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A prismatic uniform polyhedron is a uniform polyhedron with dihedral symmetry. They exist in two infinite families, the uniform prisms and the uniform antiprisms. All have their vertices in parallel planes and are therefore prismatoids. A uniform polyhedron is a polyhedron with regular polygons as faces and identical vertices. ...
This article deals with three infinite series of point groups in three dimensions which have a symmetry group which as abstract group is a dihedral group Dihn ( n ≥ 2 ). See also point groups in two dimensions. ...
In geometry, an n-sided prism is a polyhedron made of an n-sided polygonal base, a translated copy, and n faces joining corresponding sides. ...
An antiprism is a polyhedron composed of two parallel copies of some particular polygon, connected by an alternating band of triangles. ...
A prismatoid is a polyhedron where all vertices lie in two parallel planes. ...
Vertex configuration and symmetry groups
Because they are isogonal (vertex-transitive), their vertex arrangement uniquely corresponds to a symmetry group. In geometry, a polyhedron (or tiling) is isogonal or vertex-transitive if all its vertices are the same. ...
A discrete point group in 3D is a finite symmetry group in 3D that leaves the origin fixed. ...
The difference between the prismatic and antiprismatic symmetry groups is that Dph has the vertices lined up in both planes, which gives it a reflection plane perpendicular to its p-fold axis (parallel to the {p/q} polygon); while Dpd has the vertices twisted relative to the other plane, which gives it a rotatory reflection. Each has p reflection planes which contain the p-fold axis.
Enumeration There are: - prisms, for each rational number p/q > 2, with symmetry group Dph;
- antiprisms, for each rational number p/q > 3/2, with symmetry group Dpd if q is odd, Dph if q is even.
If p/q is an integer, i.e. if q = 1, the prism or antiprism is convex. (The fraction is always assumed to be stated in lowest terms.) In geometry, an n-sided prism is a polyhedron made of an n-sided polygonal base, a translated copy, and n faces joining corresponding sides. ...
An antiprism is a polyhedron composed of two parallel copies of some particular polygon, connected by an alternating band of triangles. ...
An antiprism with p/q < 2 is crossed or retrograde; its vertex figure resembles a bowtie. If p/q ≤ 3/2 no antiprism can exist, as its vertex figure would have to violate the triangle inequality. In geometry, a vertex figure is most easily thought of as the cut surface exposed when a corner of a polytope is cut off in a certain way. ...
In mathematics, triangle inequality is the theorem stating that for any triangle, the measure of a given side must be less than the sum of the other two sides but greater than the difference between the two sides. ...
Images Note: The cube and octahedron are listed here with dihedral symmetry (as a square prism and triangular antiprism respectively), although if uniformly colored, they also have octahedral symmetry. A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ...
An octahedron (plural: octahedra) is a polyhedron with eight faces. ...
// List of symmetry groups on the sphere Spherical symmetry groups are also called point groups (in 3D). ...
Image File history File links Triangular_prism. ...
In geometry, a triangular prism or three-sided prism is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides. ...
Image File history File links Download high-resolution version (1000x1000, 226 KB) Summary Uniform Trigonal antiprism (Octahedron) Licensing I, the creator of this work, hereby release it into the public domain. ...
An octahedron (plural: octahedra) is a polyhedron with eight faces. ...
Image File history File links Download high-resolution version (1000x1000, 201 KB)Image of Cube, colored with dihedral symmetry Licensing File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...
A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ...
Image File history File links Square_antiprism. ...
In geometry, the Square antiprism is the second in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. ...
Image File history File links Pentagonal_prism. ...
In geometry, the pentagonal prism is the third in an infinite set of prisms formed by square sides and two regular polygon caps. ...
Image File history File links Pentagrammic_prism. ...
In geometry, the pentagrammic prism is the first in an infinite set of concave prisms formed by square sides and two regular star polygon caps, in this case two pentagrams. ...
Image File history File links Download high resolution version (639x640, 15 KB) Summary Uniform pentagrammic prism 3. ...
In geometry, the pentagrammic antiprism is one in an infinite set of nonconvex antiprisms formed by triangle sides and two regular star polygon caps, in this case two pentagrams. ...
Image File history File links Pentagonal_antiprism. ...
In geometry, the pentagonal antiprism is the third in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. ...
Image File history File links Download high-resolution version (1000x1000, 347 KB)Pentagrammic crossed antiprism File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...
Alternative representation with hollow pentagrams In geometry, the pentagrammic crossed-antiprism is one in an infinite set of nonconvex antiprisms formed by triangle sides and two regular star polygon caps, in this case two pentagrams. ...
Image File history File links Hexagonal_prism. ...
In geometry, the hexagonal prism is a prism with hexagonal base. ...
Image File history File links Hexagonal_antiprism. ...
In geometry, the hexagonal antiprism is the third in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. ...
Image File history File links Size of this preview: 600 × 600 pixel Image in higher resolution (1000 × 1000 pixel, file size: 180 KB, MIME type: image/png) I, the creator of this work, hereby release it into the public domain. ...
Image File history File links Download high resolution version (1000x1000, 279 KB) Licensing Robert Webb produced this image for Wikipedia upon my request, and offered the credit statement below: File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...
In geometry, the {7/2} heptagrammic prism is one of an infinite set of nonconvex prisms formed by square sides and two regular star polygon caps, in this case two {7/2} heptagrams. ...
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Alternative representation with hollow heptagram centers In geometry, the {7/3} heptagrammic prism is one of an infinite set of nonconvex prisms formed by square sides and two regular star polygon caps, in this case two {7/3} heptagrams. ...
Image File history File links Octagonal_prism. ...
In geometry, the octagonal prism is the sixth in an infinite set of prisms formed by square sides and two regular polygon caps. ...
Image File history File links Size of this preview: 533 × 599 pixel Image in higher resolution (889 × 999 pixel, file size: 337 KB, MIME type: image/png)antiprism See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old...
Image File history File links Octagonal_antiprism. ...
In geometry, the octagonal antiprism is the fifth in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. ...
Image File history File links Size of this preview: 668 × 600 pixel Image in higher resolution (1000 × 898 pixel, file size: 210 KB, MIME type: image/png)antiprism See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old...
Image File history File links Size of this preview: 586 × 600 pixel Image in higher resolution (924 × 946 pixel, file size: 277 KB, MIME type: image/png) I, the creator of this work, hereby release it into the public domain. ...
Image File history File links Download high-resolution version (1000x1000, 154 KB)Decagonal prism Licensing File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...
In geometry, the decagonal prism is the eighth in an infinite set of prisms formed by square sides and two regular polygon caps. ...
Image File history File links Size of this preview: 541 × 600 pixel Image in higher resolution (899 × 997 pixel, file size: 332 KB, MIME type: image/png)prism See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old...
Image File history File links Download high-resolution version (1000x1000, 206 KB)Decagonal antiprism Licensing File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...
Image File history File links Size of this preview: 600 × 600 pixel Image in higher resolution (1000 × 1000 pixel, file size: 166 KB, MIME type: image/png)antiprism See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old...
Image File history File links Download high-resolution version (1000x1000, 159 KB)Dodecagonal prism Licensing File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...
In geometry, the dodecagonal prism is the tenth in an infinite set of prisms formed by square sides and two regular polygon caps. ...
Image File history File links Download high-resolution version (1000x1000, 190 KB)Dodecagonal antiprism Licensing File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...
See also A uniform polyhedron is a polyhedron with regular polygons as faces and identical vertices. ...
In geometry, an n-sided prism is a polyhedron made of an n-sided polygonal base, a translated copy, and n faces joining corresponding sides. ...
An antiprism is a polyhedron composed of two parallel copies of some particular polygon, connected by an alternating band of triangles. ...
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