A missing Shape

After two days of not posting here on Steemit – sorry for that (I should say thank you instead, someday you'll probably get to know why) – today I write about something I don't have a name for up to now.

I write about a shape that has been accompanying me & my structural developments for several years now – a shape that is important to me and that seems to also be important in general (..'important' in its geometric relevance, no comparison to other aspects).
It's a shape that happened to appear both planned and unplanned. After initially designing it as a rotating variation of a cell-like structure, it recurred unforeseen as a folding state of structures from a different line of development.

Let's start this whole thing geometrically. It's about lines and knots forming shapes. Due to the material I work on and which is bent and punctually joint, a so-called 'knot' connecting linear elements from 3 sides can for example also have the shape of a triangle to allow elements to pass through (...just look for those triangles in the pictures..).

To explain the initial appearance of the shape being introduced here, I need to illustrate one of the first developments based on this material including accompanying experimentation. Trying to simply construct a vertical structure without any further reasoning I designed a tower-like cellular system. To simplify materialization, on all upper levels knots were dispersed into triangles. Characteristics of the physical structure were quite unexpected. It happened to be stable regarding its basic shape, but to directly respond to external impacts. When pushed or hit with something the structure starts vibrating quite intense and settles only very slowly. Moreover, its elasticity cannot only be observed in oscillation but also regarding foldability; the top knot can easily be pushed all the way down down to the bottom. Each cell is collapsing separately while the chronological order of this process is not to be predicted.
Later and as derivation from this development also an elastically transformable single cell structure was materialized. At its folding, the highest counterforce is being observed in the middle of the transformation.
Eventually, with the aim of initiating oscillation in a different structural system, a further variation of this cell structure was designed that would rotate during compression. For this purpose the geometry was modified by leaving only one curve per side reaching from one knot to another – that's the shape I'm talking about.

Now let's look at the cube. Someday I will write an article just on that. Some years ago, with the aspiration to further variate geometric approaches, I started thinking about how to possibly fold a cubic structure. I could think about 2 or 3 possibilities before it became obvious that I need to actually materialize it to explore the dynamic potential by interacting directly with the physical object. It soon got clear that the range of transformations is much wider than I could ever have prescribed – dozens of symmetrical shapes and an endless amount of random ones can be formed.
Somehow strange, but it was more than 1 year and loads of experimentation later that for the first time it came to my mind to try what the cubic structure's smallest possible state could be. It then took only about 30 seconds, only some movements to find it – and it was astonishing: Via some stages the cube was transformed to this same geometry I designed more than a year before. Sure, I already knew the last rotating step to make the former cube completely flat.

Based on the cube and its transformations, many different structures have been developed since then. Some were directly designed to be foldable to the same small shape – supplementary to the undefined number of new possible states. Sometimes this possibility could quite easily be predicted, the potential of other structures is even now nowhere near from being explored completely. If for example this – in top view – square-shaped form is varied to any other even-numbered polygon, then the same transformation can be initiated. If though any of the faces shows an odd-numbered polygon, then this can be logically excluded.

So let's follow this link to go back to the importance of this one shape. Polyhedra are usually defined regarding their faces whereas lines are meant to be straight. In the contrary, my own approach though them differently. Due to bending by joining linear elements in knots and assembling them to bigger arrangements, lines are usually constructed as curves.

Let's look at the cube again – sure, due to its triangled corner points it's actually a truncated hexahedron: 12 edges, 8 corner knots; each face is framed by 4 edges and 4 corner knots. Now there is to be said that this cube can be transformed to a tetrahedron – the transformation initially happened on accident, it took pretty much time to replicate and to be able to conceptualize the movements. So let's look at such a tetrahedron; 6 edges, 4 corner knots, faces framed by 3 corner knots. The transformed linear structure is doubled everywhere. Finally, let's go back and look at this shape being discussed from the beginning of this article: 3 edges, 2 corner knots, 2 corner knots per side; the linear structure is quadrupled everywhere.

Conclusively, it seems like those shapes are actually part of the same series. It seems like this still widely unknown shape should be treated as substantially as cubes, tetrahedra and other prominent ones. What could be the reason for that? How could it be 'forgotten'? The curved lines seem to be the answer to these questions. If – regardless of knots being truncated – lines were straight, then this shape, consisting of 2 connected points simply couldn't exist.

Let's see, maybe one day it will get the attention it obviously deserves.

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