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, elementary
Net

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

Properties

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. The sphenocorona is among them, enumerated as the 86th Johnson solid $J_{86}$ .^[3] It is an elementary polyhedron, meaning it cannot be separated by a plane into two small regular-faced polyhedra.^[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

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

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

^ 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. 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): 718, doi:10.1007/s10958-009-9655-0, S2CID 120114341

External links

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. 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): 718, doi:10.1007/s10958-009-9655-0, S2CID 120114341

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