Description

A tetrahedral solid composed of twelve identical minimal surfaces, formed as an interplay between a convex and a concave frame made of parabolic arcs with complementary nature. 

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The idea

It was imagined as a doubly-modified regular tetrahedron, which will both expand and contract at the same time, with the expansion happening along the edges, while the contraction according to the faces. The magnitude of the two antithetical distortions must reach the limit inside the tetrahedral frame, which in case of the convex one means even merging (fluidity), while in case of the concave one tangential joining of the constituents. 

In the beginning I falsely supposed that the two arcs will have the same curvature, moreover I hoped that a convex composition made of three neighboring arcs will smoothly overlay with a concave triad, concomitantly the minimal surfaces formed between them will match and could be evenly joined if placed accordingly. While at first this mistake was interpreted as a project failure, later I realized that there are even more interesting correlations this way. 

The simplistic, intuitive approach regarding the complementarity asks for exact coincidence regarding the opposite curvatures, however another peculiar situation happens when the two arcs correspond according to the full rotation, which may even increase, not weaken the general coherence of the structure. The certitude in this regard came when I understood that the sum of all the convex and concave arcs is 1080 degrees – thus 3 x 360 degrees –  which means a full rotation according to the three main axes together. And indeed that was the case as the distribution of the arcs is in perfect accordance with the XYZ coordinates: 2 x 180 degree curvature for each axis. Beside this I observed that the total perimeter curvature of one of the twelve minimal surfaces is 180 degrees, thus exactly “one triangle size”.

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Construction

1. The frame

As already mentioned, there will be two different arc ensembles which are intertwined according to the double tetrahedral frame. The convex cluster have to be built along the defining planes of one skeletal tetrahedron, while the concave cluster along the other (complementary) skeletal tetrahedron.

The convex part can be imagined as if a tetrahedron’s edges will bulge outwardly to such extent that the four vertices will completely disappear, while in case of the concave one the arcs have to be positioned in such manner that the vertices will change into inwardly curved arc trios. In this latter case the straight edges will split mid-length into two tangential curved halves, with the cusp points overlapping with the midpoints of the neighboring convex arcs, to which they are perpendicular in pairs (think of the crosses on a stella octangula for reference).

Regarding the concrete numerical aspects, in case of the convex part of the frame the six components have to be 109.471 degree Bézier curves, while in case of the concave cluster 70.528 degree curves divided in 35.264 degree halves. The two are complementary, summing up 180 degree along each “cross”, while the relation between their tangents (which defines their eccentricity) is 2/1 in favor of the convex half arcs.

The outer edges have the exact same curvature as the two big arcs (parabola sectors) on the surface of a classic hyperbolic paraboloid and if we invert all of them in relation to the original tetrahedron’s six edges, they will tangentially join in the very center of the solid as the earlier mentioned big curvatures of three different intersecting hypars aligned with the XYZ coordinates. On the other hand, if we invert  the four triplets formed by the inner edges according to the reference octahedron’s faces, their joints will overlap with the focus points of the hypothetical paraboloids to which the concave arcs belong. 

There is an interesting correlation regarding the eccentricity of the arc trios as the outward bulging reaches the exact same height as the depth of the inward sagging. The cause of this coincidence is the fact that the convex half arcs with bigger curvatures (54.736 degrees) are forming the trios meeting with their rear ends, while the concave arcs with smaller curvatures (35.264 degrees) are joining in triplets with their front ends. The special nature of the parabola makes that while structurally different, both the span and the depth of the antithetical clusters will be identical at these angles, resulting that the convex and concave “vertices” will be located at the same distance from the intersections.

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2. The surface creation

The alignment of the parabolic curves compartmentalizes the skeletal structure into twelve identical “pseudo-rhombic” sectors each made up by 2 convex 54.73 degree and 2 concave 35.26 degree arcs. According to the starting vectors of the curves these arched rhombs have two 90 degree angles (between one convex and one concave component) and two 120 degree angles (between two convex, respectively two concave components), summing up 420 degree total angle. The minimal surfaces formed on these twelve frames will be the faces of the solid in question. 

These surfaces can be grouped in two types of triplets depending on which of the two tetrahedral symmetries we take into consideration. According to one of them the faces will join along the concave components and will present a more open aspect, resembling a classic parabola dish, where the convex arcs will form the rim, while the concave ones will give the depth of the structure. According to the other tetrahedral symmetry the three faces will join along the convex components and will look more restrained as in this case the rim is formed by six concave components. 

Despite the angle advantage of the convex constituents (54.74 degrees vs 35.26 degrees), the general aspect of the solid is clearly on the concave side compared to a reference octahedron with the six vertices situated at the points where 2 convex and 2 concave components are joining. Concretely, the octahedron will have significantly more volume than our solid, even if the outward bulging of the four convex arc triplets is superior to the inward sagging of the concave triplets.

This counter-intuitive thing is the consequence of the fact that a closed shape is inherently convex, not neutral. Imagine that you try to close a loop formed by an alternation of convex and concave arcs of the same size. If you want to maintain the fluidity of the curve you have to use more convex than concave parts, otherwise the formed waves will go into infinity along a straight line. The same way, if our structure would be about joining the convex and concave arc triplets following a plane, than the convex ones indeed would have the “upper hand” regarding the displaced space on their side. However, in case of a closed loop with a very restricted number of parts (8 in this case) that advantage is insufficient, inclining the balance towards concavity. 

Regarding the similarity with classical polyhedrons, it shares 8 common points with the reference tetrahedron (4 vertices and 4 face centers), 6 common points with the reference octahedron (6 vertices) and 10 common points (6 quadruple and 4 triple vertices) with the reference rhombic dodecahedron. Its volume is around 2.273 times bigger than the tetrahedron, 1.76 times smaller than the octahedron (remember the “concavity dilemma”), 2.64 times smaller than the rhombic dodecahedron and 5.53 times smaller than the circumscribing sphere. One curved rhombic face has around 1.206 times smaller surface area than a reference triangular face and 1.016 times more than a plane rhombic face. The surface-to-volume ratio is bigger than all the platonic solids, including the tetrahedron to which is close. No conclusive formula have been found regarding both the solid’s surface and volume. 

The structure can also be imagined as a symbiotic bond between the planes of two tetrahedrons of different sizes, where the inscribed sphere of the bigger one is the circumscribing sphere of the smaller one. This way, the planes of the bigger tetrahedron will bend inwardly (convex), while the planes of the smaller tetrahedron will bend outwardly (concave), joining in the six vertices of a reference octahedron. The ratio between the edges of the two tetrahedrons is 3/1. The earlier mentioned “full rotations” will happen because the sum of the two antithetical curvatures is 180 degrees as both distortions are starting from the original 70.528 degree tetrahedral face angle, but while the convex ones are adding 109.471 (2 x 54.736) degrees, the concave ones are losing 70.528 (2 x 35.264) degrees.

This pattern can be taken into infinity in a fractal-like manner, where every reference tetrahedron will contain a 27 (3 x 3 x 3) times smaller one inside. This way the joints of the convex triplets belonging to one scale will overlap with the joints of the concave triplets belonging to the next (higher) scale. 

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Relatedness with 2D shapes

Based on the double correlation between the hexagon and the rhombic dodecahedron respectively the octahedron, the 2D equivalent of this solid must be a plane structure made up by an alternation of 3 convex and 3 concave 60 degree circle arcs (sextants). While in case of the 3D shape the convex and concave components aren’t identical but complementary, their distribution inside the limiting frame follows the same pattern (perpendicular, respectively tangent to the major axes). Compared to the reference hexagon, the composition made up by the six alternated sextants has the exact same surface area because the 3 convex arcs are adding the same amount what the 3 concave ones are displacing. Notice that here the joining of the antithetical curves is angled (90 degree), thus the “convex superiority” for fluidly closing the loop – as we discussed in the previous paragraph – isn’t a premise. 

While these 2D shapes can tessellate the plane according to the hexagonal partitioning, their 3D counterparts can’t form a honeycomb on their own. However, there is a specific, quasi-regular arrangement if we join together the pieces along the concave arcs. This way will appear certain “pseudo-loops” between every 8 pieces, the components of which are aligned along the edges of a cubic pattern. If we select both these cubes, respectively the original tetrahedrons before the double-distortion, the two components will have a rather interesting distribution inside the space-net, namely the 8 corners of the cubes will touch the tetrahedron’s 4 face centers. If spreading this honeycomb into infinity, the rapport between the constituent cubes and tetrahedrons will be exactly 1/2, as every cube is surrounded by 8 tetrahedrons and every tetrahedron is surrounded by 4 cubes. Note: their size differs, the constituent cubes having 2/sqrt3 times longer edges than the tetrahedrons.

And finally about the name in a nutshell: “parabolic” because the frame is exclusively made up of parabolic arcs and “coherence” because there is a deep, manifold connection between the constituents.

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Honorable mentions

There are two other creations closely related to the parabolic coherence which were discovered a little earlier. The similarity is mostly given by the fact that they are sharing the same convex (outer) edges.

In case of one of them the minimal surfaces will form between pairs of three 109.471 degree parabolic arcs, without counterbalancing the structure with inner edges. This way, following the tetrahedral arrangement, there will form 4 “monkey saddle” type minimal surfaces, resembling another former solid, the elementary tetrahedral tessellator.

In case of the second variation the inner edge triplets will be distributed again inside the tetrahedral arrangement, but they will join the convex constituents at the triple joints, not at mid-edge. As such, while the span (related to the reference tetrahedron’s edges) is identical, the concavity will be only half as deep as previously – the Bézier-curves have 19.47 instead of 35.26 degrees – being more restricted by the neighboring axes.

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