Elongated square gyrobicupola | |
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Type | Canonical, Johnson J36 – J37 – J38 |
Faces | 8 triangles 18 squares |
Edges | 48 |
Vertices | 24 |
Vertex configuration | |
Symmetry group | |
Dual polyhedron | Pseudo-deltoidal icositetrahedron |
Properties | convex, singular vertex figure |
Net | |
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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 constructed 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 solids, [1] or Miller–Askinuze solid. [2]
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 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 the Johnson solid, and the elongated square gyrobicupola is among them, enumerated as the 37th Johnson solid . [4]
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]
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]
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 solids. [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]
With faces colored by its D4d symmetry, it can look like this:
The pseudo-deltoidal icositetrahedron (right) is the dual polyhedron. | |
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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.
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 is a nonconvex analog of the pseudo-rhombicuboctahedron, constructed in a similar way from the nonconvex great rhombicuboctahedron.
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]
In geometry, an Archimedean solid is one of 13 convex polyhedra whose faces are regular polygons and whose vertices are all symmetric to each other. They were first enumerated by Archimedes. They belong to the class of convex uniform polyhedra, the convex polyhedra with regular faces and symmetric vertices, which is divided into the Archimedean solids, the five Platonic solids, and the two infinite families of prisms and antiprisms. The pseudorhombicuboctahedron is an extra polyhedron with regular faces and congruent vertices, but it is not generally counted as an Archimedean solid because it is not vertex-transitive. An even larger class than the convex uniform polyhedra is the Johnson solids, whose regular polygonal faces do not need to meet in identical vertices.
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.
In geometry, rhombicuboctahedron is an Archimedean solid with 26 faces, consisting of 8 equilateral triangles and 18 squares. It is 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.
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.
In geometry, the square cupola the cupola with octagonal base. In the case of edges are equal in length, it is the Johnson solid, a convex polyhedron with faces are regular. It can be used to construct many polyhedrons, particularly in other Johnson solids.
In geometry, the elongated square cupola is one of the Johnson solids. As the name suggests, it can be constructed by elongating a square cupola by attaching an octagonal prism to its base. The solid can be seen as a rhombicuboctahedron with its "lid" removed.
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.
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.
In geometry, the gyrobifastigium is the 26th Johnson solid. It can be constructed by joining two face-regular triangular prisms along corresponding square faces, giving a quarter-turn to one prism. It is the only Johnson solid that can tile three-dimensional space.
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.
In geometry, the triangular orthobicupola is one of the Johnson solids. As the name suggests, it can be constructed by attaching two triangular cupolas along their bases. It has an equal number of squares and triangles at each vertex; however, it is not vertex-transitive. It is also called an anticuboctahedron, twisted cuboctahedron or disheptahedron. It is also a canonical polyhedron.
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.
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.
In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive. It follows that all vertices are congruent.
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.
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.