What's the name or class of the following tetrahedron-like shape?

Tetrahedron Geometry and related geometries for construction Sketchup Model

WebGL 3D-viewer

It's apparently some sort of (not strictly convex) shell of a tetrahedron and it's scaled spherical projection dual compound, which both share the same outer cube.

It has some interesting properties, since it's still unfoldable to a flat net, and when used as a projection surface, as in the Authagraph world map has a less abrupt change in curvature, and therefore less angular distortion than a regular tetrahedron.

It's construction can be described of placing 6 intersecting oblique solid bicones with their straight long edge (apex-circle-apex is a straight line normal to the base circle's plane) on the edges of the tetrahedron, and circle centers placed at the centroid.

It's therefore not! constructable from 4 right circular cones placed with apexes on the tetrahedron's vertices.

If the cone base circle is a unit circle, the height of the cone is $\sqrt2$ or double that for the bicone.

The surface is the sum of 6 bicone segments with an angle of 109.5° or $\cos^{−1}(-\frac13)$ encompassing the edge on the orthogonal circle.

Construction of bicone

One of the 6 bi-cones placed on a tetrahedronal edge

Note the adjoined cones don't meet at flat surface, but something like 163.5° (~16°) making the shape effectively concave.

concave angle closeup

A google (image) search for "inflated tetrahedron" hits quite near, yet yields no satisfying mathematical descriptions to me.

Related terms I was able to find so far:

It apparently doesn't have the properties of a Reuleaux or Meissner tetrahedron, since the plain tetrahedron is still present in the rounded out edges, which are still straight lines between the tetrahedrons vertices.

Can this be generalized for other simplexes?

  • $\begingroup$ I suspect I won't be of much help to you in any case, but ... Did I interpret the illustration correctly: Each triangular face, flat in a normal tetrahedron, are "puffed out" to a non-flat, roundish patch, keeping the triangular outer perimeter straight and flat? Do you have any equations describing this? Especially the .. um.. puffiness of the faces, the distortion, would be interesting to see parametrized. $\endgroup$ Oct 30, 2016 at 19:34
  • $\begingroup$ I have parametrization yet, just a geometric construction. I'm trying to make sense of the Authagraph process. In my ideal model there would be four intersecting surfaces(cones?) that are created by connection the Tetrahedral tiling of a unit sphere in dual position to a tetrahedron constructed in a unit cube. en.wikipedia.org/wiki/Tetrahedron#/media/… $\endgroup$
    – mxfh
    Oct 30, 2016 at 19:42
  • $\begingroup$ If they were cones, one could project directly to those four, this sounds way more straightforward than than process described by Authagraph, authagraph.com/projects/description/… and possibly also generalizable to other platonic solids. Prepare for traditional Pinata style world map projections ;) $\endgroup$
    – mxfh
    Oct 30, 2016 at 19:56
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    $\begingroup$ P.S. I believe the contours in the picture are a little misleading -- the shape is actually smooth along the circular arcs, but nonsmooth along the lines (not drawn) joining the tetrahedron vertices to the junctions where the arcs meet. The contours in the figure suggest a slightly different shape, formed by four circular cones emerging from the tetrahedon vertices which meet at the circular arcs. Have you considered this construction? $\endgroup$
    – user856
    Oct 31, 2016 at 0:43
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    $\begingroup$ If you're collecting similar-looking shapes, the central piece of Cayley's cubic surface, (which can be described by $\begin{vmatrix}1&z&y\\z&1&x\\y&x&1\end{vmatrix}=0$) is another one. Like your shape, it also contains six straight lines along the edges of a tetrahedron. $\endgroup$
    – user856
    Oct 31, 2016 at 1:36

1 Answer 1


I think the object is $Type -D- Coxeter-associahedra$. http://www.math.uakron.edu/~sf34/kpd_poly.gif


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