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# 3-3 duoprism

In geometry of 4 dimensions, a 3-3 duoprism or triangular duoprism, the smallest p-q duoprism, is a 4-polytope resulting from the Cartesian product of two triangles. [1]

## Birectified 16-cell honeycomb

In four-dimensional Euclidean geometry, the birectified 16-cell honeycomb (or runcic tesseractic honeycomb) is a uniform space-filling tessellation (or honeycomb) in Euclidean 4-space.

## Cartesian product

In set theory (and, usually, in other parts of mathematics), a Cartesian product is a mathematical operation that returns a set (or product set or simply product) from multiple sets.

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## 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.

## Convex polytope

A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn.

## Coxeter notation

In geometry, Coxeter notation (also Coxeter symbol) is a system of classifying symmetry groups, describing the angles between with fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a Coxeter-Dynkin diagram, with modifiers to indicate certain subgroups.

## Coxeter–Dynkin diagram

In geometry, a Coxeter–Dynkin diagram (or Coxeter diagram, Coxeter graph) is a graph with numerically labeled edges (called branches) representing the spatial relations between a collection of mirrors (or reflecting hyperplanes).

## Disphenoid

In geometry, a disphenoid (from Greek sphenoeides, "wedgelike") is a tetrahedron whose four faces are congruent acute-angled triangles.

## Dual polyhedron

In geometry, any polyhedron is associated with a second dual figure, where the vertices of one correspond to the faces of the other and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other.

## Duocylinder

The duocylinder, or double cylinder, is a geometric object embedded in 4-dimensional Euclidean space, defined as the Cartesian product of two disks of respective radii r1 and r2: It is analogous to a cylinder in 3-space, which is the Cartesian product of a disk with a line segment.

## 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.

## Duopyramid

In geometry of 4 dimensions or higher, a duopyramid or fusil is a polytope constructed by 2 orthogonal polytopes with edges connecting all pairs of vertices between the two.

## Four-dimensional space

A four-dimensional space or 4D space is a mathematical extension of the concept of three-dimensional or 3D space.

## Geometry

Geometry (from the γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.

## Harold Scott MacDonald Coxeter

Harold Scott MacDonald "Donald" Coxeter, FRS, FRSC, (February 9, 1907 &ndash; March 31, 2003) was a British-born Canadian geometer.

## Isogonal figure

In geometry, a polytope (a polygon, polyhedron or tiling, for example) is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure.

## Isohedral figure

In geometry, a polytope of dimension 3 (a polyhedron) or higher is isohedral or face-transitive when all its faces are the same.

## Isosceles triangle

In geometry, an isosceles triangle is a triangle that has two sides of equal length.

## John Horton Conway

John Horton Conway FRS (born 26 December 1937) is an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory.

## Norman Johnson (mathematician)

Norman Woodason Johnson (November 12, 1930 – July 13, 2017) was a mathematician, previously at Wheaton College, Norton, Massachusetts.

## Projection (linear algebra)

In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that.

## Rectified 5-simplexes

In five-dimensional geometry, a rectified 5-simplex is a convex uniform 5-polytope, being a rectification of the regular 5-simplex.

## Regular 4-polytope

In mathematics, a regular 4-polytope is a regular four-dimensional polytope.

## 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.

## Runcinated 5-cubes

In five-dimensional geometry, a runcinated 5-cube is a convex uniform 5-polytope that is a runcination (a 3rd order truncation) of the regular 5-cube.

## Runcinated 5-orthoplexes

In five-dimensional geometry, a runcinated 5-orthoplex is a convex uniform 5-polytope with 3rd order truncation (runcination) of the regular 5-orthoplex.

## Runcinated 5-simplexes

In six-dimensional geometry, a runcinated 5-simplex is a convex uniform 5-polytope with 3rd order truncations (Runcination) of the regular 5-simplex.

## Schläfli symbol

In geometry, the Schläfli symbol is a notation of the form that defines regular polytopes and tessellations.

## Schlegel diagram

In geometry, a Schlegel diagram is a projection of a polytope from R^d into R^ through a point beyond one of its facets or faces.

## Square

In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, or (100-gradian angles or right angles). It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices ABCD would be denoted.

## Tesseract

In geometry, the tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square.

## Three utilities problem

The classical mathematical puzzle known as the three utilities problem; the three cottages problem or sometimes water, gas and electricity can be stated as follows: The problem is an abstract mathematical puzzle which imposes constraints that would not exist in a practical engineering situation.

## Triangle

A triangle is a polygon with three edges and three vertices.

## 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.

## Uniform 5-polytope

In geometry, a uniform 5-polytope is a five-dimensional uniform polytope.

## Vertex arrangement

In geometry, a vertex arrangement is a set of points in space described by their relative positions.

## Vertex figure

In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.

## 3-3 duoprism

In geometry of 4 dimensions, a 3-3 duoprism or triangular duoprism, the smallest p-q duoprism, is a 4-polytope resulting from the Cartesian product of two triangles.

## 3-4 duoprism

In geometry of 4 dimensions, a 3-4 duoprism, the second smallest p-q duoprism, is a 4-polytope resulting from the Cartesian product of a triangle and a square.

## 4-polytope

In geometry, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope.

## 5-5 duoprism

In geometry of 4 dimensions, a 5-5 duoprism or pentagonal duoprism is a polygonal duoprism, a 4-polytope resulting from the Cartesian product of two pentagons.

## 5-simplex

In five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope.

## References

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