The Bilunabirotunda Pseudo­pyramid

The bilunabirotunda pseudopyramid, or J91 pseudopyramid for short, is a CRF polychoron belonging to the family of pseudopyramids, polytopes that are derived from true pyramids by a modified form of Stott expansion, consisting of a base facet and lateral facets that taper to a subdimensional apex that is not necessarily a point. It consists of a bilunabirotunda (J91) base, and 4 tetrahedra, 4 square pyramids, 4 pentagonal pyramids, and 2 triangular prisms that taper to a single edge above the J91. It has a total of 15 cells, 42 polygons (32 triangles, 6 squares, 4 pentagons), 43 edges, and 16 vertices.

The J91

It was discovered on 23 Feb 2014 by W. Gevaert of Netherlands (aka student91).


One possible construction is via the so-called EKF process (Expanded Kaleido-Faceting), applied to the icosahedral pyramid. This is the same process that derives the bilunabirotunda from the icosahedron:

Partial Stott expansion of
icosahedron into bilunabirotunda

The icosahedron is faceted according to one of its subsymmetries, yielding a non-convex polyhedron, and then Stott expansion is applied to make it convex again. Applied to the icosahedral pyramid, this process turns the icosahedral base into a bilunabirotunda, while the apex of the pyramid is lengthened from a point to a unit edge. Four of the 20 tetrahedra are preserved, while the rest are deformed into or replaced with the other cells found in the J91 pseudopyramid.


We shall explore the structure of the J91 pseudopyramid via its parallel projections into 3D.

projection of the J91 pseudopyramid, highlighting apex

The red vertical edge in the above image is the apex of the pseudopyramid, and is the part closest to the 4D viewpoint.

Surrounding this edge are two triangular prisms:

projection of the J91 pseudopyramid, showing two triangular prisms

and 4 square pyramids:

projection of the J91 pseudopyramid, showing 4 square pyramids

Touching either end of this apical edge are 4 tetrahedra:

projection of the J91 pseudopyramid, showing 4 tetrahedra

as well as 4 pentagonal pyramids, two of which are shown next:

projection of the J91 pseudopyramid, showing 2/4 pentagonal pyramids

and the other two:

projection of the J91 pseudopyramid, showing 4/4 pentagonal pyramids

Finally, of course, here is the bilunabirotunda itself, as the base of the pyramid:

projection of the J91 pseudopyramid, showing the bilunabirotunda


The Cartesian coordinates of the J91 pseudopyramid, having edge length 2, are:

where φ=(1+√5)/2 is the Golden Ratio.

Like the icosahedral pyramid from which it derives, the J91 pseudopyramid is quite shallow, having a height of only 1/φ (approx. 0.61803) for an edge length of 2.

Last updated 02 Feb 2023.

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