The Metabidiminished Rhombicosidodecahedron

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The metabidiminished rhombicosidodecahedron (J81), also known by its Bowers Acronym mabidrid, is the 81st Johnson solid. It has 50 vertices, 90 edges, and 42 faces (10 triangles, 20 squares, 10 pentagons, 2 decagons). It can be constructed by removing 2 pentagonal cupolae from the rhombicosidodecahedron such that 2 non-parallel decagonal faces are formed.

Due to the non-equivalence of its vertices, the metabidiminished rhombicosidodecahedron only has a single axis of symmetry: a 2-fold symmetry around the line passing through the top and bottom square faces.

It is a diminishing of the rhombicosidodecahedron in an analogous way to the metabidiminished icosahedron (J62) being a diminishing of the icosahedron.

Projections

In order to be able to identify the metabidiminished 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, centered on square between decagons.
	Front view, showing its wedge-like shape.
	Side view.

Coordinates

Cartesian coordinates for the metabidiminished 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 2-fold axis of symmetry.

The following coordinates are obtained the second way, and yield a J82 in a nice orientation, with its axis of symmetry parallel to the Z axis, and having edge length 2:

• (±1, ±1, ±φ³)
• (±φ, ±φ², ±2φ)
• (0, ±(2+φ), ±φ²)
• (±φ², ±2φ, ±φ)
• (±1, ±(2φ+1), ±1)
• (±(φ+2), ±φ², 0)
• (±(2φ+1), ±1, −1)
• (±2φ, ±φ, −φ²)
• (±φ², 0, −(φ+2))

where φ=(1+√5)/2 is the Golden Ratio.