The Tridiminished Icosahedron
The tridiminished icosahedron is one of the Johnson solids. It is bounded by 5 regular triangles and 3 regular pentagons, for a total of 8 faces, 15 edges, and 9 vertices. 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.
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.
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 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.
Projection centered on top and bottom triangles. This is the most symmetric projection of the tridiminished icosahedron.
Projection centered on vertex surrounded by two pentagons and a lateral triangle.
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.
Projection centered on a lateral triangle.
Projection centered on a pentagonal face.
The Cartesian coordinates of the tridiminished icosahedron are:
- (0, 1, φ)
- (0, ±1, −φ)
- ( 1, φ, 0)
- (±1, −φ, 0)
- ( φ, 0, 1)
- (−φ, 0, ±1)
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.
The tridiminished icosahedron appears as cells in a good number of CRF polychora, generalizations of the Johnson solids to 4D, especially those related to the 600-cell family of uniform polychora. Its occurrences include (but are not limited to) the following polytopes: