Five-dimensional space#Other five-dimensional geometries

5D Euclidean geometry designated by the mathematical sign: $\backslash mathbb\{E\}$⁵ {{Cite journal|last1=Güler|first1= Erhan|location=Bartın University|date=2024|title= A helicoidal hypersurfaces family in five-dimensional euclidean space|journal=Filomat|volume= 38 |issue=11|page=3814 (4^th para.;1^st sent.)|doi=10.2298/FIL2411813G |doi-access=free}} is dimensions beyond two (planar) and three (solid). Shapes studied in five dimensions include counterparts of regular polyhedra and of the sphere.

{{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, {{CDD|node_1|split1|nodes|3ab|nodes|split2|node}}. It and has a doubled symmetry from its symmetric Coxeter diagram. The kissing number of the lattice, 30, is represented in its vertices.{{Cite web|url=https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A5.html|title=The Lattice A5|website=www.math.rwth-aachen.de}} The rectified 5-orthoplex is the vertex figure of the D₅ lattice, {{CDD|nodes_10ru|split2|node|3|node|split1|nodes}}. Its 40 vertices represent the kissing number of the lattice and the highest for dimension 5.Sphere packings, lattices, and groups, by John Horton Conway, Neil James Alexander Sloane, Eiichi Bannai

[https://books.google.com/books?id=upYwZ6cQumoC&dq=Sphere%20Packings%2C%20Lattices%20and%20Groups&pg=PR19]

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.{{cite book|first=Barton |last=Zwiebach |author-link=Barton Zwiebach |title=A First Course in String Theory |publisher=Cambridge University Press |year=2004 |isbn=0-521-83143-1 |pages=14–16, 399}}

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 $\backslash mathbb\{C\}^5$. In quantum information theory, quantum systems described by quantum states belonging to $\backslash mathbb\{C\}^5$ are sometimes called ququints.{{cite journal|last1=Jain |first1=Akalank |last2=Shiroman |first2=Prakash |title=Qutrit and ququint magic states |journal=Physical Review A |volume=102 |number=4 |year=2020 |page=042409 |arxiv=2003.07164 |doi=10.1103/PhysRevA.102.042409}}{{cite website|first=Davide |last=Castelvecchi |title=Meet ‘qudits’: more complex cousins of qubits boost quantum computing |website=Nature |date=2025-03-25 |access-date=2025-05-11 |url=https://www.nature.com/articles/d41586-025-00939-x}}

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

{{Reflist|30em}}

• {{cite book |title=Space-Time-Matter, Modern Kaluza-Klein Theory |last=Wesson |first=Paul S. |year=1999 |publisher=World Scientific |location=Singapore |isbn=981-02-3588-7 |url=https://archive.org/details/spacetimematterm0000wess |url-access=registration}}
• {{cite book |title=Five-Dimensional Physics: Classical and Quantum Consequences of Kaluza-Klein Cosmology |last=Wesson |first=Paul S. |year=2006 |publisher=World Scientific |location=Singapore |isbn=981-256-661-9 |url=https://archive.org/details/fivedimensionalp0000wess |url-access=registration}}
• Weyl, Hermann, [https://archive.org/details/raumzeitmateriev00weyl Raum, Zeit, Materie], 1918. 5 edns. to 1922 ed. with notes by Jūrgen Ehlers, 1980. trans. 4th edn. Henry Brose, 1922 [https://archive.org/details/spacetimematter00weyluoft Space Time Matter], Methuen, rept. 1952 Dover. {{ISBN|0-486-60267-2}}.

• [https://robertinventor.com/software/virtualflower/virtualflower/tesseract.htm Anaglyph of a five dimensional hypercube in hyper perspective]

{{Dimension topics}}

Category:Dimension

Category:Multi-dimensional geometry

Category:5 (number)