Triangular cupola

Triangular cupola
Triangular cupola
Type	Johnson; J2 – J3 – J4
Faces	4 triangles; 3 squares; 1 hexagon
Edges	15
Vertices	9
Vertex configuration
Symmetry group
Properties	convex
Net

In geometry, the triangular cupola is one of the Johnson solids ( $J 3$ ). It can be seen as half a cuboctahedron.

Properties

The triangular cupola has four equilateral triangles, three squares, and one regular hexagon. as their faces; the regular hexagon is the base and one of the four triangles is the top. The edge length of that hexagon is equal to the edge length of both squares and triangles. All of its faces are regular.^[1]^[2] The dihedral angle between each triangle and the hexagon is approximately $70.5^{\circ }$ , that between each square and the hexagon is $54.7^{\circ }$ , and that between square and triangle is $125.3^{\circ }$ .^[3] A convex polyhedron in which all of the faces are regular is a Johnson solid, and the triangular cupola is among them, enumerated as the third Johnson solid $J_{3}$ .^[2]

Given that $a$ is the edge length of a triangular cupola. Its surface area $A$ can be calculated by adding the area of four equilateral triangles, three squares, and one hexagon:^[1] $A=\left(3+{\frac {5{\sqrt {3}}}{2}}\right)a^{2}\approx 7.33a^{2}.$ Its height $h$ and volume $V$ is:^[4]^[1] ${\begin{aligned}h&={\frac {\sqrt {6}}{3}}a\approx 0.82a,\\V&=\left({\frac {5}{3{\sqrt {2}}}}\right)a^{3}\approx 1.18a^{3}.\end{aligned}}$

It has an axis of symmetry passing through the center of its both top and base, which is symmetrical by rotating around it at one- and two-thirds 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 $C_{3v}$ of order 6.^[3]

Related polyhedra and honeycombs

The dual of the triangular cupola is the polyhedron with six triangular and three kite faces.

The triangular cupola can be augmented by 3 square pyramids, leaving adjacent coplanar faces. This isn't a Johnson solid because of its coplanar faces. Merging those coplanar triangles into larger ones, topologically this is another triangular cupola with isosceles trapezoidal side faces. If all the triangles are retained and the base hexagon is replaced by 6 triangles, it generates a coplanar deltahedron with 22 faces.

The triangular cupola can form a tessellation of space with square pyramids and/or octahedra,^[5] the same way octahedra and cuboctahedra can fill space.

The family of cupolae with regular polygons exists up to n=5 (pentagons), and higher if isosceles triangles are used in the cupolae.

Family of convex cupolae
v
t
e
n	2	3	4	5	6	7	8
Schläfli symbol	${2} \|\| t{2}$	${3} \|\| t{3}$	${4} \|\| t{4}$	${5} \|\| t{5}$	${6} \|\| t{6}$	${7} \|\| t{7}$	${8} \|\| t{8}$
Cupola	Digonal cupola	Triangular cupola	Square cupola	Pentagonal cupola	Hexagonal cupola (Flat)	Heptagonal cupola (Non-regular face)	Octagonal cupola (Non-regular face)
Related uniform polyhedra	Rhombohedron	Cuboctahedron	Rhombicuboctahedron	Rhombicosidodecahedron	Rhombitrihexagonal tiling	Rhombitriheptagonal tiling	Rhombitrioctagonal tiling

References

^ ^a ^b ^c 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.
^ ^a ^b 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.
^ ^a ^b 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.
^ Sapiña, R. "Area and volume of the Johnson solid $J_{3}$ ". Problemas y Ecuaciones (in Spanish). ISSN 2659-9899. Retrieved 2020-09-08.
^ "J3 honeycomb".

External links

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

This polyhedron-related article is a stub. You can help Wikipedia by expanding it.

[berman-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.

[uehara-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.

[johnson-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.

[pye-4] Sapiña, R. "Area and volume of the Johnson solid $J_{3}$ ". Problemas y Ecuaciones (in Spanish). ISSN 2659-9899. Retrieved 2020-09-08.

[5] "J3 honeycomb".

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v t e Johnson solids
Pyramids, cupolae and rotundae	square pyramid pentagonal pyramid triangular cupola square cupola pentagonal cupola pentagonal rotunda
Modified pyramids	elongated triangular pyramid elongated square pyramid elongated pentagonal pyramid gyroelongated square pyramid gyroelongated pentagonal pyramid triangular bipyramid pentagonal bipyramid elongated triangular bipyramid elongated square bipyramid elongated pentagonal bipyramid gyroelongated square bipyramid
Modified cupolae and rotundae	elongated triangular cupola elongated square cupola elongated pentagonal cupola elongated pentagonal rotunda gyroelongated triangular cupola gyroelongated square cupola gyroelongated pentagonal cupola gyroelongated pentagonal rotunda gyrobifastigium triangular orthobicupola square orthobicupola square gyrobicupola pentagonal orthobicupola pentagonal gyrobicupola pentagonal orthocupolarotunda pentagonal gyrocupolarotunda pentagonal orthobirotunda elongated triangular orthobicupola elongated triangular gyrobicupola elongated square gyrobicupola elongated pentagonal orthobicupola elongated pentagonal gyrobicupola elongated pentagonal orthocupolarotunda elongated pentagonal gyrocupolarotunda elongated pentagonal orthobirotunda elongated pentagonal gyrobirotunda gyroelongated triangular bicupola gyroelongated square bicupola gyroelongated pentagonal bicupola gyroelongated pentagonal cupolarotunda gyroelongated pentagonal birotunda
Augmented prisms	augmented triangular prism biaugmented triangular prism triaugmented triangular prism augmented pentagonal prism biaugmented pentagonal prism augmented hexagonal prism parabiaugmented hexagonal prism metabiaugmented hexagonal prism triaugmented hexagonal prism
Modified Platonic solids	augmented dodecahedron parabiaugmented dodecahedron metabiaugmented dodecahedron triaugmented dodecahedron metabidiminished icosahedron tridiminished icosahedron augmented tridiminished icosahedron
Modified Archimedean solids	augmented truncated tetrahedron augmented truncated cube biaugmented truncated cube augmented truncated dodecahedron parabiaugmented truncated dodecahedron metabiaugmented truncated dodecahedron triaugmented truncated dodecahedron gyrate rhombicosidodecahedron parabigyrate rhombicosidodecahedron metabigyrate rhombicosidodecahedron trigyrate rhombicosidodecahedron diminished rhombicosidodecahedron paragyrate diminished rhombicosidodecahedron metagyrate diminished rhombicosidodecahedron bigyrate diminished rhombicosidodecahedron parabidiminished rhombicosidodecahedron metabidiminished rhombicosidodecahedron gyrate bidiminished rhombicosidodecahedron tridiminished rhombicosidodecahedron
Other elementary solids	snub disphenoid snub square antiprism sphenocorona augmented sphenocorona sphenomegacorona hebesphenomegacorona disphenocingulum bilunabirotunda triangular hebesphenorotunda
(See also List of Johnson solids, a sortable table)