The Truncated Cube

The truncated cube is a 3D uniform polyhedron bounded by 8 triangles and 6 octagons. It may be constructed by truncating the cube's vertices at (2-√2) of its edge length. More simply, it may be constructed by radially expanding the cube's edges outwards by 1/√2 of its edge length and taking the convex hull. It may also be constructed by radially expanding the triangular faces of the cuboctahedron outwards.

The truncated


In order to be able to identify the truncated cube in various projections of 4D objects, it is useful to know how it appears from various viewpoints. The following are some of the commonly-encountered views:

Projection Envelope Description

Parallel projection centered on an octagonal face.

Regular dodecagon

Parallel projection centered on a triangular face.


Parallel projection centered on an edge shared between two octagons.


Vertex-centered parallel projection.


The Cartesian coordinates of the truncated cube, centered on the origin and having edge length 2, are all permutations of coordinates and changes of sign of:


The truncated cube occurs as cells in the following uniform polychora:

It also occurs in the following CRF polychora (amongst others):

Last updated 05 Jul 2018.

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