The Icosahedron

The regular icosahedron is one of the five Platonic solids. It is bounded by 20 equilateral triangles, and has 12 vertices and 30 edges. Its dual is the dodecahedron.


The icosahedron occurs as cells in the snub 24-cell, the truncated 600-cell, and the rectified 600-cell.


In order to be able to identify the icosahedron 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
Regular decagon

Vertex-first parallel projection.


Edge-first parallel projection. Four of the icosahedron's faces project onto the top two and bottom two edges of the projection envelope.

Regular hexagon

Face-first parallel projection.

Non-uniform octagon

Parallel projection, parallel to two opposite faces. The two opposite faces project to the top and bottom edges of the projection envelope.


The Cartesian coordinates of an origin-centered regular icosahedron of edge length 2 are:

where φ=(1+√5)/2 is the Golden Ratio.

Last updated 08 Aug 2012.

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