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 ...
, an octahedral prism is a convex
uniform 4-polytope. This 4-polytope has 10
polyhedral cells: 2
octahedra connected by 8
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''. A unif ...
s.
Alternative names
*Octahedral dyadic prism (
Norman W. Johnson
Norman Woodason Johnson () was a mathematician at Wheaton College, Norton, Massachusetts.
Early life and education
Norman Johnson was born on in Chicago. His father had a bookstore and published a local newspaper.
Johnson earned his unde ...
)
*Ope (Jonathan Bowers, for octahedral prism)
*Triangular antiprismatic prism
*Triangular antiprismatic hyperprism
Coordinates
It is a
Hanner polytope with vertex coordinates, permuting first 3 coordinates:
:(
±1,0,0 ±1)
Structure
The octahedral prism consists of two octahedra connected to each other via 8 triangular prisms. The triangular prisms are joined to each other via their square faces.
Projections
The octahedron-first
orthographic projection
Orthographic projection (also orthogonal projection and analemma) is a means of representing Three-dimensional space, three-dimensional objects in Two-dimensional space, two dimensions. Orthographic projection is a form of parallel projection in ...
of the octahedral prism into 3D space has an octahedral
envelope
An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter or card.
Traditional envelopes are made from sheets of paper cut to one of three shapes: a rhombus, a shor ...
. The two octahedral cells project onto the entire volume of this envelope, while the 8 triangular prismic cells project onto its 8 triangular faces.
The triangular-prism-first orthographic projection of the octahedral prism into 3D space has a
hexagonal prismic envelope. The two octahedral cells project onto the two hexagonal faces. One triangular prismic cell projects onto a triangular prism at the center of the envelope, surrounded by the images of 3 other triangular prismic cells to cover the entire volume of the envelope. The remaining four triangular prismic cells are projected onto the entire volume of the envelope as well, in the same arrangement, except with opposite orientation.
Related polytopes
It is the second in an infinite series of
uniform antiprismatic prisms.
It is one of 18 uniform polyhedral prisms created by using uniform
prism
Prism usually refers to:
* Prism (optics), a transparent optical component with flat surfaces that refract light
* Prism (geometry), a kind of polyhedron
Prism may also refer to:
Science and mathematics
* Prism (geology), a type of sedimentary ...
s to connect pairs of parallel
Platonic solid
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges c ...
s and
Archimedean solid
In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are composed ...
s.
It is one of four four-dimensional
Hanner polytopes; the other three are the
tesseract, the
16-cell
In geometry, the 16-cell is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mi ...
, and the dual of the octahedral prism (a
cubical bipyramid
In 4-dimensional geometry, the cubical bipyramid is the direct sum of a cube and a segment, + . Each face of a central cube is attached with two square pyramids, creating 12 square pyramidal cells, 30 triangular faces, 28 edges, and 10 vertices. A ...
).
[https://bendwavy.org/klitzing/explain/hanner.htm]
References
*
John H. Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English people, English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to ...
, Heidi Burgiel, Chaim Goodman-Strass, ''The Symmetries of Things'' 2008, (Chapter 26)
*
Norman Johnson ''Uniform Polytopes'', Manuscript (1991)
External links
*
*
{{DEFAULTSORT:Octahedral Prism
4-polytopes