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Eight-dimensional space, the Glossary

Index Eight-dimensional space

In mathematics, a sequence of n real numbers can be understood as a location in n-dimensional space.[1]

Table of Contents

  1. 43 relations: Algebra of physical space, Biquaternion, Clifford algebra, Complex number, Coxeter element, Coxeter group, Coxeter–Dynkin diagram, Dimension (vector space), Division algebra, E8 (mathematics), Euclidean distance, Euclidean space, Field (mathematics), Harold Scott MacDonald Coxeter, Hurwitz's theorem (composition algebras), Isomorphism, Kissing number, Lattice (group), Manifold, Mathematics, N-sphere, Neil Sloane, Pauli matrices, Point (geometry), Polytope, Quaternion, Real number, Regular polytope, Spacetime algebra, Special relativity, Springer Science+Business Media, Uniform 8-polytope, Vector space, Warren Siegel, William Rowan Hamilton, 1 42 polytope, 2 41 polytope, 240 (number), 4 21 polytope, 8-cube, 8-demicube, 8-orthoplex, 8-simplex.

  2. 8 (number)
  3. Dimension
  4. Multi-dimensional geometry

Algebra of physical space

In physics, the algebra of physical space (APS) is the use of the Clifford or geometric algebra Cl3,0(R) of the three-dimensional Euclidean space as a model for (3+1)-dimensional spacetime, representing a point in spacetime via a paravector (3-dimensional vector plus a 1-dimensional scalar).

See Eight-dimensional space and Algebra of physical space

Biquaternion

In abstract algebra, the biquaternions are the numbers, where, and are complex numbers, or variants thereof, and the elements of multiply as in the quaternion group and commute with their coefficients.

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Clifford algebra

In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra with the additional structure of a distinguished subspace.

See Eight-dimensional space and Clifford algebra

Complex number

In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted, called the imaginary unit and satisfying the equation i^.

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Coxeter element

In mathematics, a Coxeter element is an element of an irreducible Coxeter group which is a product of all simple reflections.

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Coxeter group

In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors).

See Eight-dimensional space and Coxeter group

Coxeter–Dynkin diagram

In geometry, a Coxeter–Dynkin diagram (or Coxeter diagram, Coxeter graph) is a graph with numerically labeled edges (called branches) representing a Coxeter group or sometimes a uniform polytope or uniform tiling constructed from the group.

See Eight-dimensional space and Coxeter–Dynkin diagram

Dimension (vector space)

In mathematics, the dimension of a vector space V is the cardinality (i.e., the number of vectors) of a basis of V over its base field. Eight-dimensional space and dimension (vector space) are dimension.

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Division algebra

In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible.

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E8 (mathematics)

In mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8.

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Euclidean distance

In mathematics, the Euclidean distance between two points in Euclidean space is the length of the line segment between them.

See Eight-dimensional space and Euclidean distance

Euclidean space

Euclidean space is the fundamental space of geometry, intended to represent physical space.

See Eight-dimensional space and Euclidean space

Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers.

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Harold Scott MacDonald Coxeter

Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician.

See Eight-dimensional space and Harold Scott MacDonald Coxeter

Hurwitz's theorem (composition algebras)

In mathematics, Hurwitz's theorem is a theorem of Adolf Hurwitz (1859–1919), published posthumously in 1923, solving the Hurwitz problem for finite-dimensional unital real non-associative algebras endowed with a nondegenerate positive-definite quadratic form.

See Eight-dimensional space and Hurwitz's theorem (composition algebras)

Isomorphism

In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping.

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Kissing number

In geometry, the kissing number of a mathematical space is defined as the greatest number of non-overlapping unit spheres that can be arranged in that space such that they each touch a common unit sphere.

See Eight-dimensional space and Kissing number

Lattice (group)

In geometry and group theory, a lattice in the real coordinate space \mathbb^n is an infinite set of points in this space with the properties that coordinate-wise addition or subtraction of two points in the lattice produces another lattice point, that the lattice points are all separated by some minimum distance, and that every point in the space is within some maximum distance of a lattice point.

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Manifold

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.

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Mathematics

Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.

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N-sphere

In mathematics, an -sphere or hypersphere is an -dimensional generalization of the -dimensional circle and -dimensional sphere to any non-negative integer. Eight-dimensional space and n-sphere are multi-dimensional geometry.

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Neil Sloane

Neil James Alexander Sloane FLSW (born October 10, 1939) is a British-American mathematician.

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Pauli matrices

In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices that are traceless, Hermitian, involutory and unitary.

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Point (geometry)

In geometry, a point is an abstract idealization of an exact position, without size, in physical space, or its generalization to other kinds of mathematical spaces.

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Polytope

In elementary geometry, a polytope is a geometric object with flat sides (faces).

See Eight-dimensional space and Polytope

Quaternion

In mathematics, the quaternion number system extends the complex numbers.

See Eight-dimensional space and Quaternion

Real number

In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature.

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Regular polytope

In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry.

See Eight-dimensional space and Regular polytope

Spacetime algebra

In mathematical physics, spacetime algebra (STA) is the application of Clifford algebra Cl1,3(R), or equivalently the geometric algebra to physics.

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Special relativity

In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between space and time.

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Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.

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Uniform 8-polytope

In eight-dimensional geometry, an eight-dimensional polytope or 8-polytope is a polytope contained by 7-polytope facets.

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Vector space

In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', can be added together and multiplied ("scaled") by numbers called ''scalars''.

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Warren Siegel

Warren Siegel is a theoretical physicist specializing in supersymmetric quantum field theory and string theory.

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William Rowan Hamilton

Sir William Rowan Hamilton (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist.

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1 42 polytope

In 8-dimensional geometry, the 142 is a uniform 8-polytope, constructed within the symmetry of the E8 group.

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2 41 polytope

In 8-dimensional geometry, the 241 is a uniform 8-polytope, constructed within the symmetry of the E8 group.

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240 (number)

240 (two hundred forty) is the natural number following 239 and preceding 241.

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4 21 polytope

In 8-dimensional geometry, the 421 is a semiregular uniform 8-polytope, constructed within the symmetry of the E8 group.

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8-cube

In geometry, an 8-cube is an eight-dimensional hypercube.

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8-demicube

In geometry, a demiocteract or 8-demicube is a uniform 8-polytope, constructed from the 8-hypercube, octeract, with alternated vertices removed.

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8-orthoplex

In geometry, an 8-orthoplex or 8-cross polytope is a regular 8-polytope with 16 vertices, 112 edges, 448 triangle faces, 1120 tetrahedron cells, 1792 5-cells 4-faces, 1792 5-faces, 1024 6-faces, and 256 7-faces.

See Eight-dimensional space and 8-orthoplex

8-simplex

In geometry, an 8-simplex is a self-dual regular 8-polytope.

See Eight-dimensional space and 8-simplex

See also

8 (number)

Dimension

Multi-dimensional geometry

References

[1] https://en.wikipedia.org/wiki/Eight-dimensional_space

Also known as 8-dimensional space, 8th dimension, Eight-dimensional, Eighth dimension.