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QUATERNIONIC POLYTOPE

Quaternionic polytope

Concept in geometry

In geometry, a quaternionic polytope is a generalization of a polytope in real space to an analogous structure in a quaternionic module, where each real

Quaternionic polytope

Quaternionic_polytope

Complex polytope

Generalization of a polytope in real space

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

Complex polytope

Complex_polytope

Quaternion

Four-dimensional number system

Quaternionic manifold – Concept in geometry Quaternionic matrix – Concept in linear algebra Quaternionic polytope – Concept in geometry Quaternionic projective

Quaternion

Truncated 24-cells

(2002), No. 1, 243-259 ] Table 2, page 252 Quaternionic Construction of the W(F4) Polytopes with Their Dual Polytopes and Branching under the Subgroups W(B4)

Truncated 24-cells

Truncated_24-cells

120-cell

Four-dimensional analog of the dodecahedron

Al-Ajmi, Mudhahir; Ozdes Koca, Nazife (2011). "Quaternionic representation of snub 24-cell and its dual polytope derived from E8 root system". Linear Algebra

120-cell

600-cell

Four-dimensional analog of the icosahedron

Al-Ajmi, Mudhahir; Ozdes Koca, Nazife (2011). "Quaternionic representation of snub 24-cell and its dual polytope derived from E8 root system". Linear Algebra

600-cell

24-cell

Regular object in four dimensional geometry

In four-dimensional geometry, the 24-cell is a convex regular 4-polytope, a four-dimensional analogue of a Platonic solid. It is named for the 24 octahedra

24-cell

24 (number)

Natural number

24-cell, consisting of 24 octahedra and having 24 vertices, is a special polytope that only exists in four dimensions. The vertices of the 24-cell are the

24 (number)

24_(number)

Sedenion

Hypercomplex number system

Degrees of freedom Polytopes and shapes Hyperplane Hypersurface Hypercube Hyperrectangle Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid

Sedenion

Dual snub 24-cell

Al-Ajmi, Mudhahir; Ozdes Koca, Nazife (2011). "Quaternionic representation of snub 24-cell and its dual polytope derived from E 8 {\displaystyle E_{8}} root

Dual snub 24-cell

Dual_snub_24-cell

Three-dimensional space

Geometric model of the physical space

5. ISBN 978-0-19-960139-4. Morais, João Pedro; et al. (2014). Real Quaternionic Calculus Handbook. Springer Science & Business Media. pp. 1–13. ISBN 978-3-0348-0622-0

Three-dimensional space

Three-dimensional_space

Tridiminished icosahedron

63rd Johnson solid (8 faces)

Al-Ajmi, Mudhahir; Koca, Nazife Ozdes (2011). "Quaternionic representation of snub 24-cell and its dual polytope derived from E 8 {\displaystyle \mathrm {E}

Tridiminished icosahedron

Tridiminished_icosahedron

Enneahedron

Polyhedron with 9 faces

Al-Ajmi, Mudhahir; Ozdes Koca, Nazife (2011), "Quaternionic representation of snub 24-cell and its dual polytope derived from E 8 {\displaystyle E_{8}} root

Enneahedron

Complex geometry

Study of complex manifolds and several complex variables

complex structures I , J , K {\displaystyle I,J,K} which satisfy the quaternionic relations I 2 = J 2 = K 2 = I J K = − Id {\displaystyle

Complex geometry

Complex_geometry

Snub 24-cell

Al-Ajmi, Mudhahir; Ozdes Koca, Nazife (2011). "Quaternionic representation of snub 24-cell and its dual polytope derived from E8 root system". Linear Algebra

Snub 24-cell

Snub_24-cell

Hypercomplex number

Element of a unital algebra over the field of real numbers

{\displaystyle \mathbb {H} ^{\otimes 3}=M(4,\mathbb {H} )} yields a quaternionic matrix and its even subalgebra H ⊗ 2 ⊗ R C {\displaystyle \mathbb {H}

Hypercomplex number

Hypercomplex_number

N-sphere

Generalized sphere of dimension n (mathematics)

⁠-sphere, Lie group structure Sp(1) = SU(2). 4-sphere Homeomorphic to the quaternionic projective line, ⁠ H P 1 {\displaystyle \mathbf {HP} ^{1}} ⁠. ⁠ SO ⁡

N-sphere

Sporadic group

Finite simple group type not classified as Lie, cyclic or alternating

a type 2-3-3 triangle J2 is the group of automorphisms preserving a quaternionic structure (modulo its center). Consists of subgroups which are closely

Sporadic group

Sporadic_group

Octonion

Hypercomplex number system

basis with signature (− − − −) and is given in terms of the following 7 quaternionic triples (omitting the scalar identity element): ( I , j , k ) , ( i

Octonion

Glossary of areas of mathematics

geometry used to describe the physical phenomena of quantum physics Quaternionic analysis Ramsey theory the study of the conditions in which order must

Glossary of areas of mathematics

Glossary_of_areas_of_mathematics

Spin group

Double cover Lie group of the special orthogonal group

group (symmetries of the hypercube, or equivalently of its dual, the cross-polytope). For point groups that reverse orientation, the situation is more complicated

Spin group

Spin_group

Norman Johnson (mathematician)

American mathematician (1930–2017)

of Toronto in 1966 with a dissertation title of The Theory of Uniform Polytopes and Honeycombs under the supervision of H. S. M. Coxeter.[T66] From there

Norman Johnson (mathematician)

Norman_Johnson_(mathematician)

Projective space

Completion of the usual space with "points at infinity"

naturally to the case where K is a division ring; see, for example, Quaternionic projective space. The notation PG(n, K) is sometimes used for Pn(K).

Projective space

Projective_space

Coxeter notation

Classification system for symmetry groups in geometry

elements can be seen in ringed nodes Coxeter-Dynkin diagram for uniform polytopes and honeycomb are related to hole nodes around the + elements, empty circles

Coxeter notation