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).

**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.

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 fluid contrastoid

Another solid with similar conceptual background preceded the tangential cohesion. In short, both are based on the “ambivalent funnels” discussed in the previous chapter, the main difference is that while in case of the former the number one priority was to maintain the 50-50 rapport between the convex and concave parts, in case of the latter it was the connectedness (continuity) between the constituents.

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Construction

1. The frame

As my purpose is to generate a closed system (solid with volume) it is inevitable that the convex generatrices will be longer than the concave ones, otherwise their coherence could attain only linear or planar results as in the case of the corrugated cardboard or the egg box. In a tetrahedral arrangement this is impossible. Then what could be preserved at the level of complementarity?

Maybe is possible that the shorter concave sectors be the exact inverse of those segments of the convex ones, which represent the central parts of the funnels. Yes, this way the elements can be combined well and at the same time turns out soon that there are only two right solutions to the problem.

Note: the special case when the origins of the four funnels are overlapping in the solid’s center falls out as there the convex parts would mask the concave ones.

a) The original method: We know that the four origins are situated on the vertices of an imaginary regular tetrahedron and the radial curves which starts from those points are belonging to well defined planes (3×4=12 in all, but reduced to six because they are overlapping in pairs). As the tetrahedral axes form 109.47 degree angles and the funnels main curves are coplanar in twos with the axis pairs, there are only two possibilities when the neighboring arcs convex ends will form continuity, in other words they will join without a fault. That will be attainable using certain kind of elliptic arcs.

The below image represents one of the two solutions, when the joining will happen inside the 109.47 degree axis angle (the other one would be the 70.52 degree complementary angle).

In this case the necessary curves are segments of a special kind of ellipse, whose main characteristics are the sqrt(2) ratio between the big and small diameter, respectively the equality between the latter and the distance between the two focal points. This can be obtained in the simplest way by intersecting a cylinder with a 45 degree plane.

In the below image we can see the ellipse (left) and the structure formed by joining the convex ends of the neighboring arcs (right), with the origins O1 and O2, and the appearance on the opposite poles of O1′ and O2′ after mirroring the upper half to the lower one to close the curve. The green convex arcs belonging to the painted segments are exact complementaries of the blue concave ones, marked the same way. We get six “open lemniscate” shapes like this one after fluidly joining all the generatrices of the four funnels in the convex ends (J1, J1′). These united pairs will be sorted out along the twelve edges of a stella octangula (stellated octahedron).

The concave ends will mirror the continuity in the origins (O1, O2), creating alike attachments on the other side (O1′, O2′). This way the former tetrahedral symmetry will be doubled, transforming to octahedral, consequently there will form eight interconnected funnel frames. Here the concave generatrices will be simultaneously part of two funnels, while the convex ones, beside their main role inside the structure will also represent the boundaries of other funnels. Actually every tierce will be part of two different funnels (overlapping).

The other solution will be represented by the more elongated eight shaped curve (unpainted ellipses) perpendicular to the vertical one, which we were describing before. As can be seen, in this case the curvatures are starting more gentle from the origins and even if they are compensating in part at the joinings, in case of identical center-origin distance the used perimeter is slightly bigger (sqrt(2) in function of the radius), hence less “ambitious” than the first method. So we stay with the efficiency.

b) The cube method: The same result can be achieved more straightforward by slicing a cube along one of its diagonals (intersecting with a 45 degree plane), sliding the resulting rectangle in his own plane simultaneously in all four cardinal directions by exactly one length, than drawing tangent ellipses around each new projection. Thus the four ellipses will be tangent on the corners of the initial rectangle.

In case of the two rectangles obtained by the sliding along the shorter edges we keep only the ellipse arcs situated outside of the tangencies, while in case of the other two we keep only the arcs situated inside the tangencies. This way the four curves (two bigger convex and two shorter concave) will form the “open lemniscate”. Doing this operation with the other five diagonal rectangles of the cube you will get the complete frame of the wanted solid.

This can be imagined as if you take a cube and bend its edges inward in a manner that the initial 90 degree vectors in the corner points will change to 0 degrees (tangent to the symmetry axes), then bend the diagonals of the cube’s faces outward in a manner that the initial 60 degree vectors in the corner points will change to 360 degrees (tangent to the symmetry axes from the opposite direction). This way you will got a complete edge continuity through the vertices, given by the joining of the convex and concave arcs created before.

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2. The surface creation

The six tangent “pseudo lemniscates” will divide the frame into twenty four identical parts, each delimited by one smaller concave and two bigger convex arcs (J1O3’O1 in the below image). The convex parts have the same size and are longer than the concave ones, but they are including the exact complementary of the latter in their structure.

None of the three curves of a cadre are coplanar. Regarding the surface creation, the smooth transition between the convex and concave arcs will be given by the minimal surfaces bordered by the formerly mentioned three ellipse sectors. This way the convex arc will gradually transform into the concave one by doing a 60 degree revolution around its fixed inner endpoint, while the outer endpoint will trace a curve identical to the arc itself.

Regarding their directions related to the axes, the minimal surfaces of a given funnel composition (kaleidocycle) can be sorted into two antagonistic triplets, as we can see in the below image: the greens are the “clock-wise team”, the reds are the “counter clock-wise team”. However, as every minimal surface is simultaneously part of two funnels, their directions are ambivalent, it depends on which connection are we take into consideration.

The solid can be painted in a chess-board fashion, following a simple rule: after choosing a color for one face, every neighbor (sharing a convex or concave arc) must have the other color. Because of the overlapping, in four funnels (O1, O2, O3, O4) the greens will be the clock-wise, the reds the counter clock-wise ones, while in the remaining four (O1′, O2′, O3′, O4′) inversely.

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Homeomorphism with the contrastoid

This shape shows upfront similarities with my older creation, hence their namesake. It can be perceived kind of a further development of this, during which the axes are contracting to the midpoints, therefore the straight edges are curving inward and are creating the convex generatrices, while the concave ones are forming between these points from the simultaneous upflexions of the axes from the center, overlapping in twos. At the same time it preserves many basic properties of this solid: octahedral symmetry, kaleidocycle disposition of the minimal surfaces and the dual inherence of the tierces.

The differences between them can be sorted in two main categories in function of the regularity. On the one hand the shape with the rhombic dodecahedral cadre represents a higher level of symmetry summation. The 90 degree hyperbolic paraboloid faces of the contrastoid are symmetry compages on their own. Due to this and to their specific arrangement one has the exact same view from the center to the triple vertices as it has vice versa.

On the other hand the new creation has more connections with respect to the edges, as the generatrices are merging fluidly (no corners). Thus the angled triple edge meetings will turn into funnel centers, while the quadruple ones to rounded attachment points. Unlike the contrastoid, which actually isn’t a genuine solid, but a composite of six identical pieces joined together along the edges, its fluid version is a compact structure.

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Honorable mention

Following the same logic of combining the antagonistic tangencies in a different setting (also based on the rhombic dodecahedral frame) resulted in the composition with tetrahedral symmetry below. Here the positive generatrices are not fluidly merging into the negative ones, but the two are parts of perpendicular planes.

This was kind of a “bold project” as the essences of all the four main curved 3D shapes (outer dome, inner dome, outer funnel, inner funnel) were present in its structure. For a short period of time I believed that the convex (left image) and concave (right image) positionings will be perfect complementaries, consequently the solid will have tessellation properties, but it turned out that this was a false presumption. Nonetheless, the shape still has an aesthetic appeal.

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Tangential cohesion

Description

A tetrahedral space-filling solid with twenty four identical hyperbolic faces, built on a frame given by an ensemble of joined convex and concave tangent arc triads.

It’s the space closed by such sides, which represent the gradual-symmetric transition between the outer and inner parts of two centrally interconnected identical funnel surfaces with the biggest possible curvatures.

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The idea

It is based on an inversion rapport with the saddle surfaces. While the main curves of the latter have their vectors in the pole perpendicular to the vertical axis, the ones of the solid in question must be tangent to this. Therefore its constituents can be considered kind of “inverted monkey saddles” as in both cases the main curves are “S” shaped, the major difference is given by their positioning in relation to the axis. To keep it simple, in the below figure I will use 90 degree circle arcs instead of parabola sectors to represent the general leaning of the two curves.

The monkey saddle is an undulated surface similar to the hyperbolic paraboloid (“pringle shape”), but instead of two it has threefold symmetry. It contains three “S” shaped big curvatures (blue) and three straight lines (red), while the rest of the structure is given by a fluid transition between these two. All the radial intersections have their vectors in the pole perpendicular to the vertical axis.

In the right of the below image is a horn torus with both the outer and inner surfaces partially visible. If we take into consideration only the generatrices situated on the upper half, we can identify three convex arcs (green) converging to the center, which are part of a hypothetical vortex type surface of revolution, resembling a funnel interior or “black hole”. If we look only to the lower half there are also three, but this time concave and divergent arcs (blue), which are part of a horn type surface of revolution, resembling a funnel exterior or spike. As can be easily concluded, one is the inversion of the other. The “S” curves here are the a+a-, b+b- and c+c- half green-half blue arcs, which are all tangent to the vertical axis.

How could we create such a shape, which forms a uniform transition between these inner and outer funnel surfaces? Theoretically, in a sense this would be the opposite of the saddle surfaces, as there is a “crossing” between the plain convex (outer dome) and plain concave (inner dome) structures. This “ambivalent funnel” would be our solid’s base motif. Let the details of this task for later and move farther with the big lines.

The other priority is to obtain tetrahedral symmetry, while reaching the biggest contrast between the convex and concave constituents, but also preserving the exact 50-50 ratio between them. Thus the range of possibilities narrows considerably and bobs up the idea of the single correct solution.

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Construction

1. The frame

Regarding the formerly mentioned “biggest contrast”, the limiting factor will came from the side of the concave components, as the four axes of the tetrahedral setting will not allow to exceed a certain curvature. Theoretically the convex generatrices could make much bigger deflexions, including the particular case, when they will form continuities with their peers from another side.

Despite I am always looking for extremes, at first I didn’t noticed the special connection between the concave tangencies along the axes and the corresponding convex curvatures, as the latter didn’t seem to reach any concrete limits (continuity or tangency) at those angles. Only later I realized that these positive deflexions are limited by the edges of a hypothetical rhombic dodecahedron, therefore they are representing the maximum possible bending inside the tetrahedral tessellation frame. As a matter of fact, the imagined solid will also have space filling properties, like the ones I’ve described in the previous articles.

Now let’s determine the exact nature of this arc. In short it will be embodied by the Bézier curves of the given rhombic dodecahedral structure (edges and axes). This by the way entirely overlaps with the contrastoid’s frame, thus it can be imagined as a derivative of the former composition. Actually both are resulting from the farther development of the tetrahedral tessellator, the difference is that in case of the contrastoid the convex and concave generatrices transformed into an angled segment which overlaps with the frame, while the new solid’s components will represent the evenly rounded version of this (see the above image).

In this respect, the tangential cohesion’s generatrices are intermediates between the tetrahedral tessellator’s and the contrastoid’s, not only because of the spatial positioning between the two, but also regarding the mutually shared properties: it is even like the tetrahedral tessellator’s straight lines, but crosses the origin perpendicularly like the contrastoid’s inner segments.

As compared with the elementary tetrahedral tessellator, the common trait is that both have smooth curvatures, but the origin crossing is quite different (horizontal vs vertical). Also the deflexions will be contrary and more significant (109.47 degrees vs 35.36 degrees), thus the biggest convex arcs will be where the other has the biggest concave ones, while the biggest concave arcs where the other has the biggest convex ones.

Note: Every tetrahedral space filler have those specific O’ points on its surface, at half edge length from its center. Actually the solids have much more in common, this includes ten vertices (four tetrahedral, six octahedral) joined by twelve edges and an additional twelve straight lines arranged in radial triplets in the interior. Their volume is also equal, exactly r^3.

2. The surface creation

Let’s return now to the “vortex fusion”. From both the inner and outer funnels we have symmetrically disposed, parabolic arc triads, which are fluidly joining in the origins and these generatrices will remain the only constituents what we will keep from the surfaces of revolution. The “S” shaped curvatures will divide the frame into twenty four identical parts, each delimited by one convex, one concave and one straight sector.

Regarding the surface creation, the smooth transition between the convex and concave arcs will be given by the minimal surfaces bordered by the formerly mentioned three different sectors.

This way the convex arc will gradually transform into the concave one by doing a 60 degree revolution around its fixed inner endpoint, while the outer endpoint will follow the edge of the rhombic dodecahedral frame. Then the concave one viceversa and after six transformations inside the skew hexagon the kaleidocycle is closed. The same combination of maneuvres will happen with the generatrices associated with the other three skew hexagons of the tetrahedral setting and the solid is finalized.

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Relatedness with 2D shapes

Based on the conclusions stated in the description of the elementary tetrahedral tessellator, the tangential cohesion’s 2D correspondent must be a certain kind of “distorted triangle”, which forms a tessellation derived from the regular triangular tiling.

The edges of these triangles are deviating from the original straight lines the same way as the tangential cohesion’s kaleidocycles from the minimal surfaces bordered by the skew hexagon: both are reaching the maximum possible even curvature inside their limiting cell (convex generatrices), respectively fixed by the axes (concave generatrices). In case of the 2D shape this cell is the regular hexagon, while in case of its 3D version the rhombic dodecahedron.

As in the plane the setting includes also an orientation of the distorted triangles (clockwise or counter-clockwise) we can say that the genuine 2D pair of the new 3D tessellator in question are both directions together. This can be easily represented on the two sides of a transparent paper or by mirroring.

If we want to relate the concentric structure of the ambivalent funnels with those of the monkey saddles, then the below linear representation will show us the essence of this rapport. Here the fluid red curves of the “a” version illustrates the peculiarities of the saddle surfaces, while the green, “onion dome” pattern of the “b” version (the 2D equivalents of the alternating ridges and trenches) those of the funnels.

And last but not least the meaning of the solid’s name in short: “tangential” as the convex and concave generatrices are all osculating to the axes (their vectors are meeting in 0 degree angles) and “cohesion” as because of the significant fragmentation it has a big surface area in relation to the volume, therefore the adhesion between the cells of this honeycomb is strong.

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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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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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The tetrahedral tessellator

This shape is the first result what according to my judgement can be categorized as “archetypal”, which in short means something what reached a certain finalized state with some outstanding and well defined particularity.

At the same time it’s also kind of an “anomaly” as all the other previous designs are exclusively made of curved surfaces, while this one has only plain faces. No wonder that it was discovered by accident, while working on the level 4 continuity expansion.

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Generalities

A star polyhedron with tetrahedral symmetry, presenting 3D tessellation properties (honeycomb). It is composed of twenty four isosceles triangles, which are making up four sextet structures with three convex and three concave radial edges. It has ten vertices (four triple edge meetings, six quadruple edge meetings) and thirty-six edges (twenty-four convex and twenty-four concave). Every sextet is composed of three coplanar triangle pairs, whose constituents are touching in the vertices.

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The creation

1. The original method: I searched those points of a regular tetrahedron’s four symmetry axes (the lines joining the vertices and the face-centers), which are situated at equal distance from both the vertices of the faces they are belonging to and from the points situated somewhere on the lines joining the center with the edge midpoints (E in the below image). For the ABC face that point is O’.

It turned out that there is only a single possibility for my idea, namely when the formerly mentioned points are situated at sqrt(2)/2 distance from the center (if the edges are 1 unit). As the reach between the vertices and the center, the sqrt(3/8), coincides with the edge belonging 109.47 degree circle arcs top’s reach, I figured out that these points are corresponding to the edge belonging 141.057 degree ( 2*(180-109.47) ) circle arc top distances from the center.

Separately joining the six new points with all four vertices of the tetrahedron will give a structure composed of twenty-four isosceles triangles.

In the above image the 1 unit represents the radius of the circumsphere, thus the tetrahedron’s edges are 2*sqrt(2/3), approximately 1.633 in length. In the right figure the plane formed by the E, F, G points is closer, while the ABC triangle is farther from the observer. The O’ is situated halfway between them.

2. Starting from a tetrahedron: We construct four prisms from the faces of a regular tetrahedron, all with the height identical with the tetrahedron’s height. The composition made by the six double intersections of these prisms (this includes also the central quadruple intersection) is our solid.

3. Deducing from the rhombic dodecahedron: Four out of eight vertices represented by the triple edge meetings (one of the two tetrahedral compositions) of the mentioned catalan solid will be “sunk” twice as close toward the center. This is the easiest way for direct visualization and also tells about the close relationship between the two polyhedrons (more details in the final part).

The outcome of this maneuvre will be the disappearance of these vertices, as the new points will wind up on the lines joining the four unchanged triple edge meetings with the six quadruple edge meetings, namely on the edge midpoints of these (see the “original method”). The resulting twelve lines are creating four triple intersections, which are dividing the solid into twenty-four identical isosceles triangles.

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Formulas

Surface: S=12*sqrt(2)/ sqrt(3)*a^, around 9.7979*a^

Volume: V=8/3*sqrt(3)*a^3, around 1.5396*a^3

Surface in function of the circumscribed sphere: S=9*sqrt(2)/ sqrt(3)

Volume in function of the circumscribed sphere : V=r^3

V-sphere/ V-TetrTess=4/3*PI, around 4.18878

S-sphere/ S-TetrTess=4*sqrt(3)/ 9*sqrt(2) *PI, around 1.71006

(V-sphere/ V-TetrTess)/ (S-sphere/ S-TetrTess)=sqrt(6)

S-TetrTess/ V-TetrTess=9/sqrt(2), around 6.3639

One triangle: A=1; B,C=sqrt(11)/ sqrt(12), around 0.9574; alfa-angle: 62.965 degrees, beta-angle, gamma-angle: 58.518 degrees; S=sqrt(2)/ 2*sqrt(3)*a^

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The tessellation

During the joining of three individuals, one triangle of the first and one of the third solid will became coplanar forming a rhomb, whose characteristics are:

a=sqrt(11)/ sqrt(12); D=2*sqrt(2)/ sqrt(3); d=1; D/ d=2*sqrt(2)/ sqrt(3), around 1.633, the same as the ratio between the edge length and the radius of a circumscribed regular tetrahedron: 1/ 0.6123; A, C (acute) angles: 62.964 degrees; B, D (obtuse) angles: 117.036 degrees; S=sqrt(2)/ sqrt(3)*a^; the angle formed by two neighboring rhombs: 146.443 degrees.

Though similar, these rhombs are not the same as the ones which make up the rhombic dodecahedron. They are a little “thinner”, so if their small diagonals are the same, the ratio between the big diagonals is 2/sqrt(3), the same relation as the one between the sides and the height of an equilateral triangle.

Every vertex is shared by ten individuals (represented by six quadruple and four triple corners). Every edge is shared by six individuals. If this is vertical, than it’s the common part of three standing and three upside down solids, situated on the rotational axis of the sextet. Every individual is bordered by other four. If the central is a “standing” one, than all the surrounding four are “upside down” and vice versa. In other words: every individual can be “glued” only to the faces of the inversely positioned ones, thus the layers are forming chessboard-like 3D patterns. This can be illustrated well while using two different colors.

Inside the tessellation we can discern twelve non-parallel planes, represented by one triangle pair on every individual’s surface. The triangles are meeting in one vertex, forming a “hour-glass” shape, an individual is made up by twelve hour-glasses. To every plane belongs a rhombic pattern. The distance between two parallel rhombic rows belonging to the same plane is 4a, namely four times the small diameter of the rhombs, consequently the distance between two cell centers belonging to neighboring vertical rows is also 4a.

If we take into account only one “level” than we can observe that the individuals are joining each other alternating between the standing and upside down positions in a zigzag fashion, forming 120 degree angles with their centers.

This tessellation is heading towards a regular tetrahedron with infinitely long edges, during while the cells are forming bigger and bigger tetrahedral compositions, following the “square piramidal” sequence (1, 5, 14, 30, 55, 91…). Thus a “level four” structure is made up by thirty individuals, for example.

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Relatedness with regular polyhedra

If we join the triple edge meetings with the center, the solid can be divided into four identical parts, which if attached with their outer sides to the faces of a solid with similar size, their inverted inner sides together will form a rhombic dodecahedron.

This property can be found also in the case of the cube, where this inverting action results in six identical parts. The main difference between them is that the cube, the rhombic dodecahedron and the stellated rhombic dodecahedron (Escher’s solid) can turn into each other with the previously mentioned maneuvre during a cyclic process of three steps: cube, rhombic dodecahedron, Escher’s solid, cube, … and so on. In this “fractal sequence” the edge lengths of the similar solids (cubes for example) will successively double, while their volume will increase eight times.

The same process can be started from the tetrahedral tessellator, but there is “no return” to him. In turn, our shape can be constructed from the stellation of a certain kind of triakis tetrahedron. This is the shape represented by the quadruple (central) intersection of the four prisms described in the second method of the solid’s creation (see above). It can be also mentioned that if we circumscribe a tetrahedral tessellator inside a cube (the vertices tangent to the cube’s six faces) than its volume will be exactly one octa of the cube’s volume.

The further development of this shape lead me to the discovery of two other interesting solids: the contrastoid and the elementary tetrahedral tessellator.

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The quest for the tetrahedral centrum-continuity

A big chapter of my plannings was dedicated to the perfectioning of a particular kind of solid, what in a nutshell can be described as a tetrahedral structure, which has the center point on its surface, while there is the maximum possible continuity through this point.

The first and simplest member of this long lasting “dynasty” was a special case of intersection between four spheres, therefore strongly related to the first two self-designed 3D shapes (see the previous post ).

This new one was represented by the combined six double intersections (including the four triple ones) formed by the spheres arranged in a manner to have one single common (quadruple) point. Actually it’s the 3D representation of another pattern formed by sextants, which is the one when three circles are meeting in one common point.

In the rightmost part of the above image we can notice one continuity through the center represented by the vertical line (in fact a circle arc), partially masked by a lobe. There are six arcs of this kind on the surface of the solid, intersecting in the middle. One can think that these are also sextants, but the truth is they are somewhere between 1/5 and 1/6.

The next step was to reach more continuities, while keeping the former one unaltered. There are two ways to exceed the initial setting: one is to have concentric continuity inside the “triple petals”, the other is to reach evenness in the intersection of the edges.

In the case of the “concentric type” there will form four funnels in the place of the petals, as the initial generatrix is used for the creation of surfaces of revolution, while in the “vertex type” case the meeting in the four triple points will happen in the form of 0 (zero) degree vectors related to the tangent plane as the six edges are part of a sphere’s surface (109.47 degree circle arcs).

The following development was to change the generatrices of the concentric type’s funnels in a manner to eliminate the outer sectors by reducing them into a single point, thus transforming into vertices . Of course, the central continuity remains (see the image below).

From this shape originated two others with even higher cumulated coherence: in the first one’s case appeared new continuities perpendicular to the edge midpoints, while in the second one’s case the vertices disappeared, reaching evenness by the zero degree vectors. Theoretically this second one can also be considered the evolution of the vertex type level 1 shape.

While different, at the same time both are constituted by the intersection of four surfaces of revolution like their predecessor, only having distinct curvatures, each of these calculated to reach the specific property improvement. The first one still has discontinuities in the tops (dents in this case), while the second one remained with the sharp edges. The strength of one is the weakness of the other and vice versa.

The “final level” was reached by a shape, which possess the advantages of both while lacking their deficits as mentioned in the previous paragraph. I surmounted the dilemma with the construction of “wavy funnels“, by following an undulating revolution path of threefold symmetry with the generatrices, therefore preserving both the edge-transverse and the vertex evennesses unchanged.

Nonetheless, it can’t be considered a finalized, “archetypal” solid as it still remained with edges, even if less marked ones. But it’s likely the closest possible approximation of the original idea.

Actually, I was suspecting from a pretty long time, that what I’m intuitively searching for is an already existing shape named Roman surface, which can be considered the 3D correspondent of the regular trifolium and has many structural similarities with my creations, except that it’s self-intersecting.

What I’ve tried is to attain the same complete surface continuity without the self-intersecting property, using only one single central passage, which finally turned out to be impossible. With all the respect to reality, we can still call the level 4 result “the possible maximum” in this context.

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Honorable mentions

There are some secondary designs which show up marked esthetic appearances in the form of YinYang-like complementarity.

Two of them resulted from the reduction of the edge-transverse/ concentric type of level 3 shape: one can be considered 1/3, the other 2/3 part of this solid. The first is kind of “pseudo hyperbolic paraboloid“, while the second one has similarities with the Enneper surface and the cross-cap, depending on the view.

However, unlike the mentioned three, my shapes are all derived from joining different sections of some surfaces of revolution, therefore can’t be considered genuine continuous structures.

As expected, intersecting two pieces of type 1 will result in type 2 (compare the first and middle images from the upper row with the rightmost image from the lower row). The convex parts of both originated from the outer side, while the concave ones from the inner side of the funnels of which the level 3 solid is constituted.

One of the main cross-sections of the first shape is a quadrifolium as it can be seen in the below image. On the right side is a symmetric combination of two transverse-edge/ concentric type of level 3 solids, which surprisingly still has the center point on its surface (reachable from twelve directions).

Another design was inspired by the former 1/3, changing the two “onion dome” curvatures to reach complete tangency (180 degree vector meeting) in the center point, resulting in an oppositely-doubly folded torus appearance. Their close relatedness can be appreciated the best by looking at the principal cross-sections.

While the generatrices are halves of Bernoulli lemniscates, it’s still only a “pseudo-homogeneous” surface, having the same quartette partitioning (180 degree segments in this case) as the previous two.

During the time when I started to experiment with the sinuous surfaces of revolution, the specific maneuver which finally was utilized in the creation of the level 4 continuity-expansion, I created a few other models with partial coherence and varying degrees of complementarity.

The first solid in the below image is constituted of two, the second one of four and the third one of eight intersecting “wave surfaces”.

While the first one show up constant fluidity, it’s not tetrahedral, the second one is tetrahedral, but has no continuity (or tangency) in the concave triple meetings and the third one with octahedral symmetry, built on a frame formed by six lemniscates doesn’t really have any complementarity effect as the concave parts became masked by the convex ones (partly valid for the first one too).

Still decent compositions.

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The beginnings

The first concrete step regarding the creative work in geometry was when I intuitively discovered the special properties of the sixth part of a circle, also known as sextant. It was more an aesthetic drive, than a thirst for knowledge. I observed that this simple shape is kind of a “base ingredient” from which one can build different structures in the plane with markedly harmonious appearance.

It is not an exaggeration to call obsession the strong positive feeling what made me motivated to think about and experiment with this “abstract cell”.

The primary compositions I identified were represented by three arcs surrounding an equilateral triangle, what can happen in four ways: three concave, three convex, two convex and one concave, one convex and two concave. Starting from the relation with the inscribed triangle I named these four as “-3, +3, +1, and -1“, referring to the disposition of the circular segments which make them different.

The first case is when three circles are tangent, the “-3” will be the interstice between them. The second and third case appears when three circles are touching the others center, where the “+3” is the central triple intersection and the “+1” are the three double intersections. The fourth case is when two circles are tangent and a central third touches the center of both, where the “-1” represents the two non-intersecting parts of the middle circle. Joining the curved triangles in a symmetrical manner you can reach beautiful patterns, which are the most expressive when painted in a chessboard fashion.

Note: only afterwards I heard about the well known Flower of Life and realized that it’s made from the same units as my compositions.

The YinYang-like design above is composed of six white and six black +3’s and -3’s, or six +1’s and -1’s, depending on which curve you consider the limit between the two possibilities along each triangle side. As the summation of the convex and concave arcs are the same whichever you choose, the surface area of both the white and the black sectors are the same as it would be using equilateral triangles or rhombs for the partitioning.

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Although it’s not directly related, I want to mention here a specific quest of reduced dimensions, but of big personal significance. Namely the idea to found the “most imperfect triangle“. The conception was the result of an inversion rapport with the equilateral triangle, which can be considered the “most perfect” in the category. As there all sides are equal, I concluded that in the reverse case those must be the “least equal”. As a triangle has three sides, there are also three relations: a/b, b/c and a/c. To reach the most unequal complex, even the most balanced of the three ratios must be the farthest possible from equal.

The puzzle was solved by the double rapport a/b=b/c and a=b+c, which leads to a quadratic equation with the solution 1.618033…, known as (surprise!) the golden ratio. The resulting “least perfect triangle” is actually flattened to a straight line section, where two vertices are at the endpoints and the third 1.618033… times closer to one end than to the other.

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Later on I moved to the third dimension and tried to figure out what can be the correspondent of the sextant. It was a little disappointing to realize that there is no such a thing, as you can’t combine the concave and convex parts of a sphere’s surface in the same complementary way as in the case of the 2D tilings. Still I was searching for some insight.

The first solid conceived was the one I named “the space between the spheres“, which is the interstice formed by four tangent spheres. As it’s not a completely closed space I concluded that it must be delimited “artificially” along the openings and the correct way to do this is to cut it at the narrowest parts.

At that time I thought that a closely packed bunch of spheres will have solely these tetrahedral spaces separating the neighbors, only later understood that there are two kind of voids (the other one is the octahedral). Actually it’s the same mistake as the presumption that one can fill the space without gaps using solely regular tetrahedrons (including Aristotle). That’s not possible, while it is with a tetrahedral-octahedral combo.

Following the 2D relations, the second (complementary) idea was to intersect four spheres in a manner that each will touch the center of the other three. This way the central, quadruple intersection will be a spherical tetrahedron. Unlike the tetrahedral interstice, which can’t be considered the genuine 3D representation of the triangular interstice, this tetrahedron delimited only by curved convex faces it’s the exact 3D correspondent of the Reuleaux triangle.

Some years later I heard for the first time about the 3D printing technology and with the assistance of virtual designers, respectively dental technicians soon I could hold in my hands the concrete objects. Consequently I involved myself even more into the creation of interesting solids.

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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.