In the

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 ...

of 4 dimensions, the 3-3 duoprism or triangular duoprism is a four-dimensional convex polytope. It can be constructed as the

Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ti ...

of two triangles and is the simplest of an infinite family of four-dimensional polytopes constructed as Cartesian products of two polygons, the

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 ...

s. It has 9 vertices, 18 edges, 15 faces (9

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 adj ...

s, and 6

triangle A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), 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, an ...

s), in 6

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 ...

cells. It has

Coxeter diagram Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington to ...

, and symmetry , order 72. Its vertices and edges form a

3\times 3

rook's graph In graph theory, a rook's graph is a graph that represents all legal moves of the rook chess piece on a chessboard. Each vertex of a rook's graph represents a square on a chessboard, and each edge connects two squares on the same row (rank) or on ...

Hypervolume

The

hypervolume A four-dimensional space (4D) is a mathematical extension of the concept of three-dimensional or 3D space. Three-dimensional space is the simplest possible abstraction of the observation that one only needs three numbers, called ''dimensions'', ...

of a

uniform A uniform is a variety of clothing worn by members of an organization while participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, se ...

3-3 duoprism, with edge length ''a'', is

V_4 = a^4

. This is the square of the area of an equilateral triangle,

A = a^2

Graph

The graph of vertices and edges of the 3-3 duoprism has 9 vertices and 18 edges. Like the

Berlekamp–van Lint–Seidel graph In graph theory, the Berlekamp–Van Lint–Seidel graph is a locally linear strongly regular graph with parameters (243,22,1,2). This means that it has 243 vertices, 22 edges per vertex (for a total of 2673 edges), exactly one shared neighbor per ...

and the unknown solution to

Conway's 99-graph problem In graph theory, Conway's 99-graph problem is an unsolved problem asking whether there exists an undirected graph with 99 vertices, in which each two adjacent vertices have exactly one common neighbor, and in which each two non-adjacent vertices ...

, every edge is part of a unique triangle and every non-adjacent pair of vertices is the diagonal of a unique square. It is a

toroidal graph In the mathematical field of graph theory, a toroidal graph is a graph that can be embedded on a torus. In other words, the graph's vertices can be placed on a torus such that no edges cross. Examples Any graph that can be embedded in a plane ...

, a

locally linear graph In graph theory, a locally linear graph is an undirected graph in which every edge belongs to exactly one triangle. Equivalently, for each vertex of the graph, its neighbors are each adjacent to exactly one other neighbor, so the neighbors can be ...

, a

strongly regular graph In graph theory, a strongly regular graph (SRG) is defined as follows. Let be a regular graph with vertices and degree . is said to be strongly regular if there are also integers and such that: * Every two adjacent vertices have commo ...

with parameters (9,4,1,2), the

3\times 3

, and the

Paley graph In mathematics, Paley graphs are dense undirected graphs constructed from the members of a suitable finite field by connecting pairs of elements that differ by a quadratic residue. The Paley graphs form an infinite family of conference graphs, whic ...

of order 9. This graph is also the

Cayley graph In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem (named after Arthur Cayle ...

of the group

G=\langle a,b:a^3=b^3=1,\ ab=ba\rangle\simeq C_3\times C_3

with generating set

S=\

Images

Symmetry

In 5-dimensions, the some

uniform 5-polytope In geometry, a uniform 5-polytope is a five-dimensional uniform polytope. By definition, a uniform 5-polytope is vertex-transitive and constructed from uniform 4-polytope Facet (geometry), facets. The complete set of convex uniform 5-polytopes ...

s have 3-3 duoprism

vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...

s, some with unequal edge-lengths and therefore lower symmetry: The birectified 16-cell honeycomb also has a 3-3 duoprism

s. There are three constructions for the honeycomb with two lower symmetries.

Related complex polygons

The

regular complex polytope In geometry, a complex polytope is a generalization of a polytope in real space to an analogous structure in a complex Hilbert space, where each real dimension is accompanied by an imaginary one. A complex polytope may be understood as a collecti ...

₃₂, , in

\mathbb^2

has a real representation as a 3-3

in 4-dimensional space. ₃₂ has 9 vertices, and 6 3-edges. Its symmetry is ₃ sub>2, order 18. It also has a lower symmetry construction, , or ₃×₃, with symmetry ₃ sub>3, order 9. This is the symmetry if the red and blue 3-edges are considered distinct.

Related polytopes

3-3 duopyramid

The dual of a ''3-3 duoprism'' is called a 3-3

duopyramid In geometry of 4 dimensions or higher, a double pyramid or duopyramid or fusil is a polytope constructed by 2 orthogonal polytopes with edges connecting all pairs of vertices between the two. The term fusil is used by Norman Johnson as a rhom ...

or triangular duopyramid. It has 9

tetragonal disphenoid In geometry, a disphenoid () is a tetrahedron whose four Face (geometry), faces are Congruence (geometry), congruent acute-angled triangles. It can also be described as a tetrahedron in which every two Edge (geometry), edges that are opposite ea ...

cells, 18 triangular faces, 15 edges, and 6 vertices. It can be seen in orthogonal projection as a 6-gon circle of vertices, and edges connecting all pairs, just like a

5-simplex In five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope. It has six vertices, 15 edges, 20 triangle faces, 15 tetrahedral cells, and 6 5-cell facets. It has a dihedral angle of cos−1(), or approximately 78.46°. The 5-s ...

seen in projection. : 3-3 duopyramid ortho

orthogonal projection In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it wer ...

Related complex polygon

The

regular complex polygon In geometry, a regular complex polygon is a generalization of a regular polygon in real space to an analogous structure in a complex Hilbert space, where each real dimension is accompanied by an imaginary one. A regular polygon exists in 2 real ...

₂₃ has 6 vertices in

\mathbb^2

with a real representation in

\mathbb^4

matching the same

vertex arrangement In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes. For example, a ''square vertex arrangement'' is understood to mean four points in a plane, equ ...

of the 3-3 duopyramid. It has 9 2-edges corresponding to the connecting edges of the 3-3 duopyramid, while the 6 edges connecting the two triangles are not included. It can be seen in a hexagonal projection with 3 sets of colored edges. This arrangement of vertices and edges makes a

complete bipartite graph In the mathematical field of graph theory, a complete bipartite graph or biclique is a special kind of bipartite graph where every vertex of the first set is connected to every vertex of the second set..Electronic edition page 17. Graph theory ...

with each vertex from one triangle is connected to every vertex on the other. It is also called a

Thomsen graph __NOTOC__ Thomsen is a Danish patronymic surname meaning "son of Tom (or Thomas)", itself derived from the Aramaic תום or ''Tôm'', meaning "twin". There are many varied surname spellings, with the first historical record believed to be found ...

or 4-''cage''.Regular Complex Polytopes, p.110, p.114

Notes

References

*''Regular Polytopes'',

H. S. M. Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...

, Dover Publications, Inc., 1973, New York, p. 124. *

Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington to ...

, ''The Beauty of Geometry: Twelve Essays'', Dover Publications, 1999, (Chapter 5: Regular Skew Polyhedra in three and four dimensions and their topological analogues) ** Coxeter, H. S. M. ''Regular Skew Polyhedra in Three and Four Dimensions.'' Proc. London Math. Soc. 43, 33-62, 1937. *

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) ** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. Dissertation, University of Toronto, 1966 * {{PolyCell , urlname =section6.html, title = Catalogue of Convex Polychora, section 6
Apollonian Ball Packings and Stacked Polytopes
Discrete & Computational Geometry, June 2016, Volume 55, Issue 4, pp 801–826

External links

The Fourth Dimension Simply Explained
mdash;describes duoprisms as "double prisms" and duocylinders as "double cylinders"

– glossary of higher-dimensional terms
Exploring Hyperspace with the Geometric Product
4-polytopes