Triaprism
Encyclopedia
In geometry
of 6 dimensions or higher, a triaprism (or triprism) is a polytope
resulting from the Cartesian product
of three polytopes, each of two dimensions or higher. The Cartesian product of an a-polytope, a b-polytope, and a c-polytope is an (a+b+c)-polytope, where a, b and c are 2-polytopes (polygon
) or higher.
The term triaprism is coined by George Olshevsky
, shortened from triple prism, similar to duoprism
for the product of two polytopes. Conway
proposed a similar name proprism for product prism, thus applying to any number of products.
The lowest dimensional triaprisms exist in 6-dimensional space as 6-polytope
s being the Cartesian product
of three polygon
s in 2-dimensional Euclidean space
.
The smallest is a 3-3-3 triaprism or (triangle-triangle-triangle-triaprism), being the product of three triangles. It has 5-faces (3-3 duoprism prisms), 36 4-faces (9 3-3 duoprisms, 27 3-4 duoprism), 81 cells (27 cube
s, 54 triangular prism
s), 108 faces (81 squares
, 27 triangle
s), 81 edges
, and 27 vertices
.
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
of 6 dimensions or higher, a triaprism (or triprism) is a polytope
Polytope
In elementary geometry, a polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions...
resulting from the Cartesian product
Cartesian product
In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
of three polytopes, each of two dimensions or higher. The Cartesian product of an a-polytope, a b-polytope, and a c-polytope is an (a+b+c)-polytope, where a, b and c are 2-polytopes (polygon
Polygon
In geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...
) or higher.
The term triaprism is coined by George Olshevsky
George Olshevsky
George Olshevsky is a freelance editor, writer, publisher, amateur paleontologist, and mathematician living in San Diego, California.Olshevsky maintains the comprehensive online Dinosaur Genera List...
, shortened from triple prism, similar to duoprism
Duoprism
In geometry of 4 dimensions or higher, a duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher...
for the product of two polytopes. Conway
Conway
-Surname:* Alan Conway, impersonator of Stanley Kubrick* Albert Conway , Chief Judge of the New York Court of Appeals * Anne Conway, Viscountess Conway , English philosopher* Arthur Conway , any of several men...
proposed a similar name proprism for product prism, thus applying to any number of products.
The lowest dimensional triaprisms exist in 6-dimensional space as 6-polytope
6-polytope
In six-dimensional geometry, a uniform polypeton is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform polytera....
s being the Cartesian product
Cartesian product
In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
of three polygon
Polygon
In geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...
s in 2-dimensional Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
.
The smallest is a 3-3-3 triaprism or (triangle-triangle-triangle-triaprism), being the product of three triangles. It has 5-faces (3-3 duoprism prisms), 36 4-faces (9 3-3 duoprisms, 27 3-4 duoprism), 81 cells (27 cube
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. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...
s, 54 triangular prism
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....
s), 108 faces (81 squares
Square (geometry)
In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...
, 27 triangle
Triangle
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
s), 81 edges
Edge (geometry)
In geometry, an edge is a one-dimensional line segment joining two adjacent zero-dimensional vertices in a polygon. Thus applied, an edge is a connector for a one-dimensional line segment and two zero-dimensional objects....
, and 27 vertices
Vertex (geometry)
In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...
.
See also
- Uniform 6-polytope#Uniform prismatic families
- 7-polytope#Uniform triaprismatic forms
- 8-polytope#Uniform triaprismatic forms