Pentagonal cupola

Pentagonal cupola
Type	Johnson J4 – J5 – J6
Faces	5 triangles 5 squares 1 pentagon 1 decagon
Edges	25
Vertices	15
Vertex configuration	${\displaystyle 10\times (3\times 4\times 10)}$ ${\displaystyle 5\times (3\times 4\times 5\times 4)}$
Symmetry group	${\displaystyle C_{\mathrm {v} }}$
Properties	convex, elementary
Net

In geometry, the pentagonal cupola is one of the Johnson solids (J₅). It can be obtained as a slice of the rhombicosidodecahedron. The pentagonal cupola consists of 5 equilateral triangles, 5 squares, 1 pentagon, and 1 decagon.

Properties

The pentagonal cupola's faces are five equilateral triangles, five squares, one regular pentagon, and one regular decagon. ^[1] It has the property of convexity and regular polygonal faces, from which it is classified as the fifth Johnson solid. ^[2] This cupola cannot be sliced by a plane without cutting within a face, so it is an elementary polyhedron. ^[3]

The following formulae for circumradius ${\displaystyle R}$, and height ${\displaystyle h}$, surface area ${\displaystyle A}$, and volume ${\displaystyle V}$ may be applied if all faces are regular with edge length ${\displaystyle a}$: ^[4] $${\displaystyle {\begin{aligned}h&={\sqrt {\frac {5-{\sqrt {5}}}{10}}}a&\approx 0.526a,\\R&={\frac {\sqrt {11+4{\sqrt {5}}}}{2}}a&\approx 2.233a,\\A&={\frac {20+5{\sqrt {3}}+{\sqrt {5\left(145+62{\sqrt {5}}\right)}}}{4}}a^{2}&\approx 16.580a^{2},\\V&={\frac {5+4{\sqrt {5}}}{6}}a^{3}&\approx 2.324a^{3}.\end{aligned}}}$$

3D model of a pentagonal cupola

It has an axis of symmetry passing through the center of both top and base, which is symmetrical by rotating around it at one-, two-, three-, and four-fifth of a full-turn angle. It is also mirror-symmetric relative to any perpendicular plane passing through a bisector of the hexagonal base. Therefore, it has pyramidal symmetry, the cyclic group ${\displaystyle C_{5\mathrm {v} }}$ of order ten. ^[3]

Related polyhedron

The pentagonal cupola can be applied to construct a polyhedron. A construction that involves the attachment of its base to another polyhedron is known as augmentation; attaching it to prisms or antiprisms is known as elongation or gyroelongation. ^{[5] [6]} Some of the Johnson solids with such constructions are:

• elongated pentagonal cupola ${\displaystyle J_{20}}$
• gyroelongated pentagonal cupola ${\displaystyle J_{24}}$
• pentagonal orthobicupola ${\displaystyle J_{30}}$
• pentagonal gyrobicupola ${\displaystyle J_{31}}$
• pentagonal orthocupolarotunda ${\displaystyle J_{32}}$
• pentagonal gyrocupolarotunda ${\displaystyle J_{33}}$
• elongated pentagonal orthobicupola ${\displaystyle J_{38}}$
• elongated pentagonal gyrobicupola ${\displaystyle J_{39}}$
• elongated pentagonal orthocupolarotunda ${\displaystyle J_{40}}$
• gyroelongated pentagonal bicupola ${\displaystyle J_{46}}$
• gyroelongated pentagonal cupolarotunda ${\displaystyle J_{47}}$
• augmented truncated dodecahedron ${\displaystyle J_{68}}$
• parabiaugmented truncated dodecahedron ${\displaystyle J_{69}}$
• metabiaugmented truncated dodecahedron ${\displaystyle J_{70}}$
• triaugmented truncated dodecahedron ${\displaystyle J_{71}}$
• gyrate rhombicosidodecahedron ${\displaystyle J_{72}}$
• parabigyrate rhombicosidodecahedron ${\displaystyle J_{73}}$
• metabigyrate rhombicosidodecahedron ${\displaystyle J_{74}}$
• trigyrate rhombicosidodecahedron ${\displaystyle J_{75}}$

Relatedly, a construction from polyhedra by removing one or more pentagonal cupolas is known as diminishment ^[1] :

• diminished rhombicosidodecahedron ${\displaystyle J_{76}}$
• paragyrate diminished rhombicosidodecahedron ${\displaystyle J_{77}}$
• metagyrate diminished rhombicosidodecahedron ${\displaystyle J_{78}}$
• bigyrate diminished rhombicosidodecahedron ${\displaystyle J_{79}}$
• parabidiminished rhombicosidodecahedron ${\displaystyle J_{80}}$
• metabidiminished rhombicosidodecahedron ${\displaystyle J_{81}}$
• gyrate bidiminished rhombicosidodecahedron ${\displaystyle J_{82}}$
• tridiminished rhombicosidodecahedron ${\displaystyle J_{83}}$

References

1. 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. Uehara, Ryuhei (2020). Introduction to Computational Origami: The World of New Computational Geometry. Springer. p. 62. doi:10.1007/978-981-15-4470-5. ISBN   978-981-15-4470-5. S2CID   220150682.
3. Johnson, Norman W. (1966). "Convex polyhedra with regular faces". Canadian Journal of Mathematics . 18: 169–200. doi: 10.4153/cjm-1966-021-8 . MR   0185507. S2CID   122006114. Zbl   0132.14603.
4. Braileanu1, Patricia I.; Cananaul, Sorin; Pasci, Nicoleta E. (2022). "Geometric pattern infill influence on pentagonal cupola mechanical behavior subject to static external loads". Journal of Research and Innovation for Sustainable Society. 4 (2). Thoth Publishing House: 5–15. doi: 10.33727/JRISS.2022.2.1:5-15 (inactive 1 July 2025). ISSN   2668-0416.
5. Demey, Lorenz; Smessaert, Hans (2017). "Logical and Geometrical Distance in Polyhedral Aristotelian Diagrams in Knowledge Representation". Symmetry. 9 (10): 204. Bibcode:2017Symm....9..204D. doi: 10.3390/sym9100204 .
6. Slobodan, Mišić; Obradović, Marija; Ðukanović, Gordana (2015). "Composite Concave Cupolae as Geometric and Architectural Forms" (PDF). Journal for Geometry and Graphics. 19 (1): 79–91.

External links

• Weisstein, Eric W., "Pentagonal cupola" ("Johnson solid") at MathWorld .