In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a prismatic uniform polyhedron is a
uniform polyhedron
In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent.
Uniform polyhedra may be regular (if also ...
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.
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
Inversion or inversions may refer to:
Arts
* , a French gay magazine (1924/1925)
* ''Inversion'' (artwork), a 2005 temporary sculpture in Houston, Texas
* Inversion (music), a term with various meanings in music theory and musical set theory
* ...
if and only if ''p'' is even, while D
''p''d contains inversion symmetry if and only if ''p'' is odd.
Enumeration
There are:
*
prisms, for each rational number ''p/q'' > 2, with symmetry group D
''p''h;
*
antiprisms, 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 polyhedron or polytope is sliced off.
Definitions
Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw lines ...
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 degenerate triangles, bu ...
. If ''p/q'' = 3/2 the uniform antiprism is degenerate (has zero height).
Forms by symmetry
Note: The
tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ...
,
cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the on ...
, and
octahedron
In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at e ...
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.
See also
*
Uniform polyhedron
In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent.
Uniform polyhedra may be regular (if also ...
*
Prism (geometry)
In geometry, a prism is a polyhedron comprising an polygon base, a second base which is a translated copy (rigidly moved without rotation) of the first, and other faces, necessarily all parallelograms, joining corresponding sides of the two ...
*
Antiprism
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