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 octahedronâ€™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 BĂ©zier 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.

**2. The surface creation**

After the frame of the base pattern is ready, the next step is identical as in case of many other previous shapes: it will be represented by the minimal surfaces bordered by these curves.

In this case they are forming distorted rhombic parcels with degenerated vertices, each of them joining a third of two octahedral faces along one of the edges. One such curved rhombic surface will have two zero degree “hyperbolic vertices” where the joined generatrices will be tangential to the axes (in case of two convex, respectively two concave arcs) and two 48.18 degree angles corresponding to the original octahedral face arrangement made by two medians (in case of one convex and one concave arc).

The symmetry of the shape is tetrahedral and it has the exact same volume like a regular octahedron, as the inner missing parts are of identical shape with the outer surpluses. Concomitantly, you can create the concave parts by flipping over the convex ones according to the octahedron’s faces and vice versa.

While with the appropriate positioning there will be perfect overlap in case of two opposite faces, the solid doesn’t have 3D tessellation properties (can’t form a honeycomb) for the same reason the regular octahedron can’t do this only by itself, but in a combination with the tetrahedron. You can check it here.

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**Relatedness with 2D shapes**

Based on the relation between the equilateral triangle and the regular tetrahedron, respectively between the hexagon and the octahedron, the 2D correspondent of this solid must be a certain kind of rhombic trio with 2 convex and 2 concave sides each, formed along a hexahedral frame. Actually this shape is a composite of 3 identical parts, joined only in the center point.

While in case of the octahedron the self-tessellation is impossible, for its plane version – the hexagon – it is. For this reason the 2D correspondent of this solid also owns this property.

To accordingly visualize the pattern-repetition, you have to use 4 colors instead of 2, otherwise there will be overlaying along half of the edges. This way every rhombic trio of one color will have all the other 3 colors as neighbors.

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