In 4-dimensional
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 uniform antiprismatic prism or antiduoprism is a
uniform 4-polytope with two
uniform antiprism cells in two parallel 3-space
hyperplane
In geometry, a hyperplane is a subspace whose dimension is one less than that of its '' ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hype ...
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, 2 ''p''-gonal
prism, and ''2p''
triangular prism
In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides. A right triangular prism has rectangular sides, otherwise it is ''oblique''. ...
cells. It has 4''p''
equilateral triangle
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...
, 4''p''
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
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, with two of the tetrahedral cells degenerated into squares. The ''triangular antiprismatic prism'' is the first nondegenerate form, which is also an
octahedral prism. 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 antiprisms 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 edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non- colli ...
, 16
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
and 4
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
faces. It has 40 edges, and 16 vertices.
Pentagonal antiprismatic prism
A pentagonal antiprismatic prism or pentagonal antiduoprism is a convex
uniform 4-polytope. It is formed as two parallel
pentagonal antiprisms 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 edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non- colli ...
, 20
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
and 4
pentagon
In geometry, a pentagon (from the Greek language, Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is ...
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. It is formed as two parallel
hexagonal antiprisms 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 edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non- colli ...
, 24
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
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'' h ...
faces. It has 60 edges, and 24 vertices.
Heptagonal antiprismatic prism
A heptagonal antiprismatic prism or heptagonal antiduoprism is a convex
uniform 4-polytope. It is formed as two parallel
heptagonal antiprisms connected by cubes and
triangular prism
In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides. A right triangular prism has rectangular sides, otherwise it is ''oblique''. ...
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 edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non- colli ...
, 28
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
and 4
heptagonal faces. It has 70 edges, and 28 vertices.
Octagonal antiprismatic prism
A octagonal antiprismatic prism or octagonal antiduoprism is a convex
uniform 4-polytope (four-dimensional polytope). It is formed as two parallel
octagonal antiprisms 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 edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non- colli ...
, 32
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
and 4
octagon
In geometry, an octagon (from the Greek ὀκτάγωνον ''oktágōnon'', "eight angles") 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, wh ...
al faces. It has 80 edges, and 32 vertices.
See also
*
Duoprism
References
*
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ''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
4-polytopes