geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, a prismatic uniform polyhedron is 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 ...

with

dihedral symmetry In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, g ...

. They exist in two infinite families, the uniform prisms and the uniform

antiprism In geometry, an antiprism or is a polyhedron composed of two Parallel (geometry), parallel Euclidean group, direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway po ...

s. All have their vertices in parallel planes and are therefore

prismatoid In geometry, a prismatoid is a polyhedron whose vertex (geometry), vertices all lie in two parallel Plane (geometry), planes. Its lateral faces can be trapezoids or triangles. If both planes have the same number of vertices, and the lateral faces ...

Vertex configuration and symmetry groups

Because they are isogonal (vertex-transitive), their vertex arrangement uniquely corresponds to a

symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...

. The difference between the prismatic and antiprismatic symmetry groups is that D_''p''h has the vertices lined up in both planes, which gives it a reflection plane perpendicular to its ''p''-fold axis (parallel to the polygon); while D_''p''d 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. The D_''p''h symmetry group contains inversion if and only if ''p'' is even, while D_''p''d contains inversion symmetry if and only if ''p'' is odd.

Enumeration

There are: *

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 ...

s, for each rational number ''p/q'' > 2, with symmetry group D_''p''h; *

s, for each rational number ''p/q'' > 3/2, with symmetry group D_''p''d if ''q'' is odd, D_''p''h 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.) An antiprism with ''p/q'' < 2 is ''crossed'' or ''retrograde''; its

vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...

resembles a bowtie. If ''p/q'' < 3/2 no uniform antiprism can exist, as its vertex figure would have to violate the

triangle inequality In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of Degeneracy (mathematics)#T ...

. If ''p/q'' = 3/2 the uniform antiprism is degenerate (has zero height).

Forms by symmetry

Note: The

tetrahedron In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...

cube A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...

, and

octahedron In geometry, an octahedron (: octahedra or octahedrons) is any polyhedron with eight faces. One special case is the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Many types of i ...

are listed here with dihedral symmetry (as a ''digonal antiprism'', ''square prism'' and ''triangular antiprism'' respectively), although if uniformly colored, the tetrahedron also has tetrahedral symmetry and the cube and octahedron also have octahedral symmetry.

References

* *Cromwell, P.; ''Polyhedra'', CUP, Hbk. 1997, . Pbk. (1999), . p.175 *.

External links

Prisms and Antiprisms
George W. Hart {{Polyhedron navigator Prismatoid polyhedra Uniform polyhedra

Vertex configuration and symmetry groups

Enumeration

Forms by symmetry

See also

References

External links