Pentagonal orthobirotunda

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Pentagonal orthobirotunda
Pentagonal orthobirotunda.png
Type Birotunda,
Johnson
J33J34J35
Faces 2×10 triangles
2+10 pentagons
Edges 60
Vertices 30
Vertex configuration 10(32.52)
2.10(3.5.3.5)
Symmetry group D5h
Properties convex
Net
Johnson solid 34 net.png

In geometry, the pentagonal orthobirotunda is a polyhedron constructed by attaching two pentagonal rotundae along their decagonal faces, matching like faces. It is a Johnson solid.

Contents

Construction

The pentagonal orthobirotunda is constructed by attaching two pentagonal rotundas to their base, covering decagon faces. The resulting polyhedron has 32 faces, 30 vertices, and 60 edges. This construction is similar to icosidodecahedron (or pentagonal gyrobirotunda), an Archimedean solid: the difference is one of its rotundas twisted around 36°, making the pentagonal faces connect to the triangular one, a process known as gyration. [1] [2] A convex polyhedron in which all of the faces are regular polygons is the Johnson solid. The pentagonal orthobirotunda is one of them, enumerated as the 34th Johnson solid . [3]

Icosidodecahedron.png
Dissected icosidodecahedron.png
Pentagonal orthobirotunda solid.png
The difference between icosidodecahedron and pentagonal orthobirotunda, and its dissection.

Properties

The surface area of an icosidodecahedron can be determined by calculating the area of all pentagonal faces. The volume of an icosidodecahedron can be determined by slicing it into two pentagonal rotundae, after which summing up their volumes. Therefore, its surface area and volume can be formulated as: [1]

References

  1. 1 2 Berman, Martin (1971). "Regular-faced convex polyhedra". Journal of the Franklin Institute. 291 (5): 329–352. doi:10.1016/0016-0032(71)90071-8. MR   0290245.
  2. Ogievetsky, O.; Shlosman, S. (2021). "Platonic compounds and cylinders". In Novikov, S.; Krichever, I.; Ogievetsky, O.; Shlosman, S. (eds.). Integrability, Quantization, and Geometry: II. Quantum Theories and Algebraic Geometry. American Mathematical Society. p. 477. ISBN   978-1-4704-5592-7.
  3. Francis, Darryl (2013). "Johnson solids & their acronyms". Word Ways. 46 (3): 177.