Parabolic coherence

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).

**The convex (green) and concave (dark blue) parabolic arcs. Note: the curvature’s value is 180 degrees minus the opposite angle, hence the numbers look reversed**

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.

**The double-frame made up by the convex (silver) and concave (gold) arc ensembles**

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.

**The contour of one pseudo-rhombic face highlighted with teal color**

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.

**Cross-section (red) showing the three-fold concavity of the solid**

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 three characteristic view of the solid**

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.

**Tricolor tiling with alternated sextants**

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.

**Different perspectives of the space-net made up by 8 joined frames (upper row) and the related cube-tetrahedron arrangement** **(lower row)**

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.

**CNC machined anodized titanium miniature in natural context**

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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.

**The three characteristic view of the two solids**

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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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 cohesion. While 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:

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).

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.

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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.

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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The contrastoid

Description

An ensemble made of 90 degree hyperbolic paraboloids built on a rhombic dodecahedral frame, presenting unusually high degree of symmetry.

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Construction

I modified the four sextets of the formerly discovered solid named “tetrahedral tessellator” in a way, that while the frame remained unchanged, the concave generatrices (trenches) will lean into the center and at the same time the convex ones (ridges) will rise twice as high.

This combined maneuver will cause the double bending of the isosceles triangles (their inner vertices will elongate), as a result the “star points” (the centers of the sextets) will transform into (half)axes with the length identical with the solid’s edges. Concomitantly, the former tetrahedral symmetry will turn into octahedral.

The starting point of the idea was, that I tried to maximize the contrast between the positive and negative generatrices, while preserving the tessellation property unaltered. First I was thinking of arcs, but meanwhile realized that the most extreme case will be given by some angled line segments, where the concave generatrices will overlap with the symmetry axes, with their corners reaching the shape’s center. The construction resulting from this action will be indeed kind of “limit case” as it will narrow to zero thickness along all the four axes.

The following practical steps were directly targeting the construction of 90 degree hiperbolic paraboloids as I understood in the imaginative way, that will be the result of the triangle’s transformation (see also the construction of the tetrahedral tessellator).

In the below image we can see that the O’ point of the initial polyhedron turned into the D’O section (half axis) of the saddle surface composition.

Since we took as much from the inside as we added to the outside, the volume of the contrastoid is equal with its predecessor, thus half of the rhombic dodecahedron’s volume as well (interestingly exactly r^3) and at the same time its edges are overlapping with the catalan solid’s edges.

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Properties

It has twenty four bent faces, twenty eight edges (twenty four convex, four concave) and fourteen vertices (eight triple edge meetings, six quadruple edge meetings). Two of the four margins of the hyperbolic paraboloids are situated on the edges of the rhombs, while the other two along the axes. The saddle surfaces are meeting in the solid’s center, each touching the nucleus with one end of its convex generatrix. The other endpoint of the convex arcs are found at the quadruple meetings of the rhombic dodecahedral frame. Both endpoints of the concave generatrices are situated in the triple edge meetings.

Along the four axes of the shape the bent surfaces are forming kaleidocycle-like sextets on both sides (eight), while each one simultaneously belongs to two different axes. The hyperbolic paraboloids sharing the quadruple edge meetings are forming closed spaces. These shapes are owning tessellation properties, namely if they are laid on top of each other than they will completely fill the three dimensional space.

The contrastoid can be defined also as an ensemble of six “3D tiles” of this kind, joined along their edges. The shape of the gaps between the tiles are the exact negative mold of these, thus the compositions joined along the rhombs will form such a spatial network, where the mentioned tiles will alternate with the gaps in a fifty-fifty rapport.

The saddle surfaces are meeting in four different ways:

1: In two edges: There are two subtypes: convex and concave meeting. In the first case one endpoint of each convex generatrix are touching in the solid’s center, forming 0 (zero) degree vectors. In the second case one endpoint of the concave generatrices are touching in the triple corners, forming 360 degree vectors.

2: In one edge: This has also two subtypes, in one case we can identify a single straight line across the two neighboring surfaces.

3: In two vertices: The endpoints of the convex generatrices are shared, these are meeting in the center, respectively in the quadruple corners.

4: In one vertex: This has three subtypes. In all cases one endpoint of the convex generatrices are touching in the center. In one case the two convex generatrices are forming a smooth curvature, in the second case the outer endpoints are found on the opposite quadruple corners (polar disposition), while in the third case the faces are situated on the two opposite sides along the axes joining the triple corners.

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Relatedness with 2D shapes

Starting from the correlation between the regular hexagon and the rhombic dodecahedron, the contrastoid’s 2D mate must be the radial triad of the alternated triangular tiling, as it is showed in the darker sector of the below image (maroon or olive).

Just as the hexagon’s surface is double of the triangle triads that make it up, the volume of the rhombic dodecahedron is also double of the contrastoid, namely the shape of the voids are identical with the shape of the six 3D tiles that make up the composition. Actually there are twelve pieces of “void halves”, these laid on top of each other in pairs will make up the negative molds of the ensemble’s six cells. The 3D equivalent of the hexagon’s three diagonals are the four axes of the rhombic dodecahedron.

The contrastoid has an extremely complex symmetry. Both the whole composition and its components have many symmetry planes. Beside the direct mirroring, the 90, respectively 60 degree turns are generating the pattern repetitions.

My assumption is that it’s the structure with the highest degree of formal complementarity.

The meaning of its name is “antagonistic shape”, which refers to the harmonic interlacement of the parabolas forming the convex and concave extremes (tangencies).

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About the project

Welcome!

I have a lifetime affection for geometric shapes, regarding their inherent aesthetics and complex connection between the constituents. Symmetry, ratio, complementarity and coherence are at the base of formal beauty in general. For me geometry is kind of “compressed knowledge” in the sense it can offer you spontaneous understanding, sometimes accompanied by deep, highly satisfying experiences. I am consciously avoiding the mainstream terms “sacred” or “spiritual” with the purpose not to be confused with pseudo-sciences.

The intent of this blog is to share and give an adequate description of some solids with interesting properties, which happened to be discovered by me. The more detailed descriptions will belong to the ones considered as highly “archetypal” or finalized, some others will be mentioned in a more summarized way. Though I’m not a mathematician, as accuracy is a serious issue here, the presentation will be mostly technical with only a few “personal intrusions”, like names and certain conclusions. On the whole the target is the blending of science and art, where both can be appraised separately, while also forming a symbiotic conglomerate.

With the fast development and propagation of the 3D printing technology, the emerging new domain known as “mathematical art” is becoming more and more popular. Being an intuitive visualizer, I never start with raw algebraic formulas. As a generality, the simplified process of the inventions is a sudden inner cognition represented by some vaguely contoured idea, after what is a pretty long rational interval, when I try to figure out the exact connections, followed by the isolated logical steps to reach and synchronize those and finally the “translation” of the latter to concrete maneuvres for the 3D modeling guidance.

My name is Orbán Sebestyén Zsombor (Zéró), a freethinker and independent explorer from Transylvania and this is my second blog, preceded by The Exiled Weatherman.