In 4-dimensional
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 uniform antiprismatic prism or antiduoprism is a
uniform 4-polytope
In geometry, a uniform 4-polytope (or uniform polychoron) is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedron, uniform polyhedra, and faces are regular polygons.
There are 47 non-Prism (geometry), prism ...
with two
uniform antiprism cells in two parallel 3-space
hyperplane
In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is ...
s, connected by
uniform prisms cells between pairs of faces. The symmetry of a ''p''-gonal antiprismatic prism is
''p'',2+,2 order 8''p''.
A p-gonal antiprismatic prism or p-gonal antiduoprism has 2 ''p''-gonal
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 ...
, 2 ''p''-gonal
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 ...
, and ''2p''
triangular prism
In geometry, a triangular prism or trigonal prism is a Prism (geometry), prism with 2 triangular bases. If the edges pair with each triangle's vertex and if they are perpendicular to the base, it is a ''right triangular prism''. A right triangul ...
cells. It has 4''p''
equilateral triangle
An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...
, 4''p''
square
In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
and 4
regular ''p''-gon faces. It has 10''p'' edges, and 4''p'' vertices.
Convex uniform antiprismatic prisms
There is an infinite series of convex uniform antiprismatic prisms, starting with the ''digonal antiprismatic prism'' is a
tetrahedral prism
In geometry, a tetrahedral prism is a convex uniform 4-polytope. This 4-polytope has 6 polyhedron, polyhedral cells: 2 tetrahedron, tetrahedra connected by 4 triangular prisms. It has 14 faces: 8 triangular and 6 square. It has 16 edges and 8 vert ...
, with two of the tetrahedral cells degenerated into squares. The ''triangular antiprismatic prism'' is the first nondegenerate form, which is also an
octahedral prism
In geometry, an octahedral prism is a convex uniform 4-polytope. This 4-polytope has 10 polyhedral cells: 2 octahedra connected by 8 triangular prisms.
Alternative names
*Octahedral dyadic prism ( Norman W. Johnson)
*Ope (Jonathan Bowers, for o ...
. The remainder are unique uniform 4-polytopes.
Star antiprismatic prisms
There are also star forms following the set of
star antiprisms, starting with the pentagram :
Square antiprismatic prism
A square antiprismatic prism or square antiduoprism is a convex
uniform 4-polytope. It is formed as two parallel
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 ...
s connected by cubes and triangular prisms. The symmetry of a square antiprismatic prism is
,2+,2 order 32. It has 16
triangle
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...
, 16
square
In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
and 4
square
In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
faces. It has 40 edges, and 16 vertices.
Pentagonal antiprismatic prism
A pentagonal antiprismatic prism or pentagonal antiduoprism is a convex
uniform 4-polytope
In geometry, a uniform 4-polytope (or uniform polychoron) is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedron, uniform polyhedra, and faces are regular polygons.
There are 47 non-Prism (geometry), prism ...
. It is formed as two parallel
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. It consists of two pentagons joined to each other by a ring of ten triangles fo ...
s connected by cubes and triangular prisms. The symmetry of a pentagonal antiprismatic prism is
0,2+,2 order 40. It has 20
triangle
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...
, 20
square
In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
and 4
pentagon
In geometry, a pentagon () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is 540°.
A pentagon may be simple or list of self-intersecting polygons, self-intersecting. A self-intersecting ...
al faces. It has 50 edges, and 20 vertices.
Hexagonal antiprismatic prism
A hexagonal antiprismatic prism or hexagonal antiduoprism is a convex
uniform 4-polytope
In geometry, a uniform 4-polytope (or uniform polychoron) is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedron, uniform polyhedra, and faces are regular polygons.
There are 47 non-Prism (geometry), prism ...
. It is formed as two parallel
hexagonal antiprism
In geometry, the hexagonal antiprism is the 4th in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps.
Antiprisms are similar to prism (geometry), prisms except the bases are twisted re ...
s connected by cubes and triangular prisms. The symmetry of a hexagonal antiprismatic prism is
2,2+,2 order 48. It has 24
triangle
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...
, 24
square
In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
and 4
hexagon
In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.
Regular hexagon
A regular hexagon is de ...
faces. It has 60 edges, and 24 vertices.
Heptagonal antiprismatic prism
A heptagonal antiprismatic prism or heptagonal antiduoprism is a convex
uniform 4-polytope
In geometry, a uniform 4-polytope (or uniform polychoron) is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedron, uniform polyhedra, and faces are regular polygons.
There are 47 non-Prism (geometry), prism ...
. It is formed as two parallel
heptagonal antiprisms connected by cubes and
triangular prism
In geometry, a triangular prism or trigonal prism is a Prism (geometry), prism with 2 triangular bases. If the edges pair with each triangle's vertex and if they are perpendicular to the base, it is a ''right triangular prism''. A right triangul ...
s. The symmetry of a heptagonal antiprismatic prism is
4,2+,2 order 56. It has 28
triangle
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...
, 28
square
In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
and 4
heptagon
In geometry, a heptagon or septagon is a seven-sided polygon or 7-gon.
The heptagon is sometimes referred to as the septagon, using ''Wikt:septa-, septa-'' (an elision of ''Wikt:septua-, septua-''), a Latin-derived numerical prefix, rather than ...
al faces. It has 70 edges, and 28 vertices.
Octagonal antiprismatic prism
A octagonal antiprismatic prism or octagonal antiduoprism is a convex
uniform 4-polytope
In geometry, a uniform 4-polytope (or uniform polychoron) is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedron, uniform polyhedra, and faces are regular polygons.
There are 47 non-Prism (geometry), prism ...
(four-dimensional polytope). It is formed as two parallel
octagonal 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 connected by cubes and triangular prisms. The symmetry of an octagonal antiprismatic prism is
6,2+,2 order 64. It has 32
triangle
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...
, 32
square
In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
and 4
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 faces. It has 80 edges, and 32 vertices.
See also
*
Duoprism
In geometry of 4 dimensions or higher, a double prism or duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher. The Cartesian product of an -polytope and an -polytope is an -polytope, wher ...
References
*
John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, ''The Symmetries of Things'' 2008, (Chapter 26)
*
Norman Johnson ''Uniform Polytopes'', Manuscript (1991)
External links
* {{PolyCell , urlname = section6.html, title = 6. Convex uniform prismatic polychora
Uniform 4-polytopes