Reduced tangential cohesion

Similarly to the octahedral antithesis, it was imagined last autumn during the same transatlantic flight from South America to Europe.

Description

As the name suggests, this solid is a derivation of an earlier creation named tangential cohesionWhile its frame consists of the same 24 curved edges (12 convex + 12 concave), it lacks the straight ones. Concomitantly, instead of 24, it’s made up by 12 identical minimal surfaces, each bordered by 2 convex and 2 concave arcs. 

One rhombic face highlighted with teal borders

Taking the rhombic dodecahedral skeleton out of its structure, it looses the tessellation (honeycomb) property of its predecessor. The general aspect of the solid is more delicate, “flower-like”, with both very thin and very sharp parts. The thin ones are 6 in number and are associated with the convex constituents, while the 4 needle-like spurs are formed around the concave joints. 

The three characteristic view of the tangential cohesion (upper row) and the reduced tangential cohesion (lower row)

Regarding the 12 hyperbolic “pseudo-rhombic” faces it has more similarity with the octahedral antithesis, but while there the convex and concave arc triplets are distributed in separate parcels related to the symmetry of a reference octahedron, here the arcs are fluidly intertwined, forming waves in complementary pairs. 

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Octahedral antithesis (atlantoid)

Regarding the physical context, the conceptualization of this solid has a rather interesting story: I imagined it during a long night flight from South-America to Europe, right in the middle of the Atlantic Ocean. Well, as you already know the “nick’s” origin, let’s dive deeper into the geometrical aspects.

Description

An ensemble of 12 minimal surfaces formed on rhombic frames with tangentially disposed curved edges, joining the center of the solid with the four tetrahedral vertices in an antithetical manner in relation to a regular octahedron’s edges.

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The idea

Likewise the tangential cohesion, the basic concept here was also about a smooth connection between the inner and outer parts of funnel surfaces with the biggest possible curvatures.

While in the case of its predecessor the convex and concave arc triads (the generatrices) were positioned one above the other, here they are disposed in separate compartments, following a checkered pattern inside the octahedral frame, where one tetrahedral composition will have elements curving towards the center, while the other one in the opposite direction, pointing towards the exterior.

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Construction

1. The frame

To get in line with the previous conception we have to consider the skeletal structure of the octahedron as the main boundary setter. In this regard the searched arc will be the Bézier curve related to the below triangle, where CB represents one tetrahedral axis (face center – solid center) of the octahedron:

The generatrix of the solid

We have to place 24 pieces of this arc type along the octahedron’s frame in symmetrical pairs of three. Each of the 12 edges of the platonic solid will be related to 2 inward and 2 outward curving generatrices, where the starting vector of both directions will be represented by the planes of the octahedron’s faces, while the final vector will be tangent to the tetrahedral axes.

You can imagine this by coloring the adjunct octahedron’s faces in checkered pattern, where the 4 white ones will represent outward, while the 4 black ones inward arc trios. Below you can see how the curves will be distributed on two neighboring faces.

One pair of convex (blue) and concave (green) generatrix trios

2. The surface creation

After the frame of the base pattern is ready, the next step is identical as in case of many other previous shapes: it will be represented by the minimal surfaces bordered by these curves.

In this case they are forming distorted rhombic parcels with degenerated vertices, each of them joining a third of two octahedral faces along one of the edges. One such curved rhombic surface will have two zero degree “hyperbolic vertices” where the joined generatrices will be tangential to the axes (in case of two convex, respectively two concave arcs) and two 48.18 degree angles corresponding to the original octahedral face arrangement made by two medians (in case of one convex and one concave arc).

One rhombic face highlighted with teal borders

The symmetry of the shape is tetrahedral and it has the exact same volume like a regular octahedron, as the inner missing parts are of identical shape with the outer surpluses. Concomitantly, you can create the concave parts by flipping over the convex ones according to the octahedron’s faces and vice versa.

While with the appropriate positioning there will be perfect overlap in case of two opposite faces, the solid doesn’t have 3D tessellation properties (can’t form a honeycomb) for the same reason the regular octahedron can’t do this only by itself, but in a combination with the tetrahedron. You can check it here.

The three characteristic view of the solid

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Relatedness with 2D shapes

Based on the relation between the equilateral triangle and the regular tetrahedron, respectively between the hexagon and the octahedron, the 2D correspondent of this solid must be a certain kind of rhombic trio with 2 convex and 2 concave sides each, formed along a hexahedral frame. Actually this shape is a composite of 3 identical parts, joined only in the center point.

While in case of the octahedron the self-tessellation is impossible, for its plane version – the hexagon – it is. For this reason the 2D correspondent of this solid also owns this property.

2D tapestry with 4 colors

To accordingly visualize the pattern-repetition, you have to use 4 colors instead of 2, otherwise there will be overlaying along half of the edges. This way every rhombic trio of one color will have all the other 3 colors as neighbors.

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Tetrahedral divergence

Description

The general idea of this shape was conceived much earlier, not long after the invention of the tangential cohesion, to which (despite the big visual discrepancy) is closely related. Actually it shares all its big “S” shaped curves, though nothing more. While the previous one joined the convex and concave arcs in the most direct way (through minimal surfaces), the new one is a little more complicated.

Here, during the revolution inside a kaleidocycle the neighboring arcs will not approach each other, merging halfway along the horizontal plane, but quit the opposite: they will diverge until both will overlie with the axis of rotation. The joining will be represented by the fact that the divergence will finally reach 180 degrees, when the two will merge into the same line, but from opposite directions. Their single common point is the center of the kaleidocycle, contrary with the overlapping. The below image is a figurative representation of the rapport between these two situations: in “A” case the blue and green triangles share a common edge, while in “B” case those edges are only situated on the same line.

To reach even more interesting correlations, we need to go back to the elementary tetrahedral tessellator. This new shape is kind of “complete inversion” of that solid in relation to the rhombic dodecahedral cell’s four rhombohedra. In case of the tangential cohesion the inversion was only partial, represented by the vertical “S” shaped generatrices taking the place of the horizontal ones, while the halfway joining between two neighboring curves were overlapping in the horizontal plane at every 60 degrees revolution. Here instead, also the transformation along the revolution movement (the surface creation) is “inverted”, thus not only the big “S” curves, but anything between will be tangential to the vertical axis.

In case of the elementary tetrahedral tessellator every vertical plane touching the kaleidocycle’s center will intersect the surfaces along horizontal tangencies, while in the case of the tetrahedral divergence all these will became vertical tangencies. Let’s see the concrete details of how all this will manifest.

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Construction

1. The frame

To get in line with the previous conception, at every 30 degrees revolution inside the kaleidocycle, one big curve (half of the “S” shape) will need to transform into the vertical axis. The exact length of this straight sector will be the tetrahedral radius of the rhombic dodecahedral cell, while everything between the two must remain inside this spatial unit, at the same time continuously touching its limits.

This action will divide the space twice as more as in the case of the tangential cohesion, thus there will form not 24, but 48 identical frames. Each of them will be represented by three segments of which one is the half “S” curve, one the straight (half) axis and one is the path what the moving endpoint of the first will draw while transforming into the second. Let’s identify the exact nature of this third segment.

Its equivalent in case of the tangential cohesion (but also the tetrahedral tessellator, elementary tetrahedral tessellator and the contrastoid) is half edge length of the skew hexagon frame (kaleidocycle outer limit).

Here instead, the path will be not straight, but another Bézier curve, portrayed in the upper 2 images in green. In the first image the inner end (A) will need to be tangent to the 30 degrees shift (related to the revolution), while the outer one (B) to the big diagonal of the concerned rhombic face (related to the constant rise, or descent). In the second image you can see a cluster of 12 green curves inside one of the four rhombohedra (one kaleidocycle), also the “S” curves marked with blue, respectively the axis of revolution in red.

The image above shows the side-view of one of the 48 minimal surfaces (“half leaf”) with the three limiting segments colored according to the previous description.

2. The surface creation

After the frame of the base pattern is ready, the next step is identical as in case of many other previous shapes: it will be represented by the minimal surfaces bordered by the three edges (here two béziers and one straight segment). Inside one kaleidocycle there will be 12 minimal surfaces, 6 in the upper half and 6 in the lower one (see the below image).

Every two will share a common half S curve (left and right sides), this pair resembling a “leaf”. There are 3 leaves up and 3 down, arranged in symmetric antiprismal composition: related to the vertical plane, where one triad has a leaf, the other has the gap and vice versa. The top view resembles a flower of life base motif, while the complete spatial disposition is a 3D version of this pattern with kind of a tesseract effect.

The inner four kaleidocycles (4×3 leaves) will confine a certain space shaped like an 8 pointed star with 4 shorter and 4 longer limbs, while the outer four will be separated hyperbolic surfaces without any volume. That’s before joining with other tetrahedral divergences according to the rhombic dodecahedral honeycomb structure, where each outer leaf triad will join other 3, theoretically repeating the mentioned 8 pointed star pattern into infinity. The overlapping is along the Bézier curves situated on the surface of each rhombic face, the final result being a certain 3D mesh following the tetrahedral molecular geometry.

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Relatedness with 2D shapes

The below image shows the plane equivalent of the tetrahedral divergence shape (teal color). The hexagonal tiling is the 2D correspondent of the rhombic dodecahedral honeycomb, where the flower-like pattern represents the gaps, while the interwoven mesh in between is the plane variant of the spatial net composed of the previously described 8 pointed stars (here 6 pointed ones).

In each hexagonal cell there is one six pointed star (inside the red triangle) and 3 separated thirds of other stars of the same kind (outside the red triangle), which will be centered along 3 symmetric corners of the initial hexagon. The infinite net formed by them follows the triangular tiling.

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