Elongated square gyrobicupola

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Elongated square gyrobicupola
Elongated square gyrobicupola.png
Type Canonical,
Johnson
J36J37J38
Faces 8 triangles
18 squares
Edges 48
Vertices 24
Vertex configuration
Symmetry group
Properties convex,
singular vertex figure
Net
Johnson solid 37 net.png

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 was once mistakenly considered a rhombicuboctahedron by many mathematicians. 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. For this reason, it is also known as pseudo-rhombicuboctahedron, Miller solid, [1] or MillerAskinuze solid. [2]

Contents

Construction

The elongated square gyrobicupola can be constructed similarly to the rhombicuboctahedron, by attaching two regular square cupolas onto the bases of an octagonal prism, a process known as elongation. The difference between these two polyhedrons is that one of the two square cupolas is twisted by 45 degrees, a process known as gyration, making the triangular faces staggered vertically. [3] [1] The resulting polyhedron has 8 equilateral triangles and 18 squares. [3] A convex polyhedron in which all of the faces are regular polygons is a Johnson solid, and the elongated square gyrobicupola is among them, enumerated as the 37th Johnson solid . [4]

Small rhombicuboctahedron.png
Exploded rhombicuboctahedron.png
Pseudorhombicuboctahedron.png
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. [5] It was independently rediscovered by J. C. P. Miller in 1930 by mistake while attempting to construct a model of the rhombicuboctahedron. This solid was discovered again by V. G. Ashkinuse in 1957. [1] [6] [7]

Properties

An elongated square gyrobicupola with edge length has a surface area: [3] 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: [3]

3D model of an elongated square gyrobicupola J37 elongated square gyrobicupola.stl
3D model of an elongated square gyrobicupola

The elongated square gyrobicupola possesses three-dimensional symmetry group 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 solid. [1] [7] [8]

The dihedral angle of an elongated square gyrobicupola can be ascertained in a similar way as the rhombicuboctahedron, by adding the dihedral angle of a square cupola and an octagonal prism: [2]

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. [9]

The pseudo great rhombicuboctahedron Pseudo-great rhombicuboctahedron.png
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. [10]

Related Research Articles

<span class="mw-page-title-main">Archimedean solid</span> Polyhedra in which all vertices are the same

The Archimedean solids are a set of thirteen convex polyhedra whose faces are regular polygons, but not all alike, and whose vertices are all symmetric to each other. The solids were named after Archimedes, although he did not claim credit for them. They belong to the class of uniform polyhedra, the polyhedra with regular faces and symmetric vertices. Some Archimedean solids were portrayed in the works of artists and mathematicians during the Renaissance.

<span class="mw-page-title-main">Cuboctahedron</span> Polyhedron with 8 triangular faces and 6 square faces

A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e., an Archimedean solid that is not only vertex-transitive but also edge-transitive. It is radially equilateral. Its dual polyhedron is the rhombic dodecahedron.

In geometry, a Johnson solid, sometimes also known as a Johnson–Zalgaller solid, is a strictly convex polyhedron whose faces are regular polygons. They are sometimes defined to exclude the uniform polyhedrons. There are ninety-two solids with such a property: the first solids are the pyramids, cupolas. and a rotunda; some of the solids may be constructed by attaching with those previous solids, whereas others may not. These solids are named after mathematicians Norman Johnson and Victor Zalgaller.

<span class="mw-page-title-main">Rhombicuboctahedron</span> Archimedean solid with 26 faces

In geometry, the rhombicuboctahedron is an Archimedean solid with 26 faces, consisting of 8 equilateral triangles and 18 squares. It was named by Johannes Kepler in his 1618 Harmonices Mundi, being short for truncated cuboctahedral rhombus, with cuboctahedral rhombus being his name for a rhombic dodecahedron.

<span class="mw-page-title-main">Truncated cube</span> Archimedean solid with 14 regular faces

In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces, 36 edges, and 24 vertices.

<span class="mw-page-title-main">Truncated cuboctahedron</span> Archimedean solid in geometry

In geometry, the truncated cuboctahedron or great rhombicuboctahedron is an Archimedean solid, named by Kepler as a truncation of a cuboctahedron. It has 12 square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices, and 72 edges. Since each of its faces has point symmetry, the truncated cuboctahedron is a 9-zonohedron. The truncated cuboctahedron can tessellate with the octagonal prism.

<span class="mw-page-title-main">Truncated icosidodecahedron</span> Archimedean solid

In geometry, a truncated icosidodecahedron, rhombitruncated icosidodecahedron, great rhombicosidodecahedron, omnitruncated dodecahedron or omnitruncated icosahedron is an Archimedean solid, one of thirteen convex, isogonal, non-prismatic solids constructed by two or more types of regular polygon faces.

<span class="mw-page-title-main">Square cupola</span> Cupola with octagonal base

In geometry, the square cupola is a cupola with an octagonal base. In the case of all edges being equal in length, it is a Johnson solid, a convex polyhedron with regular faces. It can be used to construct many other polyhedrons, particularly other Johnson solids.

<span class="mw-page-title-main">Square orthobicupola</span> 28th Johnson solid; 2 square cupolae joined base-to-base

In geometry, the square orthobicupola is one of the Johnson solids. As the name suggests, it can be constructed by joining two square cupolae along their octagonal bases, matching like faces. A 45-degree rotation of one cupola before the joining yields a square gyrobicupola.

<span class="mw-page-title-main">Square gyrobicupola</span> 29th Johnson solid; 2 square cupolae joined base-to-base

In geometry, the square gyrobicupola is one of the Johnson solids. Like the square orthobicupola, it can be obtained by joining two square cupolae along their bases. The difference is that in this solid, the two halves are rotated 45 degrees with respect to one another.

<span class="mw-page-title-main">Elongated triangular cupola</span> Polyhedron with triangular cupola and hexagonal prism

In geometry, the elongated triangular cupola is a polyhedron constructed from a hexagonal prism by attaching a triangular cupola. It is an example of a Johnson solid.

<span class="mw-page-title-main">Elongated triangular orthobicupola</span> Johnson solid with 20 faces

In geometry, the elongated triangular orthobicupola is a polyhedron constructed by attaching two regular triangular cupola into the base of a regular hexagonal prism. It is an example of Johnson solid.

<span class="mw-page-title-main">Elongated triangular gyrobicupola</span> 36th Johnson solid

In geometry, the elongated triangular gyrobicupola is a polyhedron constructed by attaching two regular triangular cupolas to the base of a regular hexagonal prism, with one of them rotated in . It is an example of Johnson solid.

In geometry, the term semiregular polyhedron is used variously by different authors.

<span class="mw-page-title-main">Uniform polyhedron</span> Isogonal polyhedron with regular faces

In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive—there is an isometry mapping any vertex onto any other. It follows that all vertices are congruent. Uniform polyhedra may be regular, quasi-regular, or semi-regular. The faces and vertices don't need to be convex, so many of the uniform polyhedra are also star polyhedra.

<span class="mw-page-title-main">Triangular prism</span> Prism with a 3-sided base

In geometry, a triangular prism or trigonal prism is a prism with 2 triangular bases. If the edges pair with each triangle's vertex and if they are perpendicular to the base, it is a right triangular prism. A right triangular prism may be both semiregular and uniform.

A pseudo-uniform polyhedron is a polyhedron which has regular polygons as faces and has the same vertex configuration at all vertices but is not vertex-transitive: it is not true that for any two vertices, there exists a symmetry of the polyhedron mapping the first isometrically onto the second. Thus, although all the vertices of a pseudo-uniform polyhedron appear the same, it is not isogonal. They are called pseudo-uniform polyhedra due to their resemblance to some true uniform polyhedra.

<span class="mw-page-title-main">Octadecahedron</span> Polyhedron with 18 faces

In geometry, an octadecahedron is a polyhedron with 18 faces. No octadecahedron is regular; hence, the name does not commonly refer to one specific polyhedron.

<span class="mw-page-title-main">Elongated gyrobifastigium</span> Space-filling polyhedron with 8 faces

In geometry, the elongated gyrobifastigium or gabled rhombohedron is a space-filling octahedron with 4 rectangles and 4 right-angled pentagonal faces.

References

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  2. 1 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. 1 2 3 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 .
  4. Francis, Darryl (August 2013), "Johnson solids & their acronyms", Word Ways, 46 (3): 177.
  5. 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).
  6. Ball, Rouse (1939), Coxeter, H. S. M. (ed.), Mathematical recreations and essays (11 ed.), p. 137.
  7. 1 2 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..
  8. 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 .
  9. "J37 honeycombs", Gallery of Wooden Polyhedra, retrieved 2016-03-21
  10. Greenwood, Norman N.; Earnshaw, Alan (1997). Chemistry of the Elements (2nd ed.). Butterworth-Heinemann. p. 986. ISBN   978-0-08-037941-8.

Further reading