Hyper-concave

Description

An axially self-intersecting minimal surface cluster made up of three centrally connected opposite pairs outspread on cusp forming parabolic arc frames.

The idea

The base pattern was imagined as an “inverted hyperbolic paraboloid” where instead of fluidly merging into the horizontal plane, every perpendicular cross section passing through the center osculates towards the vertical axis.

To understand this perspective, think of the hypar as a continuous transition between a convex and a concave parabolic arc, gradually loosing its curvature towards the middle where completely flattens into a line. Contrary to this scenario, the two ends of our shape are diametrically opposed perpendicular cusps which both are gradually transforming into the vertical axis.

*The relation between the principal curvatures of the hyperbolic paraboloid (red) and of the Hyper-concave (teal)*

Construction

1. The frame

First I built six centrally interconnected cusps, each coplanar with one of a regular tetrahedron’s edges, with their two endpoints situated on the tetrahedral vertices. As such, the curvatures will start from the line of the tetrahedral edges and will end tangential to the three main axes. This setting is analogous but reversed with the one made by the principal curvatures of three axially intersecting hyperbolic paraboloids where are also three pairs of diametrically opposed perpendicular elements.

*One cusp shaped generatrix pair (teal) coplanar with an edge (AB) of the reference tetrahedron, pointing towards its center (O)*. Triangle AOB is the inner half of the AEBO rhomb, part of the rhombic dodecahedral scheleton

Now we reached the trickier part. While in case of the hypar the gradual transformation of the convex parabolic arc into the concave one happens along the tetrahedral edges by doing a complete revolution with the generatrix (half arc) following a skew rectangle frame made up by 4 connected edges, in case of our shape the definition of the exact revolution path is less obvious as the vertical axis into which the transformation must be directed is coplanar with the generatrix itself (half cusp).

The first thing I concluded is that the path must be a curved segment which first will leave the plane of the cusp from the generatrix then return to this at the axis. Second conclusion was that the projection of the end vector must form a 45 degree angle with the cusp plane, as 90 degree will be covered adding it’s complementary half related to the diametrically opposite perpendicular cusp, then returning to the initial one and repeating the mirroring action another 3 times to complete a full 360 degree revolution. The tetrahedral vertices are given, but exactly where on the axes will be those “return points”?

*Two different views of the complete Bézier curve pattern on the surface of a reference rhombic dodecahedron with one cusp on each face*

The breakthrough came when I realized that the most coherent way to achieve this is when the path in question has the exact same shape as the half cusp generatrix itself, consequently everything will happen inside a regular rhombic dodecahedral lattice where two neighboring such paths will be coplanar, forming identical cusps with the original ones. This way the distance from the center of the path convergence points situated on the 3 main axes represents the small diagonals of the rhombic faces while the reference tetrahedron’s edges are the big diagonals.

2. The surface creation

The elements of the formerly described frame can be sorted into two sets of identical Bézier curves: 24 outer ones which are coplanar in pairs with the rhombic dodecahedron’s faces and 12 inner ones which are likewise coplanar with the 6 internal rhombs of the same solid. These together with the 3 main axes will divide the structure into 24 triangular segments each made up by 2 neighboring (outer and inner) curves and a half axis. The minimal surfaces formed on these will be the faces of the Hyper-concave.

*The external and internal arcs on the rhombic dodecahedron* *(left) respectively one minimal* *surface highlighted in cyan color (right*)

While this was the exact course of action I built it, actually there are not 24, but only 12 minimal surfaces as the axes are representing the intersections between 2 double-sized minimal surfaces bordered by a common inner cusp and two different but diagonally connected outer curve pairs (right side of the above image). The composition has 11 principal points of which 10 are vertices (6 double, 4 triple) and one is the center.

The peculiar characteristic of this shape is that none of the cross sections passing through the center has any convex sectors, thus outside the axial intersections is exclusively concave, hence its name “hyper-concave”. While this can be stated also about other simpler shapes like the interior of a sphere or about any ellipsoid for that matter, those are forming closed systems, lacking the external perspective. A hemispheric bowl’s inner surface is both fully concave and externally observable but is missing a complete spatial span. Regarding the hyperboloid, the concave continuity is valid only along the axis of revolution, while the perpendicular cross section is fully convex, having a circle profile.

The other hallmark of the construction is the visual coherence achieved by the contour overlap of multiple symmetrical segments with complementary nature, when there are perspectives which highlight the tight fitting of triple, respectively quadruple clusters of almond shaped constituents as their projections are the same. As such, the material mass bordered by the convex (concentric) outer curves are exactly filling the similarly shaped gaps between the concave (radial) inner ones.

Two principal perspectives showing the spatial disposition of the three “inverted hypars”

After identifying this unique characteristic I realized that in case of the quadruple overlap both the convex and concave edges are projections of cusp pairs made by 90 degree parabolic segments drawn on three perpendicularly intersecting central squares, where the original arcs are becoming distorted by the tilted setting of the rhombic faces.

While conceptually akin, this fitting property is nonexistent in case of the tetrahedral divergence, as despite the superficial resemblance that composition has a quite different structural arrangement, presenting significant gaps from all perspectives, also there the inner arcs are not identical with the outer ones.

*The four characteristic view of the solid*

Regarding the formerly mentioned quadruple pattern, if we do a perspective comparison with the composition made by three intersecting hyperbolic paraboloids we can observe that beside the common square frame there the visibility of the one seen from above is halved as the concave part will be masked by the convex sectors of the other two hypars with oblique view, while in case of the concave radiance there’s no masking, but perfect boundary overlap between the three. Here the cross made by the almond shaped parts represents the perpendicular view, while the remaining sectors are diagonally distributed between the neighboring oblique ones.

Relatedness with 2D shapes

The planar equivalent of the Hyper-concave is a certain pattern inside the rhombille tiling which needs three colors for the best representation. Because the concave edge triplets of the radial gaps enclose the exact same space between them as the outer convex boundaries will with the identical ones sitting on the neighboring cells (tessellation) two opposite colors (white and black) have to be chosen to highlight this arrangement dichotomy (empty vs full) in the plane.

*The 2D tessellation – a modified version of the rhombille tiling – (left) with the pattern inside one hexagonal cell highlighted (right)*

As such, the triple inner gap colored in white will have its same shaped complementary segments on the fringes colored in black, while grey, as the intermediary shade will fill the remaining parts where the “material thinning” between full and empty happens, which in this setup is our result. Notice that in the plane both the inner and the outer curves appear concave forming an ensemble made up of three sextant (60 degree circle arcs) based tangential pseudo-astroids.

Due to the unique structure, this solid can be interpreted as one with its exterior inside and its interior outside, as all the hyperbolic surfaces are situated inward from the cross sections which separate one rhombic dodecahedral cell from another.

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.

*****************************************************************************

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.

*****************************************************************************

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.

*****************************************************************************

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.

*****************************************************************************

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.

******************************************************************************

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.

*****************************************************************************

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.

******************************************************************************