# The Truncated Octahedron

The truncated octahedron is a 3D uniform polyhedron bounded by 14 polygons (8 hexagons and 6 squares), 36 edges, and 24 vertices. It may be constructed by truncating the octahedron at 1/3 of its edge length.

## Projections

In order to be able to identify the truncated octahedron 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
Octagon

Parallel projection centered on a square face. The top, bottom, left, and right edges of the projection envelope are images of square faces.

Dodecagon

Parallel projection centered on a hexagonal face.

Hexagon

Parallel projection centered on an edge between two hexagons. The top and bottom edges of the projection envelope are images of square faces.

Non-uniform decagon

Parallel projection centered on a vertex.

## Coordinates

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

• (0, √2, 2√2)

## Occurrences

The truncated octahedron appears as cells in the following 4D uniform polytopes:

It also occurs as cells in the following CRF polychora:

Last updated 17 Jun 2019.