Elongated square gyrobicupola: Difference between revisions

Elongated square gyrobicupola
Elongated square gyrobicupola
Type	Johnson; J36 – J37 – J38
Faces	8 triangles; 18 squares
Edges	48
Vertices	24
Vertex configuration	8+16(3.43)
Symmetry group	D4d
Dual polyhedron	Pseudo-deltoidal icositetrahedron
Properties	convex, singular vertex figure
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In geometry, the elongated square gyrobicupola or pseudo-rhombicuboctahedron is one of the Johnson solids ( $J 37$ ). It is not usually considered to be an Archimedean solid, even though its faces consist of regular polygons that meet in the same pattern at each of its vertices, because unlike the 13 Archimedean solids, it lacks a set of global symmetries that map every vertex to every other vertex (though Grünbaum has suggested it should be added to the traditional list of Archimedean solids as a 14th example). It strongly resembles, but should not be mistaken for, the small rhombicuboctahedron, which is an Archimedean solid. It is also a canonical polyhedron.

This shape 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.^[1] It was independently rediscovered by J. C. P. Miller by 1930 (by mistake while attempting to construct a model of the small rhombicuboctahedron^[2]) and again by V. G. Ashkinuse in 1957.^[3]

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.^[4]

Construction and relation to the rhombicuboctahedron

As the name suggests, it can be constructed by elongating a square gyrobicupola (J₂₉) and inserting an octagonal prism between its two halves.

Rhombicuboctahedron	Exploded sections of rhombicuboctahedron	Pseudo-rhombicuboctahedron

The solid can also be seen as the result of twisting one of the square cupolae (J₄) on a rhombicuboctahedron (one of the Archimedean solids; a.k.a. the elongated square orthobicupola) by 45 degrees. It is therefore a gyrate rhombicuboctahedron. Its similarity to the rhombicuboctahedron gives it the alternative name pseudo-rhombicuboctahedron. It has occasionally been referred to as "the fourteenth Archimedean solid".

This property does not carry over to its pentagonal-faced counterpart, the gyrate rhombicosidodecahedron.

Symmetry and classification

The pseudo-rhombicuboctahedron possesses D_4d symmetry. 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 divide 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 one of the Archimedean solids.

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.^[5]

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 [V₁₈O₄₂]¹²⁻ has a pseudo-rhombicuboctahedral structure, where each square face acts as the base of a VO₅ pyramid.^[6]

References

^ 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).
^ Rouse Ball (1939), Coxeter, H. S. M. (ed.), Mathematical recreations and essays (11 ed.), p. 137
^ 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..
^ 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.
^ "J37 honeycombs", Gallery of Wooden Polyhedra, retrieved 2016-03-21
^ Greenwood, Norman N.; Earnshaw, Alan (1997). Chemistry of the Elements (2nd ed.). Butterworth-Heinemann. p. 986. ISBN 978-0-08-037941-8.

External links

[1] 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).

[2] Rouse Ball (1939), Coxeter, H. S. M. (ed.), Mathematical recreations and essays (11 ed.), p. 137

[3] 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..

[4] 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.

[5] "J37 honeycombs", Gallery of Wooden Polyhedra, retrieved 2016-03-21

[6] Greenwood, Norman N.; Earnshaw, Alan (1997). Chemistry of the Elements (2nd ed.). Butterworth-Heinemann. p. 986. ISBN 978-0-08-037941-8.

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@@ Line 1: / Line 1: @@
+{{Short description|Johnson solid with 8 faces}}
 {{Infobox polyhedron
 |image=elongated square gyrobicupola.png
-|type=[[Johnson solid|Johnson]]<br>[[elongated triangular gyrobicupola|''J''<sub>36</sub>]] -''' J<sub>37</sub>''' – [[elongated pentagonal orthobicupola|J<sub>38</sub>]]
+|type=[[Johnson solid|Johnson]]<br>{{math|[[elongated triangular gyrobicupola|''J''{{sub|36}}]] – '''''J''{{sub|37}}''' – [[elongated pentagonal orthobicupola|''J''{{sub|38}}]]}}
 |faces=8 [[triangle]]s<br>18 [[Square (geometry)|squares]]
 |edges=48
 |vertices=24
-|symmetry=''D''<sub>4d</sub>
+|symmetry=''D''{{sub|4d}}
 |vertex_config=8+16(3.4<sup>3</sup>)
 |dual=[[Pseudo-deltoidal icositetrahedron]]
@@ Line 12: / Line 13: @@
 }}
 [[File:J37 elongated square gyrobicupola.stl|thumb|3D model of an elongated square gyrobicupola]]
-In [[geometry]], the '''elongated square gyrobicupola''' or '''pseudo-rhombicuboctahedron''' is one of the [[Johnson solid]]s (''J''<sub>37</sub>). It is not usually considered to be an [[Archimedean solid]], even though its faces consist of [[regular polygon]]s that meet in the same pattern at each of its vertices, because unlike the 13 Archimedean solids, it lacks a set of global symmetries that take every vertex to every other vertex (though [[Branko Grünbaum|Grünbaum]] has suggested it should be added to the traditional list of Archimedean solids as a 14th example). It strongly resembles, but should not be mistaken for, the [[small rhombicuboctahedron]], which ''is'' an Archimedean solid. It is also a [[Midsphere#Canonical polyhedron|canonical polyhedron]].
+In [[geometry]], the '''elongated square gyrobicupola''' or '''pseudo-rhombicuboctahedron''' is one of the [[Johnson solid]]s ({{math|''J''{{sub|37}}}}). It is not usually considered to be an [[Archimedean solid]], even though its [[Face (geometry)|faces]] consist of [[regular polygon]]s that meet in the same pattern at each of its [[Vertex (geometry)|vertices]], because unlike the 13 Archimedean solids, it lacks a set of global [[symmetries]] that map every vertex to every other vertex (though [[Branko Grünbaum|Grünbaum]] has suggested it should be added to the traditional list of Archimedean solids as a 14th example). It strongly resembles, but should not be mistaken for, the [[small rhombicuboctahedron]], which ''is'' an Archimedean solid. It is also a [[Midsphere#Canonical polyhedron|canonical polyhedron]].
 This shape 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.<ref>{{citation|first=D. M. Y.|last=Sommerville|author-link=Duncan Sommerville|title=Semi-regular networks of the plane in absolute geometry|journal=Transactions of the Royal Society of Edinburgh|volume=41|year=1905|pages=725–747|doi=10.1017/s0080456800035560|url=https://zenodo.org/record/1428682}}. As cited by {{harvtxt|Grünbaum|2009}}.</ref> It was independently rediscovered by [[J. C. P. Miller]] by 1930 (by mistake while attempting to construct a model of the [[small rhombicuboctahedron]]<ref>{{citation|last=Rouse Ball|title=Mathematical recreations and essays|page=137|edition=11|year=1939|editor-last=Coxeter|editor-first=H. S. M. }}</ref>) and again by V. G. Ashkinuse in 1957.<ref>{{citation

v t e Johnson solids
Pyramids, cupolae and rotundae	square pyramid pentagonal pyramid triangular cupola square cupola pentagonal cupola pentagonal rotunda
Modified pyramids	elongated triangular pyramid elongated square pyramid elongated pentagonal pyramid gyroelongated square pyramid gyroelongated pentagonal pyramid triangular bipyramid pentagonal bipyramid elongated triangular bipyramid elongated square bipyramid elongated pentagonal bipyramid gyroelongated square bipyramid
Modified cupolae and rotundae	elongated triangular cupola elongated square cupola elongated pentagonal cupola elongated pentagonal rotunda gyroelongated triangular cupola gyroelongated square cupola gyroelongated pentagonal cupola gyroelongated pentagonal rotunda gyrobifastigium triangular orthobicupola square orthobicupola square gyrobicupola pentagonal orthobicupola pentagonal gyrobicupola pentagonal orthocupolarotunda pentagonal gyrocupolarotunda pentagonal orthobirotunda elongated triangular orthobicupola elongated triangular gyrobicupola elongated square gyrobicupola elongated pentagonal orthobicupola elongated pentagonal gyrobicupola elongated pentagonal orthocupolarotunda elongated pentagonal gyrocupolarotunda elongated pentagonal orthobirotunda elongated pentagonal gyrobirotunda gyroelongated triangular bicupola gyroelongated square bicupola gyroelongated pentagonal bicupola gyroelongated pentagonal cupolarotunda gyroelongated pentagonal birotunda
Augmented prisms	augmented triangular prism biaugmented triangular prism triaugmented triangular prism augmented pentagonal prism biaugmented pentagonal prism augmented hexagonal prism parabiaugmented hexagonal prism metabiaugmented hexagonal prism triaugmented hexagonal prism
Modified Platonic solids	augmented dodecahedron parabiaugmented dodecahedron metabiaugmented dodecahedron triaugmented dodecahedron metabidiminished icosahedron tridiminished icosahedron augmented tridiminished icosahedron
Modified Archimedean solids	augmented truncated tetrahedron augmented truncated cube biaugmented truncated cube augmented truncated dodecahedron parabiaugmented truncated dodecahedron metabiaugmented truncated dodecahedron triaugmented truncated dodecahedron gyrate rhombicosidodecahedron parabigyrate rhombicosidodecahedron metabigyrate rhombicosidodecahedron trigyrate rhombicosidodecahedron diminished rhombicosidodecahedron paragyrate diminished rhombicosidodecahedron metagyrate diminished rhombicosidodecahedron bigyrate diminished rhombicosidodecahedron parabidiminished rhombicosidodecahedron metabidiminished rhombicosidodecahedron gyrate bidiminished rhombicosidodecahedron tridiminished rhombicosidodecahedron
Other elementary solids	snub disphenoid snub square antiprism sphenocorona augmented sphenocorona sphenomegacorona hebesphenomegacorona disphenocingulum bilunabirotunda triangular hebesphenorotunda
(See also List of Johnson solids, a sortable table)