The Triakis Octahedron

The triakis octahedron is a 3D Catalan solid bounded by 24 isosceles triangles, 36 edges (24 short, 12 long), and 14 vertices. It is the dual of the truncated cube.

The triakis

The faces are transitive; isosceles triangles with two short edges and a long edge with a (2−√2) : 1 length ratio, or approximately 0.586 : 1.

The 14 vertices are of three kinds: 6 axial vertices corresponding to an inscribed octahedron, and 8 vertices corresponding to an inscribed cube.


The following are images of the triakis octahedron from various viewpoints:

Projection Description

Front view, centered on axial vertex.

Side view, centered on a long edge.

Octant view, centered on a cube vertex.


Here's an animation of a triakis octahedron rotating around the vertical axis:

Triakis octahedron


The Cartesian coordinates for the triakis octahedron are all permutations of coordinate and changes of sign of:

These coordinates correspond with a dual truncated cube of edge length (6−4√2).

Last updated 31 May 2023.

Powered by Apache Runs on Debian GNU/Linux Viewable on any browser Valid CSS Valid HTML 5! Proud to be Microsoft-free