Sphenocorona

Sphenocorona
TypeJohnson
J85 – J86 – J87
Faces12 triangles
2 squares
Edges22
Vertices10
Vertex configuration4(33.4)
2(32.42)
2x2(35)
Symmetry groupC2v
Dual polyhedron-
Propertiesconvex, elementary
Net
3D model of a sphenocorona

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 .[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 can be calculated as:[2] and its volume as:[2]

Cartesian coordinates

Let be the smallest positive root of the quartic polynomial . Then, Cartesian coordinates of a sphenocorona with edge length 2 are given by the union of the orbits of the points 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.

See also

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, S2CID 122006114, Zbl 0132.14603
  2. 1 2 3 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
  3. ↑ Francis, Darryl (2013), "Johnson solids & their acronyms", Word Ways, 46 (3): 177
  4. ↑ Cromwell, P. R. (1997), Polyhedra, Cambridge University Press, p. 86–87, 89, ISBN 978-0-521-66405-9
  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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