The Pentagonal Rotunda


The pentagonal rotunda is the 6th Johnson solid (J6). Its surface consists of 10 equilateral triangles, 6 regular pentagons, and 1 regular decagon.

A pentagonal rotunda

It can be constructed by bisecting the icosidodecahedron along one of its decagonal circuits of edges. Thus, it may be also called the bisected icosidodecahedron. Conversely, joining a pair of pentagonal rotundae in gyro orientation at their decagonal faces produces an icosidodecahedron:

Decomposition of
icosidodecahedron into two pentagonal rotundae

Thus, the icosidodecahedron may also be named the pentagonal gyrobirotunda.

If the pentagonal rotundae are joined in ortho orientation instead, we obtain the pentagonal orthobirotunda (J34), another Johnson solid.

Decomposition of pentagonal
orthobirotunda into two pentagonal rotundae

Projections

In order to be able to identify the pentagonal rotunda 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 Envelope Description
Regular decagon

Parallel projection centered on top pentagon.

Bisected decagon

Projection parallel to a pair of opposite edges of the decagonal face.

Bisected irregular octagon

Projection parallel to a pair of opposite vertices of the decagonal face. This projection has the largest number of coincident elements.

Coordinates

The Cartesian coordinates of the pentagonal rotunda are:

  • (0, −A, 2B)
  • (±1, B, 2B)
  • (±φ, −√((5-√5)/10), 2B)
  • (0, 2B, A)
  • (±φ, −√((25+11√5)/10), A)
  • (±φ2, √((5+√5)/10), A)
  • (±1, ±√(3+4φ), 0)
  • (±φ2, ±√(2+φ), 0)
  • (±2φ, 0, 0)

where φ = (1+√5)/2 is the Golden Ratio, A = √((10+2√5)/5), and B = √((5+2√5)/5).

These coordinates give a pentagonal rotunda of edge length 2 in a “nice” orientation, with its 5-fold axis of symmetry aligned to the Z axis and its decagonal face resting on the XY plane.

Algebraically “nicer” coordinates may be obtained by taking advantage of its relationship to the icosidodecahedron, by deleting 10 vertices from the latter. The coordinates thus obtained are:

(0, 0, 2φ) (0, 2φ, 0) (±2φ, 0, 0)
(±1, ±φ, φ2) (±1, φ, −φ2) (±φ, φ2, ±1)
(±φ2, ±1, φ) (±φ2, 1, −φ)

These coordinates do not give a pentagonal rotunda in a “nice” orientation, but they are algebraically nicer because they can be simply expressed in terms of the Golden Ratio φ, and do not involve nested square roots.


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