uniform polyhedron In geometry, a uniform polyhedron has regular polygons as Face (geometry), faces and is vertex-transitive—there is an isometry mapping any vertex onto any other. It follows that all vertices are congruence (geometry), congruent. Uniform po ...

polyhedron In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal Face (geometry), faces, straight Edge (geometry), edges and sharp corners or Vertex (geometry), vertices. The term "polyhedron" may refer ...

regular polygon In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either ''convex ...

vertex configuration In geometry, a vertex configuration is a shorthand notation for representing a polyhedron or Tessellation, tiling as the sequence of Face (geometry), faces around a Vertex (geometry), vertex. It has variously been called a vertex description, vert ...

vertex-transitive In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face i ...

symmetry Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...

uniform polyhedra 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 (if also fac ...

Branko Grünbaum Branko Grünbaum (; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descent. Reprinted in . They both have D_4d symmetry, the same symmetry as a

square antiprism In geometry, the square antiprism is the second in an infinite family of antiprisms formed by an even number, even-numbered sequence of triangle sides closed by two polygon caps. It is also known as an ''anticube''. If all its faces are regular ...

. They can both be constructed from a

uniform polyhedron In geometry, a uniform polyhedron has regular polygons as Face (geometry), faces and is vertex-transitive—there is an isometry mapping any vertex onto any other. It follows that all vertices are congruence (geometry), congruent. Uniform po ...

by twisting one

cupola In architecture, a cupola () is a relatively small, usually dome-like structure on top of a building often crowning a larger roof or dome. Cupolas often serve as a roof lantern to admit light and air or as a lookout. The word derives, via Ital ...

-shaped cap.

The pseudo-uniform polyhedra

Pseudorhombicuboctahedron

The pseudorhombicuboctahedron is the only convex pseudo-uniform polyhedron. It is also a

Johnson solid In geometry, a Johnson solid, sometimes also known as a Johnson–Zalgaller solid, is a convex polyhedron whose faces are regular polygons. They are sometimes defined to exclude the uniform polyhedrons. There are ninety-two Solid geometry, s ...

(J₃₇) and can also be called the

elongated square gyrobicupola 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 a Midsphere#Canonical polyhedron, canonical polyhedron. It is not consid ...

. Its dual is the pseudo-deltoidal icositetrahedron. As the name suggests, it can be constructed by elongating a square gyrobicupola (''J''₂₉) and inserting an

octagon In geometry, an octagon () is an eight-sided polygon or 8-gon. A '' regular octagon'' has Schläfli symbol and can also be constructed as a quasiregular truncated square, t, which alternates two types of edges. A truncated octagon, t is a ...

al

prism PRISM is a code name for a program under which the United States National Security Agency (NSA) collects internet communications from various U.S. internet companies. The program is also known by the SIGAD . PRISM collects stored internet ...

between its two halves. The resulting solid 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, it is not

vertex-transitive In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face i ...

, and consequently not one of the

Archimedean solid The Archimedean solids are a set of thirteen convex polyhedra whose faces are regular polygon and are vertex-transitive, although they aren't face-transitive. The solids were named after Archimedes, although he did not claim credit for them. They ...

s, as there are pairs of vertices such that there is no isometry of the solid which maps one into the other. Essentially, the two types of vertices can be distinguished by their "neighbors of neighbors." Another way to see that the polyhedron is not vertex-regular is to note that there is exactly one belt of eight squares around its equator, which distinguishes vertices on the belt from vertices on either side. The solid can also be seen as the result of twisting one of the square cupolae (''J''₄) on a

rhombicuboctahedron 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'', w ...

(one of the

Archimedean solid The Archimedean solids are a set of thirteen convex polyhedra whose faces are regular polygon and are vertex-transitive, although they aren't face-transitive. The solids were named after Archimedes, although he did not claim credit for them. They ...

s; a.k.a. the elongated square orthobicupola) by 45 degrees. Its similarity to the rhombicuboctahedron gives it the alternative name pseudorhombicuboctahedron. It has occasionally been referred to as "the fourteenth Archimedean solid". With faces colored by its ''D''_4d symmetry, it can look like this: There are 8 (green) squares around its

equator The equator is the circle of latitude that divides Earth into the Northern Hemisphere, Northern and Southern Hemisphere, Southern Hemispheres of Earth, hemispheres. It is an imaginary line located at 0 degrees latitude, about in circumferen ...

, 4 (red) triangles and 4 (yellow) squares above and below, and one (blue) square on each pole. The construction of the uniform and pseudo rhombicuboctahedra can be seen in the following augmentations of the octagonal prism:

Pseudo-great rhombicuboctahedron

The uniform nonconvex great rhombicuboctahedron may be seen as an octagrammic prism with the octagrams excavated with crossed square cupolae, similarly to how the

rhombicuboctahedron 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'', w ...

may be seen as an octagonal prism with the octagons augmented with square cupolae. Rotating one of the cupolae in this construction results in the pseudo-great rhombicuboctahedron. The pictures below show the excavation of the octagrammic prism with crossed square cupolae taking place one step at a time. The crossed square cupolae are always red, while the square sides of the octagrammic prism are in the other colours. All images are oriented approximately the same way for clarity. The pseudo great rhombicuboctahedron has a single "belt" of squares around its equator, and can be constructed by twisting one of the crossed square cupolae on a nonconvex great rhombicuboctahedron by 45 degrees. This is analogous to the pseudorhombicuboctahedron.

Duals of the pseudo-uniform polyhedra

The

duals ''Duals'' is a compilation album by the Irish rock band U2. It was released in April 2011 to u2.com subscribers. Track listing :* "Where the Streets Have No Name" and "Amazing Grace" are studio mix of U2's performance at the Rose Bowl, ...

of the pseudo-uniform polyhedra have all faces

congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In modu ...

, but not transitive: their faces do not all lie within the same symmetry orbit and they are thus not

isohedral In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruen ...

. This is a consequence of the pseudo-uniform polyhedra having the same

vertex configuration In geometry, a vertex configuration is a shorthand notation for representing a polyhedron or Tessellation, tiling as the sequence of Face (geometry), faces around a Vertex (geometry), vertex. It has variously been called a vertex description, vert ...

at every vertex, but not being

vertex-transitive In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face i ...

. This is demonstrated by the different colours used for the faces in the images of the dual pseudo-uniform polyhedra in this article, denoting different types of faces.

Pseudo-deltoidal icositetrahedron

Pseudo-great deltoidal icositetrahedron

References

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