# The elementary tetrahedral tessellator

Description

A tetrahedral space-filling polyhedron made up of four saddle surfaces with threefold symmetry.

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Construction

Similar to the contrastoid, this solid also resulted from the further development of the tetrahedral tessellator. Concretely it’s the space interlocked between the minimal surfaces bordered by the four skew hexagons of this polyhedron. In a nutshell we can describe its rapport to the former one as if the isosceles triangles fused together without changing the bordering frame, creating a wave-like continuity. Actually so-called “monkey saddles” were forming this way.

Given the strong correlation with the above mentioned two shapes, its volume is the same, thus exactly r^3. The rapport between their surface areas is 1.03075.

Note: As there is still a significant discrepancy between my rudimentary math base and the natural-born imagination skill, the algebraic approach was out of my reach in this case. Fortunately an automatic function of the 3D program could do the needed operation by itself, shortening the otherwise probably much more time consuming way to search for and collaborate with qualified mathematicians.

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Hypothesis

As the honeycomb upbuilt by rhombic dodecahedral cells is considered the 3D version of the hexagonal tiling, the network created by these “wavy solids” can be viewed as the 3D equivalent of the triangular tiling. Therefore the shape is kind of entitled to the denomination “elementary tetrahedral tessellator“.

The basic connection between the analogies:

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Proving the hypothesis

There is a certain correlation between the two and three dimensional geometric shapes, which we can name “analogical rapport”. The more regular the shape the more unambiguous the analogy. Thus the circle’s 3D pair is the sphere, the square’s is the cube, the equilateral triangle’s is the regular tetrahedron. Following the same logic, it’s also easy to observe the correlation between the pentagram and the regular dodecahedron.

However, there are only five kind of platonic solids, which are made of only three regular polygons. We mentioned all of these by now, thus the simplest forms of analogical rapports are exploited.

Let’s move on to the regular hexagon. There is no polyhedron exclusively made of hexagons, so we need to found another law to identify its 3D correspondent. Actually we don’t need it, because others have found it before: the hexagon has a property that it can fill a plane to the infinity without any gaps only by self-replication. This property is called “tessellation” or “tiling”. There are three regular polygons which possess this faculty (equilateral triangle, square and regular hexagon), from these the latter has the smallest perimeter to area ratio. In nature we can observe in many cases this efficient land use, the most known is the structures made by bees.

From this special tiling property of the hexagon it was concluded that its 3D partner must be the rhombic dodecahedron. This catalan solid fills the space the same way as the hexagons are filling the plane, thus in the most efficient way: smallest surface to volume ratio. It is made of twelve rhombs, which likewise the hexagon’s edges are intersecting in 120 degree angles. There is also tight correlation between the edge length and the distances between the vertices and the center: in both cases these are equal. As the hexagon can be divided into three identical rhombs, the rhombic dodecahedron can be divided into four identical rhombohedrons.

One can imagine this in the most representative way by placing inside both the hexagonal tiling’s and the rhombic dodecahedral honeycomb’s constituents new “intersectional cells” of the same shape and size, which will (mutually) touch the initial one’s centers. In the case of the hexagonal pattern the intersection will result in a transformation to the rhombille tiling, while in the case of its 3D correspondent the setting will change to a rhombohedral partitioning.

Remaining inside the domain of the tessellation we can observe that there is an even more relevant connection between the square tiling and the cubic honeycomb. However, using exclusively tetrahedrons it’s impossible to fill the space without gaps, therefore the 3D match of the triangular tiling could not be a network constituted of regular tetrahedrons, as the simplistic approach would require it.

According to my knowledge there is only a single kind of officially documented honeycomb with its building cells presenting tetrahedral symmetry and that’s the triakis truncated tetrahedral honeycomb. Can we call this the 3D correspondent of the triangular tiling? Or is there another one providing even more correlations?

If we look at the relation between the hexagonal and triangular tilings we can notice that the ratio between the hexagons and triangles with identical radius is exactly 1:2, namely we need two times more triangles than hexagons to cover the same surface area.

When truncating a hexagon to a triangle, the combined surface area of the three cut off parts (isosceles triangles) will be the same as the resulting triangle’s surface, while their shape is identical with the ones resulting from the division of the central triangle along its three radial lines.

Let’s get back to the rhombic dodecahedron and try to transfer this operation to the third dimension. We observe that the truncation method can’t give a satisfactory result. After all we need to divide into equal halves the four rhombohedrons that make up the solid. This in turn needs a little more complex intervention.

We are closer to the correct solution by joining the three opposite corners (except the pair with shorter reach). This way the resulting three lines will intersect in the shape’s center, forming three dividing planes in pairs. With this method we arrive exactly to the tetrahedral tessellator.

However, the halving of the antiprisms can be realised even by a single dividing surface. This will be the minimal surface bordered by the skew hexagon, the “zigzag line” following the solid’s equator (thickened blue frame on the above image). As in the 2D tessellation every equilateral triangle is surrounded by three other identical ones with the overlapping of three sides, in the case of its 3D correspondent every cell must be surrounded by four identical ones with the overlapping of four faces.

The elementary tetrahedral tessellator is the single shape, which fully satisfies this condition.

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