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Sphenomegacorona

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Sphenomegacorona
TypeJohnson
J87J88J89
Faces16 triangles
2 squares
Edges28
Vertices12
Vertex configuration2(34)
2(32.42)
2x2(35)
4(34.4)
Symmetry groupC2v
Dual polyhedron-
Propertiesconvex, elementary
Net
3D model of a sphenomegacorona

In geometry, the sphenomegacorona is a Johnson solid with 16 equilateral triangles and 2 squares as its faces.

Properties

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The sphenomegacorona 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 -megacorona refers to a crownlike complex of 12 triangles, contrasted with the smaller triangular complex that makes the sphenocorona.[1] By joining both complexes, the resulting polyhedron has 16 equilateral triangles and 2 squares, making 18 faces.[2] All of its faces are regular polygons, categorizing the sphenomegacorona as a Johnson solid—a convex polyhedron in which all of the faces are regular polygons—enumerated as the 88th 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 sphenomegacorona is the total of polygonal faces' area—16 equilateral triangles and 2 squares. The volume of a sphenomegacorona is obtained by finding the root of a polynomial, and its decimal expansion—denoted as —is given by A334114. With edge length , its surface area and volume can be formulated as:[2][5]

Cartesian coordinates

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Let be the smallest positive root of the polynomial Then, Cartesian coordinates of a sphenomegacorona 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.[6]

References

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  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.
  2. ^ a b 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.
  3. ^ Francis, D. (August 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. ^ "A334114". The On-Line Encyclopedia of Integer Sequences. 2020.
  6. ^ 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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