different forms of the cubic pattern. One is with the octahedron center and the other has an icosahedron center. The third model to the (right) is the same as the first to the (left) except the tetrahedra have been opened to show the three planer axis of the octahedron. By connecting the twelve points you will see the vector equilibrium, also know as cubeoctahedron.
There are other options in folding circles
The circumference can be folded to the outside as well as inside to form polygons and polyhedra.
Below are two examples of folding the circumference outside using the helix forms we started with. To the (left) is a 3-cell tetrahelix seed and to the (right) is the open 4-cell twisting unit.
Below is a sequential development of polyhedra with the circumference on the outside rather than folding in. On the (left) is the tetrahedron arrangement of four circles. Next to that shows the circles rearranged forming the vector equilibrium (cuboctahedron.) Adding four more circles to the first tetrahedron reveals the rhombidodecahedron. To the (right) is another derivation of eight circles in a cubic pattern.
Below are a few more examples of reconfigurations of the tetrahedron folds using four circles each.
Below we see the evolution from the tetrahedron (4 circles) to a rhombidodecahedron (8 circles.) By opening the six vertexes of the rhombidodecahedron more space is generated adding 6 squares and 24 triangles to the 12 open rhombic planes.
Below is the contrast between the open square in the snub cube (left) and the intersecting tetrahedra of a cubic relationship (right.) The snub cube is not structural, it has little stability.
Below is the snub dodecahedron (left) and the icosadodecaheron formed with the circumference outside (right.) Twenty units are used for each figure. The 12 open pentagons are placed differently in each case showing the same arrangement of twelve circles in differently formed configurations from the same creases. Both snub figures have little stability which is greatly increased when the circumference is utilized.
Above are two views of the icosadodecahedron.
These models show connections between flat plane, straight edge Euclidean and non-Euclidean spherical geometry that have not previously been explored simply because we do not fold circles. There are multiple ways to reform the circle that free us from the traditional construction methods using polygons and making polyhedra. There is also so much more information revealed in the circle that just is not possible in more traditional approaches to modeling. These figures can be formed because they are inherently in circle unity and can be revealed by using unit circle forms. The circle is the commonality between all the differences of endless possibilities of pattern formation. This is a very small sampling where each circle remains whole reflecting unity in the circumference whether it is hidden or not.
As units are so formed, is unity informed.