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Framework for computer program optimization
The polyhedral model (also called the polytope method) is a mathematical framework for programs that perform large numbers of operations -- too large
Polytope_model
Geometric object with flat sides
In elementary geometry, a polytope is a geometric object with flat sides (faces). Polytopes are the generalization of three-dimensional polyhedra to any
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
Polytope with highest degree of symmetry
In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. In
Regular_polytope
Flat-sided three-dimensional shape
listing polyhedral books regarding the modeling, studies, and history of polyhedra Monostatic polytope, a polytope standing on one face only Gauss–Bonnet
Polyhedron
Convex hull of a finite set of points in a Euclidean space
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n {\displaystyle n} -dimensional
Convex_polytope
Polytope in 8-dimensional geometry
In 8-dimensional geometry, the 421 is a semiregular uniform 8-polytope, constructed within the symmetry of the E8 group. It was discovered by Thorold Gosset
4_21_polytope
regular polytopes in Euclidean, spherical and hyperbolic spaces. This table shows a summary of regular polytope counts by rank. There is only one polytope of
List_of_regular_polytopes
Use of the polyhedral model (also called the polytope model) within a compiler requires software to represent the objects of this framework (sets of integer-valued
Frameworks supporting the polyhedral model
Frameworks_supporting_the_polyhedral_model
the rectified 600-cell or rectified hexacosichoron is a convex uniform 4-polytope composed of 600 regular octahedra and 120 icosahedra cells. Each edge has
Rectified_600-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
Regular star 4-polytope with 600 faces
polytetrahedron is a regular star 4-polytope with Schläfli symbol {3, 3, 5/2}. It is one of 10 regular Schläfli-Hess polytopes. It is the only one with 600 cells
Grand_600-cell
a runcinated 120-cell (or runcinated 600-cell) is a convex uniform 4-polytope, being a runcination (a 3rd order truncation) of the regular 120-cell.
Runcinated_120-cells
Increasing execution speed and reducing the overheads associated with loops
research as of the time of this writing (2010). Loop nest optimization Polytope model Scalable parallelism Scalable locality In the book Reasoning About Program
Loop_optimization
Solid with 2 parallel n-gonal bases connected by n parallelograms
n-polytope elements are doubled from the (n − 1)-polytope elements and then creating new elements from the next lower element. Take an n-polytope with
Prism_(geometry)
Uniform polychoron
In four-dimensional geometry, the rectified 5-cell is a uniform 4-polytope composed of 5 regular tetrahedral and 5 regular octahedral cells. Each edge
Rectified_5-cell
Line segment joining two adjacent vertices in a polygon or polytope
polytope. In a polygon, an edge is a line segment on the boundary, and is often called a polygon side. In a polyhedron or more generally a polytope,
Edge_(geometry)
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
Plane figure bounded by line segments
single plane. A polygon is a 2-dimensional example of the more general polytope in any number of dimensions. There are many more generalizations of polygons
Polygon
Multi-dimensional generalization of triangle
dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. For example, a 0-dimensional simplex is a point
Simplex
First modern model of the atom
The plum pudding model is an obsolete scientific model of the atom. It was first proposed by J. J. Thomson in 1904 following his discovery of the electron
Plum_pudding_model
Polytope or tiling whose vertices are identical
In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries
Isogonal_figure
Solid with twenty equal triangular faces
background in the comparison mensuration. It is analogous to a four-dimensional polytope, the 600-cell. Regular icosahedra occur both in natural and human-made
Regular_icosahedron
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
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
3D shape made of polyhedra sharing a common center
regular polytopes. Coxeter lists a few of these in his book Regular Polytopes. McMullen added six in his paper New Regular Compounds of 4-Polytopes. Self-duals:
Polytope_compound
4D geometry item
four-dimensional geometry, a cantellated 120-cell is a convex uniform 4-polytope, being a cantellation (a 2nd order truncation) of the regular 120-cell
Cantellated_120-cell
Solid with eight equal triangular faces
segments. More generally, every cross-polytope and its dual, hypercube, in any higher-dimensional space are Hanner polytope. The polyhedral compounds, in which
Regular_octahedron
24-cell or rectified icositetrachoron is a uniform 4-dimensional polytope (or uniform 4-polytope), which is bounded by 48 cells: 24 cubes, and 24 cuboctahedra
Rectified_24-cell
Convex polytope of parenthesizations
In mathematics, an associahedron Kn is an (n − 2)-dimensional convex polytope in which each vertex corresponds to a way of correctly inserting opening
Associahedron
Polyhedron associated with another by swapping vertices for faces
of a polytope's dual will be the topological duals of the polytope's vertex figures. For the polar reciprocals of the regular and uniform polytopes, the
Dual_polyhedron
Geometric space with five dimensions
higher dimensions, including five-dimensional space. List of regular 5-polytopes — regular geometric shapes that exist in five-dimensional space. Four-dimensional
Five-dimensional_space
Polyhedron with 12 faces
No. 479 (Jul., 1993), pp. 220–226 [1] Stellation of Pyritohedron VRML models and animations of Pyritohedron and its stellations Klitzing, Richard. "3D
Dodecahedron
Extending the elements of a polytope to form a new figure
in two dimensions, a polyhedron in three dimensions, or, in general, a polytope in n dimensions to form a new figure. Starting with an original figure
Stellation
Shape made by slicing off a corner of a polytope
broadly speaking, is the figure exposed when a corner of a general n-polytope is sliced off. Take some corner or vertex of a polyhedron. Mark a point
Vertex_figure
Advanced method of process control
of all the regions. Every region turns out to geometrically be a convex polytope for linear MPC, commonly parameterized by coefficients for its faces, requiring
Model_predictive_control
Method to solve optimization problems
affine (linear) function defined on this polytope. A linear programming algorithm finds a point in the polytope where this function has the largest (or
Linear_programming
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
In model checking, a field of computer science, a region is a convex polytope in R d {\displaystyle \mathbb {R} ^{d}} for some dimension d {\displaystyle
Region_(model_checking)
Shape with four equal sides and angles
truncated square is an octagon. The square belongs to a family of regular polytopes that includes the cube in three dimensions and the hypercubes in higher
Square
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
In geometry, a truncated 24-cell is a uniform 4-polytope (4-dimensional uniform polytope) formed as the truncation of the regular 24-cell. There are two
Truncated_24-cells
Polygon with an infinite number of sides
polytope is a partially ordered set P (whose elements are called faces) with properties modeling those of the inclusions of faces of convex polytopes
Apeirogon
Geometry book
Regular Polytopes is a geometry book on regular polytopes written by Harold Scott MacDonald Coxeter. It was originally published by Methuen in 1947 and
Regular_Polytopes_(book)
Geometric space with six dimensions
simpler ones that model some aspect of the environment. Of particular interest is six-dimensional Euclidean space, in which 6-polytopes and the 5-sphere
Six-dimensional_space
Polyhedron with eight triangular faces
the three-dimensional case of an infinite family of regular polytopes, the cross polytopes. Although it does not tile space by itself, it can tile space
Octahedron
Type of tesseract
In geometry, a truncated tesseract is a uniform 4-polytope formed as the truncation of the regular tesseract. There are three truncations, including a
Truncated_tesseract
Shape with five sides
Richard Fitzpatrick. Lulu.com. p. 119. ISBN 978-0-615-17984-1. Mathematical Models by H. Martyn Cundy and A.P. Rollett, second edition, 1961 (Oxford University
Pentagon
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
Uniform 4-polytope
In geometry, a tetrahedral prism is a convex uniform 4-polytope. This 4-polytope has 6 polyhedral cells: 2 tetrahedra connected by 4 triangular prisms
Tetrahedral_prism
Shape with three equal sides
self-replicate itself. Equilateral triangles may also form a three-dimensional polytope, called a polyhedron. A polyhedron whose faces are all equilateral triangles
Equilateral_triangle
Type of non-Euclidean geometry
regions, where they locally resemble the hyperbolic plane. The hyperboloid model of hyperbolic geometry provides a representation of events one temporal
Hyperbolic_geometry
Geometric model of the physical space
open subset of 3D space. In three dimensions, there are nine regular polytopes: the five convex Platonic solids and the four nonconvex Kepler–Poinsot
Three-dimensional_space
Geometric structure used in certain particle interactions
algebraic geometry analogous to a convex polytope, that generalizes the idea of a simplex in projective space. A polytope is the n-dimensional analogue of a
Amplituhedron
In geometry, a truncated 5-cell is a uniform 4-polytope (4-dimensional uniform polytope) formed as the truncation of the regular 5-cell. There are two
Truncated_5-cell
Edge-joined polygons which fold into a polyhedron
"Unfolding", MathWorld Regular 4d Polytope Foldouts Editable Printable Polyhedral Nets with an Interactive 3D View Paper Models of Polyhedra Unfolder for Blender
Net_(polyhedron)
Regular Schläfli-Hess 4-polytope with 600 vertices
polydodecahedron is a regular star 4-polytope with Schläfli symbol {5/2,3,3}, one of 10 regular Schläfli-Hess 4-polytopes. It is unique among the 10 for having
Great grand stellated 120-cell
Great_grand_stellated_120-cell
Convex uniform 4-polytope
four-dimensional geometry, a cantellated tesseract is a convex uniform 4-polytope, being a cantellation (a 2nd order truncation) of the regular tesseract
Cantellated_tesseract
In geometry, a monostatic polytope or unistable polyhedron is a d {\displaystyle d} -polytope which "can stand on only one face". They were described
Monostatic_polytope
Regular star 4-polytope
regular star 4-polytope with Schläfli symbol {5/2,3,5}. It is one of 10 regular Schläfli-Hess polytopes. It is one of four regular star 4-polytopes discovered
Great_stellated_120-cell
geometry, the snub 24-cell or snub disicositetrachoron is a convex uniform 4-polytope composed of 120 regular tetrahedral and 24 icosahedral cells. Five tetrahedra
Snub_24-cell
Geometric prism
In geometry, an octahedral prism is a convex uniform 4-polytope. This 4-polytope has 10 polyhedral cells: 2 octahedra connected by 8 triangular prisms
Octahedral_prism
Representation of 3D and 4D polytopes
icosahedron edit In geometry, a Schlegel diagram is a projection of a polytope from R d {\textstyle \mathbb {R} ^{d}} into R d − 1 {\textstyle \mathbb
Schlegel_diagram
Complicated polygon
polydodecahedron is a regular star 4-polytope with Schläfli symbol {5/2,5,3}. It is one of 10 regular Schläfli-Hess polytopes. It has the same edge arrangement
Small_stellated_120-cell
four-dimensional geometry, a cantellated 24-cell is a convex uniform 4-polytope, being a cantellation (a 2nd order truncation) of the regular 24-cell.
Cantellated_24-cells
Number of windings of a polytope around its center of symmetry
ray from the center to infinity, passing only through the facets of the polytope and not through any lower dimensional features, and counting how many facets
Density_(polytope)
a runcinated tesseract (or runcinated 16-cell) is a convex uniform 4-polytope, being a runcination (a 3rd order truncation) of the regular tesseract
Runcinated_tesseracts
Compound polyhedron
vertices of all the other fifteen regular polytopes in four dimensions." Wenninger, Magnus (1974). Polyhedron Models. Cambridge University Press. ISBN 0-521-09859-9
Compound_of_five_tetrahedra
Polyhedron with regular congruent polygons as faces
face, an edge of the face, a vertex of the edge, and the null polytope. An abstract polytope is said to be regular if its combinatorial symmetries are transitive
Regular_polyhedron
In model checking, a field of computer science, a difference bound matrix (DBM) is a data structure used to represent some convex polytopes called zones
Difference_bound_matrix
Topics referred to by the same term
structure or topological triangulation E8 polytope, alternate name for the 421 semiregular (uniform) polytope Elementary abelian group of order 8 E8 Theory
E8
German geometer (1860–1934)
Elemente der vierdimensionalen Geometrie mit besonderer Berücksichtigung der Polytope, Jahresbericht des Vereins für Naturkunde zu Zwickau 1893 Vielecke und
Max_Brückner
Isogonal polyhedron with regular faces
polyhedron is a 2-dimensional abstract polytope with a non-degenerate 3-dimensional realization. Here an abstract polytope is a poset of its "faces" satisfying
Uniform_polyhedron
Uniform 4-polytope bounded by 320 cells
antiprism or pentagonal double antiprismoid is a uniform 4-polytope (4-dimensional uniform polytope) bounded by 320 cells: 20 pentagonal antiprisms, and 300
Grand_antiprism
Polytope whose vertices represent permutations
permutohedron (also spelled permutahedron) of order n is an (n − 1)-dimensional polytope embedded in an n-dimensional space. Its vertex coordinates (labels) are
Permutohedron
Shape with six sides
for these higher dimensional regular, uniform and dual polyhedra and polytopes, shown in these skew orthogonal projections: A principal diagonal of a
Hexagon
Topics referred to by the same term
geometry meaning an n-dimensional analogue of a triangle Simplicial polytope, a polytope with all simplex facets Simplicial complex, a collection of simplicies
Simplex_(disambiguation)
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 geometrical object
In four-dimensional geometry, a runcinated 5-cell is a convex uniform 4-polytope, being a runcination (a 3rd order truncation, up to face-planing) of the
Runcinated_5-cell
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
4-D uniform polytope
In geometry, a dodecahedral prism is a convex uniform 4-polytope. This 4-polytope has 14 polyhedral cells: 2 dodecahedra connected by 12 pentagonal prisms
Dodecahedral_prism
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
Space formed by the ''n''-tuples of real numbers
1\\\vdots \\|x_{n}|\leq 1\end{matrix}}} for [−1,1]. Each vertex of the cross-polytope has, for some k, the xk coordinate equal to ±1 and all other coordinates
Real_coordinate_space
Irish mathematician (1860–1940)
four-dimensional geometry from an early age, and introduced the term "polytope" for a convex solid in four or more dimensions. Alicia Boole was born in
Alicia_Boole_Stott
rhombicuboctahedral prism is a convex uniform polychoron (four-dimensional polytope). It is one of 18 convex uniform polyhedral prisms created by using uniform
Rhombicuboctahedral_prism
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
Geometric space with four dimensions
both synthetic and algebraic methods. He discovered all of the regular polytopes (higher-dimensional analogues of the Platonic solids) that exist in Euclidean
Four-dimensional_space
Pyramid with a square base
construction of other polyhedra and as the cell of a four-dimensional polytope called cubic pyramid. Square pyramidal number is a natural number that
Square_pyramid
Complex structures in matter physics
regular tetrahedra if the space is not Euclidean, but spherical. It is the polytope {3,3,5}, using the Schläfli notation, also known as the 600-cell. There
Geometrical_frustration
Rigidity theorem for convex polyhedra
theorem in geometry, named after Augustin Cauchy. It states that convex polytopes in three dimensions with congruent corresponding faces must be congruent
Cauchy's_theorem_(geometry)
Topics referred to by the same term
polygon, a polygon which encloses a convex set of points Convex polytope, a polytope with a convex set of points Convex metric space, a generalization
Convex
Closed volume that completely contains the union of a set of objects
the union of a finite set of points, its convex hull is a polytope. A discrete oriented polytope (DOP) generalizes the bounding box. A k-DOP is the Boolean
Bounding_volume
Subspace of n-space whose dimension is (n-1)
and the group of all motions is generated by the reflections. A convex polytope is the intersection of half-spaces. In non-Euclidean geometry, the ambient
Hyperplane
Type of cellular automaton
three-dimensional grid, the only regular polytope that fills the whole space is the cube, while the only regular polytopes with a sufficiently large symmetry
Lattice_gas_automaton
the rectified tesseract, rectified 8-cell is a uniform 4-polytope (4-dimensional polytope) bounded by 24 cells: 8 cuboctahedra, and 16 tetrahedra. It
Rectified_tesseract
Conic solid with a polygonal base
construction gets generalized to n dimensions. The base becomes a (n − 1)-polytope in a (n − 1)-dimensional hyperplane. A point called the apex is located
Pyramid_(geometry)
Topics referred to by the same term
line that runs between Massachusetts and Virginia Region (model checking), a convex polytope data structure Region-based memory management, a memory management
Region_(disambiguation)
four-dimensional geometry, a cantellated 5-cell is a convex uniform 4-polytope, being a cantellation (a 2nd order truncation, up to edge-planing) of the
Cantellated_5-cell
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
Isogonal honeycomb of uniform polytope facets
uniform tessellation or infinite uniform polytope, is a vertex-transitive honeycomb made from uniform polytope facets. All of its vertices are identical
Uniform_honeycomb
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