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stacked polytope is a polytope formed from a simplex by repeatedly gluing another simplex onto one of its facets. Every simplex is itself a stacked polytope
Stacked_polytope
Four-dimensional analogue of the cube
labels it the γ4 polytope. The term hypercube without a dimension reference is frequently treated as a synonym for this specific polytope. The construction
Tesseract
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
Graph theory model
k-dimensional stacked polytopes, polytopes formed by starting from a simplex and then repeatedly gluing simplices onto the faces of the polytope, are k-trees
K-tree
Polyhedron with four faces
tetrahedron of the cube is an example of a Heronian tetrahedron. Every regular polytope, including the regular tetrahedron, has its characteristic orthoscheme
Tetrahedron
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
S2CID 118034273 Chen, Hao (2016), "Apollonian Ball Packings and Stacked Polytopes", Discrete & Computational Geometry, 55 (4): 801–826, arXiv:1306.2515
3-3_duoprism
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
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
Uniform 4-polytope bounded by 320 cells
location in the 4-polytope. Pentagonal double antiprismoid Norman W. Johnson Gap (Jonathan Bowers: for grand antiprism) 20 stacked pentagonal antiprisms
Grand_antiprism
Graph-theoretic description of polyhedra
00216, MR 2776000 Demaine, Erik D.; Schulz, André (2017), "Embedding stacked polytopes on a polynomial-size grid", Discrete & Computational Geometry, 57
Steinitz's_theorem
Linear stacking of regular tetrahedra that form helices
stacking of Platonic solids, the Boerdijk–Coxeter helix is not rotationally repetitive in 3-dimensional space. Even in an infinite string of stacked tetrahedra
Boerdijk–Coxeter_helix
Non-planar polygon with infinitely many sides
geometry, an infinite skew polygon or skew apeirogon is an infinite 2-polytope with vertices that are not all colinear. Infinite zig-zag skew polygons
Infinite_skew_polygon
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
Pyramid with a square base
four-dimensional polytope called cubic pyramid. Square pyramidal number is a natural number that counts the number of spheres stacked into a square pyramid
Square_pyramid
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
Solid with four equal triangular faces
tetrahedra can be stacked face-to-face in a chiral aperiodic chain called the Boerdijk–Coxeter helix. The pentachoron is a four-dimensional polytope, a generalization
Regular_tetrahedron
Tiling of euclidean or hyperbolic space of three or more dimensions
non-Euclidean spaces, such as hyperbolic honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical
Honeycomb_(geometry)
Algebraic variety containing an algebraic torus
class of toric varieties, this information is also encoded in a convex polytope, which creates a powerful connection of the subject with convex geometry
Toric_variety
different blockchains. Hyperbridge was launched on Polkadot mainnet in 2024 by Polytope Labs. The protocol is designed with a decentralised verification system
Hyperbridge
polygon (forming a Klemperer rosette), a Platonic solid, or a regular polytope in higher dimensions. The centrality of the configuration follows from
Central_configuration
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
Class of algorithms in computational geometry
hull is a convex polyhedron in three dimensions, or in general a convex polytope for any number of dimensions, whose vertices are some of the points in
Convex_hull_algorithms
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
2D shape constructed by joining together identical basic polygons
High-dimensional polyforms Edges Basic polytope (monoform) Monohedral honeycomb Polyform Applications 12 cube Cubille Polycubes: monocube, dicube, tricube
Polyform
tetrahedral cells, seen as two trihexagonal tilings. Two tetrahedra are stacked by a vertex and a central inversion. In each trihexagonal tiling, half
Quarter_cubic_honeycomb
Komei Fukuda in 1991, for problems of generating the vertices of convex polytopes and the cells of arrangements of hyperplanes. They were formalized more
Reverse-search_algorithm
NP-hard problem in combinatorial optimization
Mark (2017). "Short combinatorial proof that the DFJ polytope is contained in the MTZ polytope for the Asymmetric Traveling Salesman Problem". Operations
Travelling_salesman_problem
parallelohedron? Does every higher-dimensional tiling by translations of convex polytope tiles have an affine transformation taking it to a Voronoi diagram? Ropelength
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Polygon that is the boundary of a convex set
Convex polyhedron – Polyhedron that is the boundary of a convex set Convex polytope – Convex hull of a finite set of points in a Euclidean space Cyclic polygon –
Convex_polygon
Property of a mathematical space
Volume 4 dimensions Spacetime Fourth spatial dimension Convex regular 4-polytope Quaternion 4-manifold Polychoron Rotations in 4-dimensional Euclidean space
Dimension
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
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
Array of processing elements specialized for parallelizable workloads
processor Loop nest optimization Manycore processor Neural processing unit Polytope model Symmetric multiprocessing Systolic array Vision processing unit Chen
Spatial_architecture
Concept from linear programming
polyhedron. If it is bounded, then it is a convex polytope. Each BFS corresponds to a vertex of this polytope. As mentioned above, every basis B defines a
Basic_feasible_solution
Catalan solid with 12 faces
Padrol, Arnau; Ziegler, Günter M. (2016), "Six topics on inscribable polytopes", in Bobenko, Alexander I. (ed.), Advances in Discrete Differential Geometry
Triakis_tetrahedron
Length in a vector space
vectors whose 1-norm is a given constant forms the surface of a cross polytope, which has dimension equal to the dimension of the vector space minus 1
Norm_(mathematics)
Topological space that locally resembles Euclidean space
manifold is its Euler characteristic. Leonhard Euler showed that for a convex polytope in the three-dimensional Euclidean space with V vertices (or corners),
Manifold
Polyhedron with 8 rhombic and 4 hexagonal faces
quadrilaterals and four pentagonal faces. Coxeter, H.S.M. (1973). Regular Polytopes (3rd ed.). Dover Publications. p. 257. ISBN 0-486-61480-8. Akiyama, Jin;
Elongated_dodecahedron
Branch of mathematics
Archimedes, Plato, Euclid, and later Kepler and Coxeter all studied convex polytopes and their properties. From the 19th century on, mathematicians have studied
Geometry
Shape in hyperbolic geometry
having all degrees in this range does not guarantee an ideal realization. A stacked polyhedron is a polyhedron that can be obtained by gluing together tetrahedra
Ideal_polyhedron
Branch of mathematics
from them (e.g. by gluing along localizations or taking noncommutative stack quotients). For example, noncommutative algebraic geometry is supposed to
Noncommutative algebraic geometry
Noncommutative_algebraic_geometry
translation. Weisstein, Eric W. "Polytope Edge". From Wolfram MathWorld. TeX software, Draw cube with dashed hidden lines. From TeX StackExchange. "...needed to
Hidden_line
Covering by shapes without overlaps or gaps
pioneered this by defining polyschemes, which mathematicians nowadays call polytopes. These are the analogues to polygons and polyhedra in spaces with more
Tessellation
Infinite polyhedron with non-planar faces
Eric W. "Honeycombs and sponges". MathWorld. Olshevsky, George. "Skew polytope". Glossary for Hyperspace. Archived from the original on 4 February 2007
Skew_apeirohedron
recharacterisation of K-stability in terms of convex functions on the moment polytope of the toric variety, as was observed by Donaldson in his first paper on
K-stability_of_Fano_varieties
Mathematical set with some added structure
for example, it may be a polygon (in the plane) or, more generally, a polytope (in arbitrary finite dimension). The parallelogram law (called also parallelogram
Space_(mathematics)
Vertex-transitive tiling of the plane by regular polygons
polygons, pp. 82–85. Norman Johnson Uniform Polytopes, Manuscript (1991) N. W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph. D. Dissertation,
Uniform_tiling
Branch of mathematics
generalized to stacks in the sense of Grothendieck, usually with some additional representability conditions leading to Artin stacks and, even finer
Algebraic_geometry
24-dimensional repeating pattern of points
the Leech lattice. Starting with a single point, Λ0, stack copies of the lattice Λn can be stacked to form an (n + 1)-dimensional lattice, Λn+1, without
Leech_lattice
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
Mathematical model of the physical space
polytopes, which are the higher-dimensional analogues of polygons and polyhedra. He developed their theory and discovered all the regular polytopes,
Euclidean_geometry
Mathematical optimization problem restricted to integers
"FPT algorithm for mixed integer program". Theoretical Computer Science Stack Exchange. Retrieved 2024-05-21. Glover, F. (1989). "Tabu search-Part II"
Integer_programming
Sequence of operations for a task
p. 245, 4.7.1 Prune-and-search. For instance, the volume of a convex polytope (described using a membership oracle) can be approximated to high accuracy
Algorithm
Historical development of geometry
Platonic solids, finding that there are exactly six such regular convex polytopes in dimension four, and three in all higher dimensions. In 1878 William
History_of_geometry
Mathematics timeline
2018. Effenberger, Felix (2011). Hamiltonian Submanifolds of Regular Polytopes. Logos Verlag Berlin GmbH. p. 20. ISBN 9783832527587. Retrieved 15 June
Timeline_of_manifolds
Awarded every year by the American Mathematical Society
Heidelberg. ISBN 978-3-642-00117-8. Grünbaum, Branko (2003) [1967]. Convex Polytopes. Graduate Texts in Mathematics. Vol. 221 (2nd ed.). Springer New York
Leroy_P._Steele_Prize
STACKED POLYTOPE
STACKED POLYTOPE
Surname or Lastname
English
English : variant or patronymic form of Stack.
Surname or Lastname
English
English : topographic name for someone who lived by a prominent post or stake, for example a boundary marker, from Middle English stake ‘post’, ‘stake’, or from the same word used as a nickname for a tall, thin person.
Male
English
English unisex pet form of Latin Anastasia and Anastasius, both STACEY means "resurrection."
Girl/Female
Greek American
Shall be reborn.
Surname or Lastname
English and Irish
English and Irish : variant of Stacey.
Girl/Female
English American Greek
resurrection; or of Eustace: fruitful.
Boy/Male
American, Australian, British, English, Greek, Jamaican
Productive; Fertile; Resurrection; To Stand
Surname or Lastname
English
English : patronymic from Stark.
Female
English
Feminine variant spelling of English unisex Stacey, STACIE means "resurrection."
Surname or Lastname
English and Irish
English and Irish : from a pet form of the medieval male personal name Stace, a reduced vernacular form of Eustace.
Surname or Lastname
English and German
English and German : variant of Stock.Probably an Americanized form of Stokke.
Female
English
Feminine variant spelling of English unisex Stacey, STACEE means "resurrection."
Surname or Lastname
English
English : occupational name for someone who made and drove in stakes, or a topographic name for someone who lived near a boundary post for example, from a derivative of Middle English stake ‘post’, ‘stake’.
Boy/Male
Greek American English
Fruitful, productive. Famous bearer: St Eustace (Eustachins) was a martyred 2nd century Roman...
Surname or Lastname
English
English : unexplained.
Girl/Female
Australian, Greek, Latin
Resurrection; Form of Anastasia
Surname or Lastname
English
English : variant of Styles.German : topographic name for someone who lived on or by a hill, from Middle High German stickel ‘hill’, ‘slope’.German : nickname from Middle High German stickel ‘prickle’, ‘spine’, ‘pointed object’.
Surname or Lastname
English
English : habitational name from Stiffkey in Norfolk (pronounced Stuckey), so named from Old English styfic ‘tree stumps’ + ēg ‘island’ or ‘higher ground in a marsh’.Americanized spelling of German Stucki.
Surname or Lastname
English
English : from a diminutive of Stark 1.Altered spelling of German Starke, written thus to preserve the second syllable.
Girl/Female
American, Australian, British, English, French, Greek, Latin
Resurrection; Fruitful; Shall be Reborn; Form of Anastasia; Giving Fruit
STACKED POLYTOPE
STACKED POLYTOPE
STACKED POLYTOPE
STACKED POLYTOPE
STACKED POLYTOPE
STACKED POLYTOPE
STACKED POLYTOPE