Sphenocorona

Sphenocorona
Sphenocorona
Type	Johnson; J85 – J86 – J87
Faces	12 triangles ; 2 squares
Edges	22
Vertices	10
Vertex configuration	4(33.4) ; 2(32.42) ; 2x2(35)
Symmetry group	C2v
Dual polyhedron	-
Properties	convex
Net

In geometry, the sphenocorona is a Johnson solid with 12 equilateral triangles and 2 squares as its faces.

Properties[edit]

The sphenocorona was named by Johnson (1966) in which he used the prefix spheno- referring to a wedge-like complex formed by two adjacent lunes—a square with equilateral triangles attached on its opposite sides. The suffix -corona refers to a crownlike complex of 8 equilateral triangles.^[1] By joining both complexes together, the resulting polyhedron has 12 equilateral triangles and 2 squares, making 14 faces.^[2] A convex polyhedron in which all faces are regular polygons is called a Johnson solid, and the sphenocorona is among them, enumerated as the 86th Johnson solid $J_{86}$ .^[3] It is elementary, meaning it does not arise from "cut-and-paste" manipulations of both Platonic and Archimedean solids.^[4]

The surface area of a sphenocorona with edge length $a$ can be calculated as:^[2]

A=\left(2+3{\sqrt {3}}\right)a^{2}\approx 7.19615a^{2},

and its volume as:^[2]

\left({\frac {1}{2}}{\sqrt {1+3{\sqrt {\frac {3}{2}}}+{\sqrt {13+3{\sqrt {6}}}}}}\right)a^{3}\approx 1.51535a^{3}.

Cartesian coordinates[edit]

Let $k\approx 0.85273$ be the smallest positive root of the quartic polynomial $60x^{4}-48x^{3}-100x^{2}+56x+23$ . Then, Cartesian coordinates of a sphenocorona with edge length 2 are given by the union of the orbits of the points

\left(0,1,2{\sqrt {1-k^{2}}}\right),\,(2k,1,0),\left(0,1+{\frac {\sqrt {3-4k^{2}}}{\sqrt {1-k^{2}}}},{\frac {1-2k^{2}}{\sqrt {1-k^{2}}}}\right),\,\left(1,0,-{\sqrt {2+4k-4k^{2}}}\right)

under the action of the group generated by reflections about the xz-plane and the yz-plane.^[5]

Variations[edit]

The sphenocorona is also the vertex figure of the isogonal n-gonal double antiprismoid where n is an odd number greater than one, including the grand antiprism with pairs of trapezoid rather than square faces.

References[edit]

^ 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
^ ^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
^ Francis, Darryl (2013), "Johnson solids & their acronyms", Word Ways, 46 (3): 177
^ Cromwell, P. R. (1997), Polyhedra, Cambridge University Press, p. 87, ISBN 978-0-521-66405-9
^ Timofeenko, A. V. (2009), "The non-Platonic and non-Archimedean noncomposite polyhedra", Journal of Mathematical Science, 162 (5): 718, doi:10.1007/s10958-009-9655-0, S2CID 120114341

External links[edit]

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

[johnson-1] 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

[berman-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

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

[cromwell-4] Cromwell, P. R. (1997), Polyhedra, Cambridge University Press, p. 87, 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): 718, 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
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)

Properties[edit]

Cartesian coordinates[edit]

Variations[edit]

See also[edit]

References[edit]

External links[edit]