The Pentagonal Orthocupolarotunda

The pentagonal orthocupolarotunda is the 32nd Johnson solid (J32). It has 25 vertices, 50 edges, and 27 faces (15 equilateral triangles, 5 squares, and 7 pentagons).

The pentagonal

The pentagonal orthocupolarotunda can be constructed by joining a pentagonal rotunda to a pentagonal cupola at their decagonal face, such that the triangular faces of the rotunda share edges with the square faces of the cupola. The ortho- in the name refers to how the orientation of the top and bottom pentagons are aligned with each other. Joining the cupola and rotunda in gyro orientation produces the pentagonal gyro­cupolarotunda (J33) instead.

There is a hidden connection between the pentagonal ortho­cupola­rotunda and the icosahedron: it is the Stott expansion of a particular partial faceting of the icosahedron according to one of its 5-fold axis of symmetry. The animation below shows this relationship.

Animation showing
interconversion between J32 and icosahedron

This relationship causes the pentagonal cupolarotunda to show up as cells in a significant number of 4D CRF polytopes derived from the uniform polychora via a similar process.


Here are some views of the pentagonal orthocupolarotunda from various angles:

Projection Description

Top view.

Front view.

Side view.


The Cartesian coordinates of the pentagonal orthocupolarotunda with edge length 2 are:

where φ = (1+√5)/2 is the Golden Ratio, approximately 1.61803.

Last updated 14 May 2018.

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