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Playing with circles I periodically get into folding and curving as a natural expression of what the circle wants to do. Every fold in the circle is a straight edge; with polygons there are no curved edges. Folding circles has both curved and straight edges without having to construct either. In light of our conditioning towards the primacy of separate polygons and polyhedra, they all come from the circle.
Above In curving the straight edges in this tetrahedron arrangement (four circles, six creases each, three diameters and an infolded equilateral triangle) something caught my attention that connected to an old model that come out of earlier explorations of the nine creases of the tetrahedron.
Below are pictures of past models, two views of a tetrahedron arrangement (six points coming together in a hexagon,) a pentagon, and a square joining. They are all made using the same folded unit reformation of the tetrahedron net. Pictured are examples of 4, 5, and 6 point vertexes. The six immediately below is a double unit forming two parallel a single edge where there is usually only one edge changing the pattern. This became my focus last month.
Below Again using my circle business cards I reformed the tetrahedron as above because these units will slid into each other and allow a good degree of angle movement in joining. The variations and joining into sets are shown to the right.
Below Twelve units in six groups of two each showing the edges of the tetrahedron in a truncated form. This would be the 3 of the 4,5,6 joining from above models. Here six edges form a triangle arrangement at joining rather than a single vertex point.
Above is a tangent direction showing another variation of the tetrahedron using a variations in reformation of the units shown above.
Below is an edge model of the icosahedron that shows a truncated form using thirty sets of two units each. Here each of the twelve vertex points are an aggregate of fives circle end point forming the pentagon star with greater complexity.
Below are two models of the same arrangement each using the same number of units. On the left a short-folded unit is used, on the right the long-folded unit is used.
Below the model on the right above (long-folded units) has other units added. The added circles are folded using only one crease, as new growth might appear from the open ends, much as cell growth might generate tendril forms.
More growth changes the look of the icosahedron seed.
Below The two icosahedra have been joined as if the extension of one becomes growth from the other, with adding a variety of tetrahedron reconfigurations generating forms suggesting differentiated functions of interaction. The tetrahedron functions in a similar manner that we find in stem cells, reforming to specific interactions within their environment as relating to the needs of the whole system.
Another view of the same model.
Going back to the models that caught my attention in the first place, I did a single sphere in an icosahedron arrangement only using the double sets (four circles.) This changed the way the end points come together, now making a decagon arrangement rather than pentagon changing the pattern to a truncated dodecahedron.
Below are the units used in the following model. Reforming the tetrahedron to two variations; one short open cell and the other a longer open cell that can be combined in three different combinations, two shorts, two longs, and one short one long. One slides unit into the other and then joined in double sets in parallel allowing for circular joining on the ends. (Bobby pins are holding the double units together.) These are the unit sets used in the following model.
Below the icosahedron expanded to show two expose layers of information revealed by variations in the folded units. Here both are seen, short units on one half, long on the other half.
This is an odd pattern of the rhombicosidodecahedron that incorporates the truncated dodecahedron. It shows decagons, squares and triangles as a function of the double-edged units, making squares and open decagons without the triangular definition of the pentagon.
Below a partial third layer is added to show the dodecahedron edging. From the inner icosahedron grows the dodecahedron through the truncation process while growing outward, combining layers. There are three views.
This relationship of the icosahedron and dodecahedron through the truncations process brings up a question about the generalization in construction where five ends join together forming one vertex. We know that five locations cannot occupy the same place at the same time and therefore it becomes convenient to accommodate scale by saying the points become one rather than an aggregate of points. Truncation is a term that works to describe cutting corners from solid-formed polyhedra, but does not cover what happens with the circle where there is no cutting and there are open units. Truncation then becomes a movement into and stellation is a movement out from. When things get small to the human eye we tend to make a generalization when in fact the sophistication of our tools and concepts about quantum space suggest something different. Reading Gulliver's Travels gets me thinking about scale as it relates to Euclidean geometry and how limited our real world observations are about the interrelatedness of order and pattern as it relates to scaling of forms through a time and space experience where everything is inherently interrelated, as demonstrated with the circle.
Do not expect the model above is finished, only another beginning in exploring the growth from the icosahedron as a seed pattern found in many natural forms. July is already suggesting design changes in growth.