The Tridiminished Rhombicosi­dodeca­hedron

The tridiminished rhombicosidodecahedron (J83), also known by its Bowers Acronym tetrid, is the 83rd Johnson solid. It has 45 vertices, 75 edges, and 32 faces (5 triangles, 15 squares, 9 pentagons, 3 decagons). It can be constructed by removing 3 pentagonal cupolae from the rhombicosidodecahedron such that 3 deca­gonal faces are formed.

A tridiminished

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

It is a diminishing of the rhombicosi­dodeca­hedron in an analogous way to the tridiminished icosahedron (J63) being a diminishing of the icosahedron.


In order to be able to identify the tridiminished rhombicosi­dodecahedron 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 Description

Top view, showing trigonal symmetry.

Front view, perpendicular to top and bottom triangles.

Side view, with many coincident edges.

Left long edge is image of a decagon; decagonal face is image of other 2 decagons.


Cartesian coordinates for the tri­diminished rhombicosi­dodecahedron can be obtained in at least two different ways: by deleting vertices from the rhombicosidodecahedron, or by constructing a series of appropriately-scaled triangles and hexagons on parallel hyperplanes along its 3-fold axis of symmetry.

The following coordinates are obtained the second way, and yield a J83 in a nice orientation, with its axis of symmetry parallel to the Z axis, and having edge length 2:

  • (±1, −1/√3, −(4φ+1)/√3)
  • (0, 2/√3, −(4φ+1)/√3)
  • (±φ2, 1/(φ√3), −(2φ+3)/√3)
  • (±φ, (φ+2)/√3, −(2φ+3)/√3)
  • (±1, −φ3/√3, −(2φ+3)/√3)
  • (±1, (2φ+3)/√3, −(2φ+1)/√3)
  • (±(φ+2), −φ/√3, −(2φ+1)/√3)
  • (±φ2, −(φ+3)/√3, −(2φ+1)/√3)
  • (±(2φ+1), −(2φ−1)/√3, −1/√3)
  • (±1, (4φ+1)/√3, −1/√3)
  • (±2φ, −2φ2/√3, −1/√3)
  • (±φ3, −1/√3, (2φ−1)/√3)
  • (±φ, (3φ+2)/√3, (2φ−1)/√3)
  • (±φ2, −(3φ+1)/√3, (2φ−1)/√3)
  • (±1, −(2φ+3)/√3, φ3/√3)
  • (±(φ+2), φ/√3, φ3/√3)
  • (±φ2, (φ+3)/√3, φ3/√3)
  • (±2φ, −2φ/√3, φ3/√3)
  • (0, 4φ/√3, φ3/√3)
  • (±φ2, −1/(φ√3), (2φ+3)/√3)
  • (±φ, −(φ+2)/√3, (2φ+3)/√3)
  • (±1, φ3/√3, (2φ+3)/√3)
  • (±1, 1/√3, (4φ+1)/√3)
  • (0, −2/√3, (4φ+1)/√3)

where φ=(1+√5)/2 is the Golden Ratio.

Last updated 08 Mar 2018.

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