The Metabidiminished Rhombicosidodecahedron
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 nonparallel decagonal faces are formed.
Due to the nonequivalence of its vertices, the metabidiminished rhombicosidodecahedron only has a single axis of symmetry: a 2fold 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 wedgelike 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 appropriatelyscaled triangles and hexagons on parallel hyperplanes along its 2fold 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, ±φ^{3})
 (±φ, ±φ^{2}, ±2φ)
 (0, ±(2+φ), ±φ^{2})
 (±φ^{2}, ±2φ, ±φ)
 (±1, ±(2φ+1), ±1)
 (±(φ+2), ±φ^{2}, 0)
 (±(2φ+1), ±1, −1)
 (±2φ, ±φ, −φ^{2})
 (±φ^{2}, 0, −(φ+2))
where φ=(1+√5)/2 is the Golden Ratio.