The Octahedron

The regular octahedron is one of the Platonic solids. It is bounded by 8 equilateral triangles joined 4 to a vertex. It has 8 faces, 6 vertices, and 12 edges.


The dual of the octahedron is the cube. The octahedron occurs as cells in many 4D polytopes, most notably the regular 24-cell. The full list of occurrences is given below.

The octahedron can be bisected to form two square pyramids.


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

Projection Envelope Description

Vertex-first parallel projection.


Edge-first parallel projection. Four of the faces project to the edges of the projection envelope.


Face-first parallel projection.


The canonical coordinates of the regular octahedron are all permutations of:

These coordinates give an octahedron of edge length √2. An octahedron with edge length 2 has the coordinates:

with all permutations of coordinates thereof.


The octahedron occurs as cells in many 4D polytopes, both regular and uniform:

Some CRF polychora also contain octahedral cells, including (but not limited to):

Last updated 03 Jun 2019.

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