The Deltoidal Icositetrahedron

The deltoidal icositetrahedron is a 3D Catalan solid bounded by 24 kites, 48 edges (12 short, 12 long), and 26 vertices. It is the dual of the rhombicuboctahedron.

The deltoidal

The kite faces are transitive, with two short edges and two long edges with a (4+√2) : 7 length ratio.

The 26 vertices are of three kinds: 6 apical vertices corresponding to an inscribed octahedron, 8 vertices corresponding to an inscribed cube, and 12 vertices corresponding to an inscribed cuboctahedron.

The deltoidal icositetrahedron is the projection envelope of its direct 4D analogue, the triangular antitegmatic hexacontatetrachoron.


The following are images of the deltoidal icositetrahedron from various viewpoints:

Projection Description

Front view, centered on an axial vertex.

Side view, centered on a cuboctahedron vertex.

Octant view, centered on a cube vertex.


Here's an animation of a deltoidal icositetrahedron rotating around the vertical axis:

icositetrahedron rotating


The Cartesian coordinates for the deltoidal icositetrahedron are all permutations of coordinate and changes of sign of:

These coordinates can be obtained by inverting a rhombicuboctahedron of edge length (3−√2)/7.

Last updated 27 Jul 2023.

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