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.

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.

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”?

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.

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.

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.

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.

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.