Roundest Polyhedra

This is a collection of solutions and putative solutions to the Isoperimetric Problem for Polyhedra also know as Roundest Polyhedra with their faces tangent to a unit sphere. They have been rotated so that usually their primary axis of symmetry, if there is one, conincides with the Z-axis, and a second axis of symmetry is in the X-Z plane. In the case of C_s symmetry the reflective plane is made to coincide with the X-Z plane.

Each polyhedron is provided in two forms. The first, minvol.n.txt, is an unordered list of the tangent points of the faces in N. J. A. Sloane's future-proof format: each point's X, Y and Z coordinates are on three consecutive lines in the file. The second, minvol.n.off, in the simple OFF polyhedron file format. The first line is the literal "OFF". The second line is three space separated integers for the number of vertices, number of faces, and number of edges of the polyhedron. The next lines are the X, Y and Z coordinates of the polyhedron vertices. Finally are lines for each of the polyhedron faces, lists of space separated integers. The first on each line is the number of sides of the polygon, followed by that number of zero-based indices into the preceding list of vertices, in counter-clockwise order.

Note for 1/18/2021: Added new result for n=185.

Note for 1/22/2021: Corrected symmetry for n=165, was C₁, corrected to C₂.

Note for 1/31/2021: Corrected symmetry identification for n=428 (was C_3h, corrected to S₆) and n=472 (was C_5h, corrected to S₁₀).

Note for 2/2/2021: New result for n=470.

Note for 10/17/2023: new results for n=228, 229, 232, 378, 379, 380, 418, 419, 422, 443, 462, 463, 464, 467, 478, 485, 487, 488, 489. Added results for n=493, 494, 495, 496, 497, 498, 499, 500, 501.

Note for 1/7/2026: converted to table.

Wayne Deeter - wrd@deetour.net

Last modified: January 7, 2026