The Tridiminished Icosahedron

The tridiminished icosahedron is one of the Johnson solids. It is bounded by 5 regular triangles and 3 regular pentagons. It may be constructed by removing 3 vertices from the regular icosahedron such that 3 pentagonal faces are formed. It is the 63rd polyhedron in Norman Johnson's list, and thus bears the label J63.

A tridiminished

There are three different kinds of triangular faces: the top triangle, which is surrounded by three pentagonal faces; the bottom triangle, which is surrounded by three triangular faces; and three lateral triangles surrounding the bottom triangle.

Due to the non-equivalence of its vertices, the tridiminished icosahedron only has a single axis of symmetry: a 3-fold symmetry around the line passing through the top and bottom triangles.

The tridiminished icosahedron is the vertex figure of the snub 24-cell, and occurs as cells in the bi-icositetradiminished 600-cell.


In order to be able to identify the tridiminished 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
Irregular pentagon

Projection parallel to an edge of the top triangle. This projection has the most number of coincident faces: two pentagonal faces project to the pentagonal area, the third pentagonal face to the upper left edge; the top triangle to the upper right edge; the bottom triangle to the lower left edge; two lateral triangles to the triangular area; the third lateral triangle to the bottom edge.

Irregular hexagon

Projection centered on top and bottom triangles. This is the most symmetric projection of the tridiminished icosahedron.

Irregular heptagon

Projection centered on vertex surrounded by two pentagons and a lateral triangle.

Irregular pentagon

Projection centered on the edge between a pentagon and a lateral triangle. The bottom left edge of the projection envelope is the image of a pentagonal face.

Irregular hexagon

Projection centered on a lateral triangle.

Irregular octagon

Projection centered on a pentagonal face.


The Cartesian coordinates of the tridiminished icosahedron are:

where φ=(1+√5)/2 is the Golden Ratio.

These coordinates are obtained by deleting 3 vertices from the regular icosahedron such that 3 pentagonal faces are formed.

Last updated 02 Jan 2013.

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