Five-dimensional space

For the musical group, see The 5th Dimension.

A 2D orthogonal projection of a 5-cube

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

Five-dimensional Euclidean geometry

5D Euclidean geometry designated by the mathematical sign: $\mathbb {E}$⁵ 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. 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.

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.

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 $\mathbb {C} ^{5}$. In quantum information theory, quantum systems described by quantum states belonging to $\mathbb {C} ^{5}$ are sometimes called ququints.

See also

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

Further reading

• Wesson, Paul S. (1999). Space-Time-Matter, Modern Kaluza-Klein Theory. Singapore: World Scientific. ISBN 981-02-3588-7.
• Wesson, Paul S. (2006). Five-Dimensional Physics: Classical and Quantum Consequences of Kaluza-Klein Cosmology. Singapore: World Scientific. ISBN 981-256-661-9.
• Weyl, Hermann, Raum, Zeit, Materie, 1918. 5 edns. to 1922 ed. with notes by Jūrgen Ehlers, 1980. trans. 4th edn. Henry Brose, 1922 Space Time Matter, Methuen, rept. 1952 Dover. ISBN 0-486-60267-2.

External links

• Anaglyph of a five dimensional hypercube in hyper perspective