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

3D model of a sphenocorona

In geometry, the sphenocorona is one of the Johnson solids (J₈₆). It is one of the elementary Johnson solids that do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids.

A Johnson solid is one of 92 strictly convex polyhedra that is composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids , Archimedean solids , prisms , or antiprisms ). They were named by Norman Johnson , who first listed these polyhedra in 1966. ^[1]

Johnson uses the prefix spheno- to refer to a wedge-like complex formed by two adjacent lunes , a lune being a square with equilateral triangles attached on opposite sides. Likewise, the suffix -corona refers to a crownlike complex of 8 equilateral triangles. Joining both complexes together results in the sphenocorona. ^[2]

Construction and properties

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

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

under the action of the group generated by reflections about the xz-plane and the yz-plane. ^[3]

The sphenocorona has 12 equilateral triangles and 2 squares as its faces. ^[4] 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 ${\displaystyle J_{86}}$. ^[5]

The surface area of a sphenocorona with edge length ${\displaystyle a}$ can be calculated as: ^[4]

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

and its volume as: ^[4]

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

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.

See also

• Augmented sphenocorona

References

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, Zbl   0132.14603 .
2. 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
3. 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
4. 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
5. Francis, Darryl (2013), "Johnson solids & their acronyms", Word Ways, 46 (3): 177

External links

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