Truncated prism . A
truncated prism is formed when prism is sliced by a plane that is not
parallel to its bases. A truncated prism's bases are not
congruent, and its sides are not parallelograms.
Twisted prism A
twisted prism is a nonconvex polyhedron constructed from a uniform -prism with each side face bisected on the square diagonal, by twisting the top, usually (but not necessarily) by radians ( degrees). If the bisectors are slanted to the left, then twisting the top base in the right direction (looking at the top of the prism) by a small angle gives nonconvex polyhedron and twisting it in the left direction, a convex polyhedron (see twisted square prism on the image). If the bisectors are slanted to the right, then twisting the top base in the left direction gives nonconvex polyhedron, in the right direction, convex one (see twisted dodecagonal prism). A twisted prism cannot be
dissected into tetrahedra without adding new vertices. The simplest twisted prism has triangle bases and is called a
Schönhardt polyhedron. An -gonal
twisted prism is topologically identical to the -gonal uniform
antiprism, but has half the
symmetry group: , order . It can be seen as a nonconvex antiprism, with tetrahedra removed between pairs of triangles. Any twisted -gonal prism is an antiprism, so the twisted square prism and twisted dodecagonal prism shown on the image are both antiprisms.
Frustum A
frustum is a similar construction to a prism, with
trapezoid lateral faces and differently sized top and bottom polygons.
Star prism A
star prism is a nonconvex polyhedron constructed by two identical
star polygon faces on the top and bottom, being parallel and offset by a distance and connected by rectangular faces. A
uniform star prism will have
Schläfli symbol {{math|{
p/
q} × { },}} with rectangles and 2 {{math|{
p/
q} }} faces. It is topologically identical to a -gonal prism.
Crossed prism A
crossed prism is a nonconvex polyhedron constructed from a prism, where the vertices of one base are
inverted around the center of this base (or rotated by 180°). This transforms the side rectangular faces into
crossed rectangles. For a regular polygon base, the appearance is an -gonal
hour glass. All oblique edges pass through a single body center. Note: no vertex is at this body centre. A crossed prism is topologically identical to an -gonal prism.
Toroidal prism A
toroidal prism is a nonconvex polyhedron like a
crossed prism, but without bottom and top base faces, and with simple rectangular side faces closing the polyhedron. This can only be done for even-sided base polygons. These are topological tori, with
Euler characteristic of zero. The topological
polyhedral net can be cut from two rows of a
square tiling (with
vertex configuration ): a band of squares, each attached to a
crossed rectangle. An -gonal toroidal prism has vertices, faces: squares and crossed rectangles, and edges. It is topologically
self-dual.
Prismatic polytope A
prismatic polytope is a higher-dimensional generalization of a prism. An -dimensional prismatic polytope is constructed from two ()-dimensional polytopes, translated into the next dimension. The prismatic -polytope elements are doubled from the ()-polytope elements and then creating new elements from the next lower element. Take an -polytope with
-face elements (). Its ()-polytope prism will have -face elements. (With , .) By dimension: • Take a
polygon with vertices, edges. Its prism has vertices, edges, and faces. • Take a
polyhedron with vertices, edges, and faces. Its prism has vertices, edges, faces, and cells. • Take a
polychoron with vertices, edges, faces, and cells. Its prism has vertices, edges, faces, cells, and hypercells.
Uniform prismatic polytope A regular -polytope represented by
Schläfli symbol {{math|{
p,
q,...,
t} }} can form a uniform prismatic ()-polytope represented by a
Cartesian product of
two Schläfli symbols: {{math|{
p,
q,...,
t}×{ }.}} By dimension: • A 0-polytopic prism is a
line segment, represented by an empty
Schläfli symbol {{math|{ }.}} • : • A 1-polytopic prism is a
rectangle, made from 2 translated line segments. It is represented as the product Schläfli symbol {{math|{ }×{ }.}} If it is
square, symmetry can be reduced: {{math|{ }×{ } {4}.}} • :Example: , Square, {{math|{ }×{ },}} two parallel line segments, connected by two line segment
sides. • A
polygonal prism is a 3-dimensional prism made from two translated polygons connected by rectangles. A regular polygon {{math|{
p} }} can construct a uniform -gonal prism represented by the product {{math|{
p}×{ }.}} If , with square sides symmetry it becomes a
cube: {{math|{4}×{ } {4,3}.}} • :Example: ,
Pentagonal prism, {{math|{5}×{ },}} two parallel
pentagons connected by 5 rectangular
sides. • A
polyhedral prism is a 4-dimensional prism made from two translated polyhedra connected by 3-dimensional prism cells. A regular polyhedron {{math|{
p,
q} }} can construct the uniform polychoric prism, represented by the product {{math|{
p,
q}×{ }.}} If the polyhedron and the sides are cubes, it becomes a
tesseract: {{math|1={4,3}×{ } = {4,3,3}.}} • :Example: ,
Dodecahedral prism, {{math|{5,3}×{ },}} two parallel
dodecahedra connected by 12 pentagonal prism
sides. • ... . The squares make a 23×29 grid
flat torus. Higher order prismatic polytopes also exist as
cartesian products of any two or more polytopes. The dimension of a product polytope is the sum of the dimensions of its elements. The first examples of these exist in 4-dimensional space; they are called
duoprisms as the product of two polygons in 4-dimensions. Regular duoprisms are represented as {{math|{
p}×{
q},}} with vertices, edges, square faces, -gon faces, -gon faces, and bounded by -gonal prisms and -gonal prisms. For example, {{math|{4}×{4},}} a
4-4 duoprism is a lower symmetry form of a
tesseract, as is {{math|{4,3}×{ },}} a
cubic prism. {{math|{4}×{4}×{ } }} (4-4 duoprism prism), {{math|{4,3}×{4} }} (cube-4 duoprism) and {{math|{4,3,3}×{ } }} (tesseractic prism) are lower symmetry forms of a
5-cube. == See also ==