Triangular hebesphenorotunda

Triangular hebesphenorotunda
Triangular hebesphenorotunda
Type	Johnson ; J91 – J92 – J1
Faces	13 triangles ; 3 squares ; 3 pentagons ; 1 hexagon
Edges	36
Vertices	18
Vertex configuration	3(33.5) ; 6(3.4.3.5) ; 3(3.5.3.5) ; 2.3(32.4.6)
Symmetry group	C3v
Dual polyhedron	-
Properties	convex, elementary
Net

Last updated September 20, 2024

In geometry, the triangular hebesphenorotunda is a Johnson solid with 13 equilateral triangles, 3 squares, 3 regular pentagons, and 1 regular hexagon, making the total of its faces is 20.

Properties

The triangular hebesphenorotunda is named by Johnson (1966), with the prefix hebespheno- referring to a blunt wedge-like complex formed by three adjacent lunes—a figure where two equilateral triangles are attached at the opposite sides of a square. The suffix (triangular) -rotunda refers to the complex of three equilateral triangles and three regular pentagons surrounding another equilateral triangle, which bears a structural resemblance to the pentagonal rotunda.^[1] Therefore, the triangular hebesphenorotunda has 20 faces: 13 equilateral triangles, 3 squares, 3 regular pentagons, and 1 regular hexagon.^[2] The faces are all regular polygons, categorizing the triangular hebesphenorotunda as the Johnson solid, enumerated the last one $J_{92}$ .^[3] It is elementary polyhedra, meaning that it cannot be separated by a plane into two small regular-faced polyhedra.^[4]

The surface area of a triangular hebesphenorotunda of edge length $a$ as:^[2] $A=\left(3+{\frac {1}{4}}{\sqrt {1308+90{\sqrt {5}}+114{\sqrt {75+30{\sqrt {5}}}}}}\right)a^{2}\approx 16.389a^{2},$ and its volume as:^[2] $V={\frac {1}{6}}\left(15+7{\sqrt {5}}\right)a^{3}\approx 5.10875a^{3}.$

Cartesian coordinates

The triangular hebesphenorotunda with edge length ${\sqrt {5}}-1$ can be constructed by the union of the orbits of the Cartesian coordinates: ${\begin{aligned}\left(0,-{\frac {2}{\tau {\sqrt {3}}}},{\frac {2\tau }{\sqrt {3}}}\right),\qquad &\left(\tau ,{\frac {1}{{\sqrt {3}}\tau ^{2}}},{\frac {2}{\sqrt {3}}}\right)\\\left(\tau ,-{\frac {\tau }{\sqrt {3}}},{\frac {2}{{\sqrt {3}}\tau }}\right),\qquad &\left({\frac {2}{\tau }},0,0\right),\end{aligned}}$ under the action of the group generated by rotation by 120° around the z-axis and the reflection about the yz-plane. Here, $\tau$ denotes the golden ratio.^[5]

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In geometry, the gyroelongated triangular cupola is one of the Johnson solids (J₂₂). It can be constructed by attaching a hexagonal antiprism to the base of a triangular cupola (J₃). This is called "gyroelongation", which means that an antiprism is joined to the base of a solid, or between the bases of more than one solid.

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References

↑ Johnson, N. 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 .
1 2 3 Berman, M. (1971), "Regular-faced convex polyhedra", Journal of the Franklin Institute, 291 (5): 329–352, doi:10.1016/0016-0032(71)90071-8, MR 0290245 .
↑ Francis, D. (August 2013), "Johnson solids & their acronyms", Word Ways, 46 (3): 177.
↑ Cromwell, P. R. (1997), Polyhedra, Cambridge University Press, p. 86–87, 89, ISBN 978-0-521-66405-9 .
↑ Timofeenko, A. V. (2009), "The non-Platonic and non-Archimedean noncomposite polyhedra", Journal of Mathematical Science, 162 (5): 717, doi:10.1007/s10958-009-9655-0, S2CID 120114341 .

External links

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

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[johnson-1] Johnson, N. 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 .

[berman-2] 1 2 3 Berman, M. (1971), "Regular-faced convex polyhedra", Journal of the Franklin Institute, 291 (5): 329–352, doi:10.1016/0016-0032(71)90071-8, MR 0290245 .

[francis-3] Francis, D. (August 2013), "Johnson solids & their acronyms", Word Ways, 46 (3): 177.

[cromwell-4] Cromwell, P. R. (1997), Polyhedra, Cambridge University Press, p. 86–87, 89, ISBN 978-0-521-66405-9 .

[timofeenko-5] Timofeenko, A. V. (2009), "The non-Platonic and non-Archimedean noncomposite polyhedra", Journal of Mathematical Science, 162 (5): 717, doi:10.1007/s10958-009-9655-0, S2CID 120114341 .

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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)

Triangular hebesphenorotunda

Type	Johnson $J 91 - J 92 - J 1$
Faces	13 triangles 3 squares 3 pentagons 1 hexagon
Edges	36
Vertices	18
Vertex configuration	$3(3 3 .5) 6(3.4.3.5) 3(3.5.3.5) 2.3(3 2 .4.6)$
Symmetry group	$C 3v$
Dual polyhedron	-
Properties	convex, elementary
Net