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Looking at the octahedron I found something missed, not for lack of attention, only that my focus prevented seeing all that was there in the first place.
Observing six points of connection in the closest packing of four spheres tells us the octahedron is a spherical relationship. Both the octahedron and tetrahedron can be separated from spherical order as individualized polyhedra. Placing four tetrahedra point to point (spherical pattern) we find the same spatial octahedron relationship. The pattern is four, structure is three.
A single tetrahedron can reveal the octahedron by opening one vertex, allowing the properties of the tetrahedron to transform to those of an octahedron; 6 points showing 12 edge relationships of which 9 are formed, and 8 triangle planes, 4 surfaces and 4 open planes. Opening a tetrahedron doubles the number of triangle planes while increasing the number of vertex points half as much. if you count the inside surfaces then there are twelve planes.
Above) A tetrahedron opened to show the octahedron patterned relationship of 6 points.
Below) Two open tetrahedron joined edge-to-edge completing the "solid" enclosed octahedron.
Below right) Octahedron opened flat to show a net of eight small triangles made from two open tetrahedra. The two large triangles are short two more open tetrahedra to make a tetrahedron pattern.
Above left) Completing the tetrahedron pattern (three triangles around one make four) by using one of the large triangles as center and adding two more joining on half of the edge lengths as the first two. This pattern of of four makes a net of sixteen smaller triangles (4x4=16) from which is formed the regular twenty-sides icosahedron; 16 surfaces and 4 open faces.
Above) The sequence from tetrahedron net, to octahedron, to icosahedron net is a geometric progression of 4, 8, 16 triangles. This is reflected by the center triangles of 1, 2, 4, the number of circles used, reflecting spherical order. There is consistency in this development of the first three primary polyhedra that we do not see in traditional construction of individual figures.
There are a number of ways the octahedron net can be reformed, all are worth exploring. That these are all found within the circle assures they are transformational.
Below) Of the 6 edges around the middle of the octahedron, three adjacent edges are joined leaving three edges unattached. This opened figure can be reconfigured in three primary ways; the octahedron, a tetrahelix, and a bi-pentacap.
Below, a, b ,c) Three possible stable reformations from three open edges of the octahedron.
a. The octahedron shows three intersecting squares
b. A tetrahelix of three tetrahedra, one is the relationship between two. It can have a right or left handed twisting depending on whether the net is right or left hand.
c. The bi-pentacap with 10 triangle planes shows each pentagon has four triangle surfaces and one open face showing both inside and outside. The two pentacaps are joined by an open pentagon plane.
Here are other ways the octahedron net can be reformed without overlapping surfaces. The tetrahedron pattern has minimum two surfaces and two open faces that is consistent with the first fold of the circle in half.
d. Octahedron/tetrahedron combination. e. 2 open, 2 closed tetrahedra in a tetrahelix.
f. 5 tetrahedra in combination. g. One octahedron and two tetrahedra
h. Bi-pentacap showing 6 tetrahedra
i. An open square face without tetrahedra. This is the one configuration I missed and for the moment find interesting. It has 8 triangles surfaces and 2 open triangle faces; 10 triangle planes, the same number as the bi-pentacap (c. above.)
The open square face has no stability and when squeezed closed the unit will reform making 3 tetrahedra in a helix form (pic a. & e. above.) Squeezing opposite points towards each diagonal forms the edges of the helix, where one spins to a right and the other to the left. This also reflects the chirality in the net itself that can be right or left handed (See illus. of nets where joining tetrahedra is half way to the right or to the left.)
What happens when joining two sets together on the open square faces?
Above left) When joined on the squares a distorted icosahedron with twenty equilateral triangles is generated; 16 triangle surfaces with 4 open triangle planes. The properties are the same as the regular icosahedron except angles change the number of planes meeting at a vertex.
Above right) A regular icosahedron.
Below left) Two views showing two ways to put the square faces together. One changes the icosahedron pattern. They both have 20 equilateral triangle faces. The left shows the top and bottom surfaces are perpendicular to each other with the open faces on the same plane. On right shows the top and bottom surfaces are parallel in direction, with the open faces in a tetrahedral pattern. The open faces are consistent to the regular icosahedron. The tetrahedron arrangement of intervals are not found in the one on the left,
Above right) Another view of the same two pictured on the left. The red and black lines show two intersecting tetrahedra and the vertical green lines show the relationship of eight points of an elongated cube. The top and bottom look like rhomboids, they are not since the two triangles are not on the same plane. The two intersecting tetrahedron that form the cube are distorted yet all the triangles are congruent.
The regular icosahedron is not chiral. This distorted icosahedron shows two individual variations from the same arrangement by attaching on the square faces. There are other possible combinations of transformation that comes with open space that cannot happen with closed static forms.
Below) Joining open triangular planes in three possible orientations, where units are angled in different directions. Other variations are possible when joining both open and closed surfaces.
Below left) Three icosahedra joined on open faces so there is an open flow through the helix system.
Below right) Two more icosahedra were joined to the helix system on the left using a parallel top/bottom unit in the center. Four units are attached to the center unit on four open faces in a tetrahedron arrangement. Consistency in orientation and handedness from the first fold through all steps is important.
Above) Two views of 4 icosahedra joined on triangle faces in a tetrahedron pattern. Other combinations are possible. When the form becomes increasingly complex and individualized deviations occur that can stop any further generation.
Below) Four distorted icosahedra, consistent in orientation and chirality, are systematically joined through open faces in a tetrahelix pattern. Very different than the helix above.
Being consistent with each step in handedness and orientation while joining units is the easiest way to keep track of the variables. Consistency in pattern is the difference between generating order and falling into disorder. Pattern is consistent; the forms pattern takes often become confusing and chaotic when inconsistencies are introduced. I have taken many models apart looking for why they eventually break down and jam up, only to find I lost consistency in the process. Consistency in pattern will generate many unexpected surprises in a variety of forms.
Two circles reconfigured in two regular tetrahedra, joined to form the octahedron can than transform into a number of configurations showing combinations of 3, 4, 5, and 6 tetrahedra while at the same time generate 3, 4, and 5 fold symmetries. The two folded circles always remain two circles, regardless of the reformations. How do we explain a three-fold increase over the two tetrahedra we started with just by moving two circles? The first fold in the circle from which all folds are generated, is a pattern of movement revealing dual tetrahedra. Numbers keep track of things, they do not explain structural generation or the transforming process of spatial organization generating what often is unexpected.
Exploring without having an identifiable objective opens areas not predictable. As soon as I tie into what I think is going on, that becomes the objective of my exploration and I miss what else is there. It is a struggle to accommodate changes in our lives, yet transformational change is so easy and fluid through the movement of a paper circle as it reforms and reorganizes in multiples. Unencumbered geometry organizes systems that can be easily demonstrated by folding and joining circles. There is much to learn from folding the circle that has been missed by drawing pictures of them, folding polygons, and constructing static models; all which can be extremely engaging, although limited in-formation.