Five-dimensional space

For the musical group, see The 5th Dimension.

A five-dimensional (5D) space is a space with five dimensions.

A 2D orthogonal projection of a 5-cube

Five-dimensional Euclidean geometry

5D Euclidean geometry designated by the mathematical sign: ${\displaystyle \mathbb {E} }$^{5 [1]} is dimensions beyond two (planar) and three (solid). Shapes studied in five dimensions include counterparts of regular polyhedra and of the sphere.

Polytopes

Main article: 5-polytope

In five or more dimensions, only three regular polytopes exist. In five dimensions, they are:

1. The 5-simplex of the simplex family, {3,3,3,3}, with 6 vertices, 15 edges, 20 faces (each an equilateral triangle), 15 cells (each a regular tetrahedron), and 6 hypercells (each a 5-cell).
2. The 5-cube of the hypercube family, {4,3,3,3}, with 32 vertices, 80 edges, 80 faces (each a square), 40 cells (each a cube), and 10 hypercells (each a tesseract).
3. The 5-orthoplex of the cross polytope family, {3,3,3,4}, with 10 vertices, 40 edges, 80 faces (each a triangle), 80 cells (each a tetrahedron), and 32 hypercells (each a 5-cell).

An important uniform 5-polytope is the 5-demicube, h{4,3,3,3} has half the vertices of the 5-cube (16), bounded by alternating 5-cell and 16-cell hypercells. The expanded or stericated 5-simplex is the vertex figure of the A₅ lattice, . It and has a doubled symmetry from its symmetric Coxeter diagram. The kissing number of the lattice, 30, is represented in its vertices.^[2] The rectified 5-orthoplex is the vertex figure of the D₅ lattice, . Its 40 vertices represent the kissing number of the lattice and the highest for dimension 5.^[3]

Regular and semiregular polytopes in five dimensions
(Displayed as orthogonal projections in each Coxeter plane of symmetry)

A5	Aut(A5)	B5			D5
5-simplex {3,3,3,3}	Stericated 5-simplex	5-cube {4,3,3,3}	5-orthoplex {3,3,3,4}	Rectified 5-orthoplex r{3,3,3,4}	5-demicube h{4,3,3,3}

Other five-dimensional geometries

The theory of special relativity makes use of Minkowski spacetime, a type of geometry that locates events in both space and time. The time dimension is mathematically distinguished from the spatial dimensions by a modification in the formula for computing the "distance" between events. Ordinary Minkowski spacetime has four dimensions in all, three of space and one of time. However, higher-dimensional generalizations of the concept have been employed in various proposals. Kaluza–Klein theory, a speculative attempt to develop a unified theory of gravity and electromagnetism, relied upon a spacetime with four dimensions of space and one of time.^[4]

Geometries can also be constructed in which the coordinates are something other than real numbers. For example, one can define a space in which the points are labeled by tuples of 5 complex numbers. This is often denoted ${\displaystyle \mathbb {C} ^{5}}$. In quantum information theory, quantum systems described by quantum states belonging to ${\displaystyle \mathbb {C} ^{5}}$ are sometimes called ququints.^[5][6]

See also

• 5-manifold
• Gravity
• Hypersphere
• List of regular 5-polytopes
• Four-dimensional space