The Cube

The cube is one of the Platonic solids. It is bounded by 6 square faces, and has 8 vertices and 12 edges. It may be thought of as the 3D analogue of a square.

A cube

The dual of the cube is the octahedron. The cube occurs as cells in the tesseract, as well as in various other 4D uniform polytopes. The full list is given below.


In order to be able to identify the cube 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. Two of the cube's faces project to the top and bottom edges of the projection envelope.


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


The Cartesian coordinates of a cube centered on the origin with edge length 2 are:


The cube occurs in the following 4D polytopes:

Last updated 10 Jan 2019.

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