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Uniform polytope in 8 dimensional geometry
In 8-dimensional geometry, the 241 is a uniform 8-polytope, constructed within the symmetry of the E8 group. Its Coxeter symbol is 241, describing its
2_41_polytope
Uniform 8 dimensional polytope
In 8-dimensional geometry, the 142 is a uniform 8-polytope, constructed within the symmetry of the E8 group. Its Coxeter symbol is 142, describing its
1_42_polytope
Four-dimensional geometric object with flat sides
In geometry, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope. It is a connected and closed figure
4-polytope
Uniform polytope
In geometry, 2k1 polytope is a uniform polytope in n dimensions (n = k + 4) constructed from the En Coxeter group. The family was named by their Coxeter
Uniform_2_k1_polytope
Seven-dimensional geometric object
7-polytope is a polytope contained by 6-polytope facets. Each 5-polytope ridge being shared by exactly two 6-polytope facets. A uniform 7-polytope is
Uniform_7-polytope
Four-dimensional analogues of the regular polyhedra in three dimensions
In mathematics, a regular 4-polytope or regular polychoron is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular
Regular_4-polytope
Four-dimensional analog of the icosahedron
In geometry, the 600-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,5}. It is also known
600-cell
Four-dimensional analog of the dodecahedron
In geometry, the 120-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {5,3,3}. It is also called
120-cell
Class of 4-dimensional polytopes
In geometry, a uniform 4-polytope (or uniform polychoron) is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra
Uniform_4-polytope
Four-dimensional analog of the octahedron
convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,4}. It is one of the six regular convex 4-polytopes first described
16-cell
In 8-dimensional geometry, there are 135 uniform polytopes with A8 symmetry. There is one self-dual regular form, the 8-simplex with 9 vertices. Each
A8_polytope
Convex polytope constructed recursively
geometry, a Hanner polytope is a convex polytope constructed recursively by Cartesian product and polar dual operations. Hanner polytopes are named after
Hanner_polytope
Four-dimensional analogue of the tetrahedron
In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol {3,3,3}. It is a 5-vertex four-dimensional object bounded by five tetrahedral cells
5-cell
Polytope contained by 7-polytope facets
eight-dimensional polytope or 8-polytope is a polytope contained by 7-polytope facets, each 6-polytope ridge being shared by exactly two 7-polytope facets. A
Uniform_8-polytope
Isogonal polytope with uniform facets
In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. Here, "vertex-transitive" means
Uniform_polytope
Maths conjecture
symmetric polytope have at least 3 d {\displaystyle 3^{d}} nonempty faces? More unsolved problems in mathematics In geometry, more specifically in polytope theory
Kalai's_3^d_conjecture
Type of geometric object
nine-dimensional polytope or 9-polytope is a polytope contained by 8-polytope facets. Each 7-polytope ridge being shared by exactly two 8-polytope facets. A
Uniform_9-polytope
Polyhedron with four faces
and Geometry. 13 (2): 379–400. doi:10.4310/cag.2005.v13.n2.a5. ISSN 1019-8385. MR 2154824. Park, Poo-Sung (2016). "Regular polytope distances" (PDF).
Tetrahedron
Group of polytopes
In 8-dimensional geometry, there are 255 uniform polytopes with B8 symmetry (to which this article adds for illustration the 8-demicube as an alternation
B8_polytope
Uniform 6-dimensional polytope
uniform 6-polytope is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform 5-polytopes. The complete
Uniform_6-polytope
Uniform 6-polytope
is cos−1(1/6), or approximately 80.41°. It can also be called a heptapeton, or hepta-6-tope, as a 7-facetted polytope in 6-dimensions. The name heptapeton
6-simplex
Uniform polytopes with D8 symmetry
In 8-dimensional geometry, there are 191 uniform polytopes with D8 symmetry, of which 64 are unique and 127 are shared with the B8 symmetry. There is
D8_polytope
Five-dimensional geometric shape
5-polytope is a five-dimensional uniform polytope. By definition, a uniform 5-polytope is vertex-transitive and constructed from uniform 4-polytope facets
Uniform_5-polytope
N-dimensional generalisation of a pyramid
construction gets generalised to n dimensions. The base becomes a (n – 1)-polytope in a (n – 1)-dimensional hyperplane. A point called apex is located outside
Hyperpyramid
six-dimensional geometry, a rectified 6-simplex is a convex uniform 6-polytope, being a rectification of the regular 6-simplex. There are three unique
Rectified_6-simplexes
In 7-dimensional geometry, there are 71 uniform polytopes with A7 symmetry. There is one self-dual regular form, the 7-simplex with 8 vertices. Each can
A7_polytope
Polyhedron formed by joining mirroring pyramids base-to-base
C_{\overline {AE}}&\leq 2\pi .\end{aligned}}} A generalized n-dimensional "bipyramid" is any n-polytope constructed from an (n − 1)-polytope base lying in a hyperplane
Bipyramid
vertices of any convex polytope into a set of vectors or points in a space of a different dimension, the Gale diagram of the polytope. It can be used to describe
Gale_diagram
algebraic combinatorics, the h-vector of a simplicial polytope is a fundamental invariant of the polytope which encodes the number of faces of different dimensions
H-vector
In 6-dimensional geometry, there are 64 uniform polytopes with B6 symmetry. There are two regular forms, the 6-orthoplex, and 6-cube with 12 and 64 vertices
B6_polytope
Combinitorics of Polyhedra
convex polytopes. Research in polyhedral combinatorics falls into two distinct areas. Mathematicians in this area study the combinatorics of polytopes; for
Polyhedral_combinatorics
Uniform 4-polytope
In geometry, a truncated 120-cell is a uniform 4-polytope formed as the truncation of the regular 120-cell. There are three truncations, including a bitruncation
Truncated_120-cells
icosahedron, is a simplicial 2-sphere. More generally, the boundary of any (d+1)-dimensional compact (or bounded) simplicial convex polytope in the Euclidean space
Simplicial_sphere
Isogonal polyhedron with regular faces
terminology, a polyhedron is a 2-dimensional abstract polytope with a non-degenerate 3-dimensional realization. Here an abstract polytope is a poset of its "faces"
Uniform_polyhedron
In 7-dimensional geometry, there are 128 uniform polytopes with B7 symmetry. There are two regular forms, the 7-orthoplex, and 8-cube with 14 and 128
B7_polytope
is an orientation of the edges of a polytope such that, in every face of the polytope (including the whole polytope as one of the faces), there is exactly
Unique_sink_orientation
Irrational system of points and lines
eight-dimensional convex polytope that cannot be given rational number coordinates and that has the fewest vertices (twelve) of any known irrational polytope. The Perles
Perles_configuration
Relation between sides of a right triangle
c 2 = a 2 + b 2 − K 3 a 2 b 2 − K 2 45 a 2 b 2 ( a 2 + b 2 ) − 2 K 3 945 a 2 b 2 ( a 2 − b 2 ) 2 + O ( K 4 c 10 ) . {\displaystyle c^{2}=a^{2}+b^{2}-{\frac
Pythagorean_theorem
Uniform 7-polytope
In geometry, a demihepteract or 7-demicube is a uniform 7-polytope, constructed from the 7-hypercube (hepteract) with alternated vertices removed. It is
7-demicube
Canadian geometer (1907–2003)
author of 12 books, including The Fifty-Nine Icosahedra (1938) and Regular Polytopes (1947). Many concepts in geometry and group theory are named after him
H.S.M._Coxeter
between the numbers of faces of different dimension of a simplicial polytope. For polytopes of dimension 4 and 5, they were found by Max Dehn in 1905. Their
Dehn–Sommerville_equations
Two raised to an integer power
n-dimensional cross-polytope is also 2n and the formula for the number of x-faces an n-dimensional cross-polytope has is 2 x ( n x ) . {\displaystyle 2^{x}{\tbinom
Power_of_two
Greek-French composer, architect and engineer (1922–2001)
Xenakis's UPIC system; and the massive multimedia performances Xenakis called polytopes, that were a summa of his interests and skills. Among the numerous theoretical
Iannis_Xenakis
Algorithm for linear programming
neighborhoods of the vertices) of a geometric object called a polytope. The shape of this polytope is defined by the constraints applied to the objective function
Simplex_algorithm
Polytope made by turning a polytope's facets into pyramids
Kleetope of a polyhedron or higher-dimensional convex polytope P is another polyhedron or polytope PK formed by replacing each facet of P with a pyramid
Kleetope
Prime number of the form 2^n – 1
prime are 2, 2, 2, 3, 2, 2, 7, 2, 2, 3, 2, 17, 3, 2, 2, 5, 3, 2, 5, 2, 2, 229, 2, 3, 3, 2, 3, 3, 2, 2, 5, 3, 2, 3, 2, 2, 3, 3, 2, 7, 2, 3, 37, 2, 3, 5, 58543
Mersenne_prime
Shape with four equal sides and angles
2 + d 3 2 = d 2 2 + d 4 2 = 2 ( R 2 + L 2 ) {\displaystyle d_{1}^{2}+d_{3}^{2}=d_{2}^{2}+d_{4}^{2}=2(R^{2}+L^{2})} and d 1 2 d 3 2 + d 2 2 d 4 2 = 2 (
Square
Any of 4 regular star polyhedra
Regular polytope Regular polyhedron List of regular polytopes Uniform polyhedron Uniform star polyhedron Polyhedral compound Regular star 4-polytope – the
Kepler–Poinsot_polyhedron
Kepler–Poinsot polyhedron with 20 faces
via the extension of the (n–1)-dimensional simplex faces of the core n-polytope (equilateral triangles for the great icosahedron, and line segments for
Great_icosahedron
Polyhedron which tiles 3D space
Computational Geometry. 41 (2): 232–248. arXiv:0710.3857. doi:10.1007/s00454-008-9086-6. Lagarias, J. C.; Moews, D. (1995). "Polytopes that fill R n {\displaystyle
Space-filling_polyhedron
Cycle graph with all opposite nodes linked
"New facets of the linear ordering polytope". SIAM Journal on Discrete Mathematics. 12 (3): 326–336. CiteSeerX 10.1.1.41.8722. doi:10.1137/S0895480196300145
Möbius_ladder
Polytope
mathematics, the permutoassociahedron is an n {\displaystyle n} -dimensional polytope whose vertices correspond to the bracketings of the permutations of n +
Permutoassociahedron
Balanced complete multipartite graph
Roberts graph. This graph is also the 1-skeleton of an n-dimensional cross-polytope; for instance, the graph T(6,3) = K2,2,2 is the octahedral graph, the graph
Turán_graph
Quasiregular space-filling tesselation
non-Euclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical
Tetrahedral-octahedral honeycomb
Tetrahedral-octahedral_honeycomb
Line constructed from a triangle
simplicial polytope is a polytope whose facets are all simplices (plural of simplex). For example, every polygon is a simplicial polytope. The Euler line
Euler_line
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)
Number, approximately 1.618
Kepler triangle and Penrose tilings too, as well as in various other polytopes. Dividing by interior division Having a line segment A B {\displaystyle
Golden_ratio
Kepler–Poinsot polyhedron
two-dimensional analogue, via the extension of the edges (1-faces) of the core polytope until a point is reached where they intersect. The small stellated dodecahedron
Small_stellated_dodecahedron
Type of mathematical set
complexes can be thought of as triangulations and provide a definition of polytopes. A facet is a maximal simplex, i.e., any simplex in a complex that is
Simplicial_complex
Number of paths between grid corners, allowing diagonal steps
{\displaystyle n} , the points in an m-dimensional integer lattice or cross polytope which are at most n steps from the origin, and, in cellular automata, the
Delannoy_number
torus topology, with Euler characteristic of zero. Density: the Density (polytope) represents the number of windings of a polyhedron around its center. This
List_of_uniform_polyhedra
Method for producing composition algebras
hypercomplex number multiplication". Przegląd Elektrotechniczny. 1 (2). Wydawnictwo SIGMA-NOT: 38–41. doi:10.15199/48.2015.02.09. ISSN 0033-2097. Richard D. Schafer
Cayley–Dickson_construction
Spatial tiling of convex uniform polyhedra
zonohedra. 1900: Thorold Gosset enumerated the list of semiregular convex polytopes with regular cells (Platonic solids) in his publication On the Regular
Convex_uniform_honeycomb
Type of metric geometry
symmetric, under the taxicab distance, the shape of a sphere is a cross-polytope, the n-dimensional generalization of a regular octahedron, whose points
Taxicab_geometry
"Finiteness property of pairs of 2 × 2 sign-matrices via real extremal polytope norms". Linear Algebra and its Applications. 432 (2–3): 796–816. doi:10.1016/j
Joint_spectral_radius
Infinitely detailed mathematical structure
and methodological implications". The Journal of Physiology. 587 (15): 3929–41. doi:10.1113/jphysiol.2009.169219. PMC 2746620. PMID 19528254. Liu, Jing Z
Fractal
Formula for area of a grid polygon
continuous volume of a polytope", pp. 76–77 Diaz, Ricardo; Robins, Sinai (1997). "The Ehrhart polynomial of a lattice polytope". Annals of Mathematics
Pick's_theorem
Geometric operation which truncates the edges of polyhedra
(geometry) Conway polyhedron notation Near-miss Johnson solid Uniform 4-polytope Uniform polyhedron Spencer 1911, p. 575, or p. 597 on Wikisource, Crystallography
Chamfer_(geometry)
Goldberg polyhedron with 42 faces
cell-centered orthogonal projection of the 120-cell, one of six convex regular 4-polytopes. The chamfered dodecahedron is the shape of the fullerene C80. Occasionally
Chamfered_dodecahedron
Solid with six equal square faces
quadrilateral faces are squares. It is a three-dimensional hypercube, a family of polytopes that also includes the two-dimensional square and four-dimensional tesseract
Cube
Exponential function of an exponential function
factors are known to be at most 2 4 n {\displaystyle 2^{4^{n}}} , a result of Nielsen (2003). The maximal volume of a polytope in a d-dimensional integer lattice
Double_exponential_function
Graph made from vertices and edges of a convex polyhedron
graphs are the 3-vertex-connected, planar graphs. The analogue concept for polytopes of general dimension are the polytopal graphs. The Schlegel diagram of
Polyhedral_graph
Only regular space-filling tessellation of the cube
non-Euclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical
Cubic_honeycomb
Group of symmetries of an n-dimensional hypercube
well as the corresponding dual polytopes (the regular octahedron and its higher-dimensional counterparts, the cross-polytopes). There is one hyperoctahedral
Hyperoctahedral_group
Graph formed by subdivision of triangles
graphs, the uniquely 4-colorable planar graphs, and the graphs of stacked polytopes. They are named after Apollonius of Perga, who studied a related circle-packing
Apollonian_network
Recursively-formed graph with two terminal vertices
operations. A 2-connected graph is series–parallel if and only if there are no R-nodes in its SPQR tree. Threshold graph Cograph Hanner polytope Series-parallel
Series–parallel_graph
the regular 4-polytope 24-cell, {3,4,3} with octahedral (3-orthoplex) cell, and cube {4,3}, with (2-orthoplex) square faces. It has a 2-dimensional analogue
16-cell_honeycomb
Computer science award
Rothvoss, Thomas (2017). "The Matching Polytope has Exponential Extension Complexity". Journal of the ACM. 64 (6): 41:1–41:19. arXiv:1311.2369. doi:10.1145/3127497
Gödel_Prize
Type of uniform space-filling tessellation
Publication, 1995, ISBN 978-0-471-01003-6 [2] (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] Conway JH
5-demicubic_honeycomb
Convex polyhedron projected from hypercube
three-dimensional; in any dimension, the Minkowski sum of line segments forms a convex polytope known as a zonotope. The original motivation for studying zonohedra is
Zonohedron
Formula for the "volume" of an n-simplex
{\displaystyle c} , 16 A 2 = | 2 a 2 a 2 + b 2 − c 2 a 2 + b 2 − c 2 2 b 2 | = 4 a 2 b 2 − ( a 2 + b 2 − c 2 ) 2 = ( a 2 + b 2 + c 2 ) 2 − 2 ( a 4 + b 4 + c 4
Cayley–Menger_determinant
Partition of a sphere's surface into polygons
McMullen, Peter; Schulte, Egon (2002). "6C. Projective Regular Polytopes". Abstract Regular Polytopes. Cambridge University Press. pp. 162–5. ISBN 0-521-81496-0
Spherical_polyhedron
Graph coloring where graph elements are assigned sets of colors
polynomial time. This is a straightforward consequence of Edmonds' matching polytope theorem. Applications of fractional graph coloring include activity scheduling
Fractional_coloring
Field of mathematics using techniques from combinatorics and commutative algebra
commutative algebra is the characterization of h-vectors of simplicial polytopes conjectured in 1970 by Peter McMullen. Known as the g-theorem, it was
Combinatorial commutative algebra
Combinatorial_commutative_algebra
four-dimensional crystal classes 1985 H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, Coxeter notation for 4D point groups 2003 John Conway and Smith, On
Point groups in four dimensions
Point_groups_in_four_dimensions
Natural number
6 edges. In four dimensions, there are a total of six convex regular polytopes. In the classification of finite simple groups, twenty of twenty-six sporadic
6
Double cover Lie group of the special orthogonal group
corresponding to the 2-fold covers of the hyperoctahedral group (symmetries of the hypercube, or equivalently of its dual, the cross-polytope). For point groups
Spin_group
Smallest convex set containing a given set
\mathbb {R} ^{d}} forms a convex polygon when d = 2 {\displaystyle d=2} , or more generally a convex polytope in R d {\displaystyle \mathbb {R} ^{d}} . Each
Convex_hull
of centrally symmetric polytopes. The Kobon triangle problem on triangles in line arrangements The Kusner conjecture: at most 2 d {\displaystyle 2d} points
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Tessellation of convex uniform polyhedron cells
regular polytopes#Tessellations of hyperbolic 3-space Uniform honeycombs in hyperbolic space P. Tumarkin, Hyperbolic Coxeter n-polytopes with n+2 facets
Paracompact uniform honeycombs
Paracompact_uniform_honeycombs
Element of a unital algebra over the field of real numbers
2 = i 2 2 = i 3 2 = − 1 {\displaystyle i_{1}^{2}=i_{2}^{2}=i_{3}^{2}=-1} , i 4 2 = i 5 2 = i 6 2 = i 7 2 = + 1 {\displaystyle i_{4}^{2}=i_{5}^{2
Hypercomplex_number
Branch of differential geometry and differential topology
Russian). 41 (6(252)): 3–18. doi:10.1070/RM1986v041n06ABEH004221. ISSN 0036-0279. S2CID 250908036 – via Russian Mathematical Surveys, 1986, 41:6, 1–21.
Symplectic_geometry
Uniform Euclidean 3D tessellations and their duals
of 3-space. Geombinatorics 4, 49 – 56. Norman Johnson (1991) Uniform Polytopes, Manuscript A. Andreini, (1905) Sulle reti di poliedri regolari e semiregolari
Architectonic and catoptric tessellation
Architectonic_and_catoptric_tessellation
Convex polyhedron with regular faces
by gyration, diminishment, or dissection. Near-miss Johnson solid Blind polytope Araki, Yoshiaki; Horiyama, Takashi; Uehara, Ryuhei (2015). "Common Unfolding
Johnson_solid
Catalan solid with 12 faces
1007/s00283-010-9138-7, hdl:1773/15593, MR 2747698 Coxeter, Harold (1973), Regular polytopes, Dover Publications Economic Mineralogy: A Practical Guide to the Study
Rhombic_dodecahedron
Rational numbers with root 5 added
2 5 = a 1 a 2 − 5 b 1 b 2 a 2 2 − 5 b 2 2 + − a 1 b 2 + b 1 a 2 a 2 2 − 5 b 2 2 5 , a 1 + b 1 φ a 2 + b 2 φ = a 1 a 2 + a 1 b 2 − b 1 b 2 a 2 2 + a 2
Golden_field
Regular Polytopes (1963), Macmillan Company Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 (Table I: Regular Polytopes, (i) The
Table of polyhedron dihedral angles
Table_of_polyhedron_dihedral_angles
Type of topological space
the necessary use of abstract set theory" (PDF). Advances in Mathematics. 41 (3): 209–280. doi:10.1016/0001-8708(81)90021-9. Retrieved 19 December 2022
Hilbert_cube
Polyhedron with 8 triangles and 6 squares
equilateral polytopes are those that can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing
Cuboctahedron
ISBN 978-0-471-01003-6 [2] Archived 2016-07-11 at the Wayback Machine (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940)
Quarter_cubic_honeycomb
2 41-POLYTOPE
2 41-POLYTOPE
Surname or Lastname
English
English : patronymic from Lakin 2.
Surname or Lastname
English
English : variant of Glad 2.
Surname or Lastname
North German variant of Laas 2.Jewish (Ashkenazic)
North German variant of Laas 2.Jewish (Ashkenazic) : unexplained.English : nickname from Middle English lesse, lasse ‘smaller’ (from Old English lǣssa ‘less’), perhaps also used in the sense ‘younger’.
Surname or Lastname
Variant of Nicolai 2.English
Variant of Nicolai 2.English : variant of Nicholas.
Surname or Lastname
English
English : variant of Land 2.
Surname or Lastname
English
English : patronymic from Lamb 2.
Surname or Lastname
English
English : patronymic from Lamb 2.
Girl/Female
Indian
Mixture of 2 Names
Surname or Lastname
English
English : variant of Mixon 2.
Surname or Lastname
English
English : variant of Haddock 2.
Surname or Lastname
English
English : variant of Maul 2.
Surname or Lastname
English
English : variant of Hackett 2.
Surname or Lastname
English
English : variant of Garrett 2.
Surname or Lastname
English
English : variant of Goodall 2.
Surname or Lastname
English
English : variant of Diamond 2.
Surname or Lastname
English
English : variant of Hayden 2.
Surname or Lastname
English
English : patronymic from Lamb 2.
Surname or Lastname
English
English : variant of Greenfield 2.
Surname or Lastname
English
English : patronymic from Lamb 2.
Boy/Male
Shakespearean
King Henry IV, Part 1' Earl of March. Scroop.
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