Elongated square gyrobicupola

Elongated square gyrobicupola
Type	Canonical, Johnson J36 – J37 – J38
Faces	8 triangles 18 squares
Edges	48
Vertices	24
Vertex configuration	${\displaystyle 8+16(3\cdot 4^{3})}$
Symmetry group	${\displaystyle D_{4\mathrm {d} }}$
Dual polyhedron	Pseudo-deltoidal icositetrahedron
Properties	convex, singular vertex figure
Net

In geometry, the elongated square gyrobicupola is a polyhedron constructed by two square cupolas attaching onto the bases of octagonal prism, with one of them rotated. It is also known as pseudo-rhombicuboctahedron because many mathematicians mistakenly constructed a rhombicuboctahedron. It is not considered to be an Archimedean solid because it lacks a set of global symmetries that map every vertex to every other vertex, unlike the 13 Archimedean solids. It is also a canonical polyhedron.

Construction

The elongated square gyrobicupola can be constructed similarly to the rhombicuboctahedron, by attaching two regular square cupolas onto the bases of octagonal prism, a process known as elongation. The difference between these two polyhedrons is that one of two square cupolas of the elongated square gyrobicupola is twisted by 45 degrees, a process known as gyro, making the triangular faces staggered vertically. ^{[1] [2]} The resulting polyhedron has 8 equilateral triangles and 18 squares. ^[1] A convex polyhedron in which all of the faces are regular polygons is the Johnson solid, and the elongated square gyrobicupola is among them, enumerated as the 37th Johnson solid ${\displaystyle J_{37}}$. ^[3]

Process of the construction of the elongated square gyrobicupola

The elongated square gyrobicupola may have been discovered by Johannes Kepler in his enumeration of the Archimedean solids, but its first clear appearance in print appears to be the work of Duncan Sommerville in 1905. ^[4] It was independently rediscovered by J. C. P. Miller in 1930 by mistake while attempting to construct a model of the rhombicuboctahedron, which is known as pseudo-rhombicuboctahedron or sometimes Miller's solid. This solid was discovered again by V. G. Ashkinuse in 1957. ^{[2] [5] [6]}

Properties

An elongated square gyrobicupola with edge length ${\displaystyle a}$ has a surface area: ^[1]

$${\displaystyle \left(18+2{\sqrt {3}}\right)a^{2}\approx 21.464a^{2},}$$

by adding the area of 8 equilateral triangles and 10 squares. Its volume can be calculated by slicing it into two square cupolas and one octagonal prism: ^[1]

$${\displaystyle {\frac {12+10{\sqrt {2}}}{3}}a^{3}\approx 8.714a^{3}.}$$

3D model of an elongated square gyrobicupola

The elongated square gyrobicupola possesses three-dimensional symmetry group ${\displaystyle D_{4\mathrm {d} }}$ of order 16. It is locally vertex-regular – the arrangement of the four faces incident on any vertex is the same for all vertices; this is unique among the Johnson solids. However, the manner in which it is "twisted" gives it a distinct "equator" and two distinct "poles", which in turn divides its vertices into 8 "polar" vertices (4 per pole) and 16 "equatorial" vertices. It is therefore not vertex-transitive, and consequently not usually considered to be the 14th Archimedean solids. ^{[2] [6] [7]}

With faces colored by its D_4d symmetry, it can look like this:

The pseudo-deltoidal icositetrahedron (right) is the dual polyhedron.

There are 8 (green) squares around its equator, 4 (red) triangles and 4 (yellow) squares above and below, and one (blue) square on each pole.

Related polyhedra and honeycombs

The elongated square gyrobicupola can form a space-filling honeycomb with the regular tetrahedron, cube, and cuboctahedron. It can also form another honeycomb with the tetrahedron, square pyramid and various combinations of cubes, elongated square pyramids, and elongated square bipyramids. ^[8]

The pseudo great rhombicuboctahedron

The pseudo great rhombicuboctahedron is a nonconvex analog of the pseudo-rhombicuboctahedron, constructed in a similar way from the nonconvex great rhombicuboctahedron.

In chemistry

The polyvanadate ion [ V ₁₈ O ₄₂]¹²⁻ has a pseudo-rhombicuboctahedral structure, where each square face acts as the base of a VO₅ pyramid. ^[9]

References

1. 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 .
2. Cromwell, Peter R. (1997), Polyhedra, Cambridge University Press, p. 91, ISBN   978-0-521-55432-9 .
3. Francis, Darryl (August 2013), "Johnson solids & their acronyms", Word Ways, 46 (3): 177.
4. Sommerville, D. M. Y. (1905), "Semi-regular networks of the plane in absolute geometry", Transactions of the Royal Society of Edinburgh, 41: 725–747, doi:10.1017/s0080456800035560 . As cited by Grünbaum (2009).
5. Ball, Rouse (1939), Coxeter, H. S. M. (ed.), Mathematical recreations and essays (11 ed.), p. 137.
6. Grünbaum, Branko (2009), "An enduring error" (PDF), Elemente der Mathematik, 64 (3): 89–101, doi: 10.4171/EM/120 , MR   2520469 Reprinted in Pitici, Mircea, ed. (2011). The Best Writing on Mathematics 2010. Princeton University Press. pp. 18–31..
7. Lando, Sergei K.; Zvonkin, Alexander K. (2004), Graphs on Surfaces and Their Applications, Springer, p. 114, doi:10.1007/978-3-540-38361-1, ISBN   978-3-540-38361-1 .
8. "J37 honeycombs", Gallery of Wooden Polyhedra, retrieved 2016-03-21
9. Greenwood, Norman N.; Earnshaw, Alan (1997). Chemistry of the Elements (2nd ed.). Butterworth-Heinemann. p. 986. ISBN   978-0-08-037941-8.

Further reading

• Anthony Pugh (1976), Polyhedra: A visual approach, California: University of California Press Berkeley, ISBN   0-520-03056-7 Chapter 2: Archimedean polyhedra, prisma and antiprisms, p. 25 Pseudo-rhombicuboctahedron

External links

• Weisstein, Eric W., "Elongated square gyrobicupola" ("Johnson solid") at MathWorld .
• George Hart: pseudo-rhombicuboctahedra