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The study of integer points in convex polyhedra is motivated by questions such as "how many nonnegative integer-valued solutions does a system of linear
Integer points in convex polyhedra
Integer_points_in_convex_polyhedra
Flat-sided three-dimensional shape
a convex polyhedron defined in three-dimensional hyperbolic space. Lattice polyhedra are the convex polyhedra that can be constructed with integers coordinates
Polyhedron
Natural number
number of sides and the number of diagonals of a convex n-gon.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Bryan Bunch, The Kingdom
5
Formula for area of a grid polygon
grid points in circles. The problem of counting integer points in convex polyhedra arises in several areas of mathematics and computer science. In application
Pick's_theorem
Mathematical set closed under positive linear combinations
positive coefficients. It follows that convex cones are convex sets. The definition of a convex cone makes sense in a vector space over any ordered field
Convex_cone
Any of the five regular polyhedra
length. In Proposition 18 he argues that there are no further convex regular polyhedra. Andreas Speiser has advocated the view that the construction of
Platonic_solid
Origami, Polyhedra. Cambridge University Press. pp. 306–338. Ghomi, Mohammad (2018-01-01). "Dürer's Unfolding Problem for Convex Polyhedra". Notices
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Convex polytope whose vertices all have integer Cartesian coordinates
Cartesian coordinates. That is, it is a polytope that equals the convex hull of its integer points. Integral polytopes are also called lattice polytopes or Z-polytopes
Integral_polytope
Cuboid with all right angles and equal opposite faces
cuboid to refer to a more general class of polyhedra with six quadrilateral faces. A rectangular cuboid is a convex polyhedron with six rectangle faces. The
Rectangular_cuboid
The five convex regular polyhedra are called the Platonic solids. The vertex figure is given with each vertex count. All these polyhedra have an Euler
List_of_regular_polytopes
Plane figure bounded by line segments
through only interior points between its endpoints. This condition is true for polygons in any geometry, not just Euclidean. Non-convex: a line may be found
Polygon
Convex hull of a finite set of points in a Euclidean space
bounded convex polytopes can be found in convex polyhedra and convex polygon. In the 2-dimensional case the full-dimensional examples of unbounded convex polytopes
Convex_polytope
Shape with equal angles and equal sides
polyhedron which has just one kind of face. The remaining (non-uniform) convex polyhedra with regular faces are known as the Johnson solids. A polyhedron having
Regular_polygon
Class of 4-dimensional polytopes
uniform polyhedra, and faces are regular polygons. There are 47 non-prismatic convex uniform 4-polytopes. There are two infinite sets of convex prismatic
Uniform_4-polytope
Topological invariant in mathematics
in 1537 in an unpublished manuscript by Francesco Maurolico. Leonhard Euler, for whom the concept is named, introduced it for convex polyhedra more generally
Euler_characteristic
Geometric object with flat sides
star polyhedra and other unusual constructions led to the idea of a polyhedron as a bounding surface, ignoring its interior. In this light convex polytopes
Polytope
Solid with twenty equal triangular faces
components are elementaries—they cannot be disintegrated into smaller convex polyhedra with regular faces again. Replacing bases of a pentagonal antiprism
Regular_icosahedron
Shape with five sides
diagonals must be rational. See also Cyclic polygon § Integer area and side lengths. For all convex pentagons with sides a , b , c , d , e {\displaystyle
Pentagon
Solid with six equal square faces
polyhedron with six quadrilaterals (four-sided polygons). As for all convex polyhedra, the cube has Euler characteristic of 2, according to the formula V
Cube
Quadrilateral symmetric across a diagonal
of several face-symmetric polyhedra and tessellations, and have been studied in connection with outer billiards, a problem in the advanced mathematics
Kite_(geometry)
Regular non-convex polygon
In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not
Star_polygon
Quadrilateral with four right angles
In Euclidean plane geometry, a rectangle is a rectilinear convex polygon or a quadrilateral with four right angles. It can also be defined as: an equiangular
Rectangle
Polyhedron with regular congruent polygons as faces
finite convex regular polyhedra (the Platonic solids), and four regular star polyhedra (the Kepler–Poinsot polyhedra), making nine regular polyhedra in all
Regular_polyhedron
Shape with four equal sides and angles
integer, or taking the area of a square with integer sides, results in a square number; these are figurate numbers representing the numbers of points
Square
Japanese-American mathematician
implementing a method of Alexander Barvinok for counting integer points in convex polyhedra by decomposing the input into cones, and her 2004 dissertation
Ruriko_Yoshida
Shape with three equal sides
square bipyramid). The last five solids are part of Johnson solids, convex polyhedra made of regular polygonal faces, and all Johnson solids have equilateral
Equilateral_triangle
Graph-theoretic description of polyhedra
vertices of three-dimensional convex polyhedra: they are exactly the 3-vertex-connected planar graphs. That is, every convex polyhedron forms a 3-connected
Steinitz's_theorem
vectors in Rn and bi are scalars. This definition of polyhedra is particularly important as it provides a geometric perspective for problems in linear
N-dimensional_polyhedron
Polyhedron with four faces
four vertices. The tetrahedron is the simplest of all the ordinary convex polyhedra. The tetrahedron is the three-dimensional case of the more general
Tetrahedron
French mathematician
the geometry of numbers; more specifically, the number of integer points in convex polyhedra. With Masaki Kashiwara, she formulated a conjecture about
Michèle_Vergne
Type of plane curve
In geometry, a convex curve is a plane curve that has a supporting line through each of its points. There are many other equivalent definitions of these
Convex_curve
Polygon in which all angles are right
lengths in sequence are consecutive integers. A rectilinear polygon which is not a rectangle is never convex, but it can be orthogonally convex. See Orthogonally
Rectilinear_polygon
Two tetrahedra crossing each other
extremely non-convex caltrop-shaped particles. The stellated octahedron appears with several other polyhedra and polyhedral compounds in M. C. Escher's
Stellated_octahedron
triangles with angles π/p, π/q, π/r, where (p q r) are integers: (Coxeter, "Uniform polyhedra", 1954) (2 2 r) - Dihedral (2 3 3) - Tetrahedral (2 3 4)
List of uniform polyhedra by Schwarz triangle
List_of_uniform_polyhedra_by_Schwarz_triangle
Application of geometry in number theory
problem of finding nonzero integer points in a suitable convex body. Inequalities involving several linear forms in integer variables can be interpreted
Geometry_of_numbers
Branch of geometry that studies combinatorial properties and constructive methods
topology. Polyhedra and tessellations had been studied for many years by people such as Kepler and Cauchy, modern discrete geometry has its origins in the late
Discrete_geometry
88th Johnson solid (18 faces)
R. (1997). Polyhedra. Cambridge University Press. p. 86–87, 89. ISBN 978-0-521-66405-9. "A334114". The On-Line Encyclopedia of Integer Sequences. 2020
Sphenomegacorona
Integer matrices with +1 or −1 determinant; invertible over the integers. GL_n(Z)
"Introduction to Integral Boundary Points of Convex Polyhedra", in M. Jünger; et al. (eds.), 50 Years of Integer Programming, 1958-2008, Springer-Verlag
Unimodular_matrix
Solid with eight equal triangular faces
solids, a set of convex polyhedra whose faces are congruent regular polygons. Platonic solids are the ancient set of five polyhedra named after Plato
Regular_octahedron
Relation of an integral polytope's volume to how many integer points it encloses
of integer points the polytope contains. The theory of Ehrhart polynomials can be seen as a higher-dimensional generalization of Pick's theorem in the
Ehrhart_polynomial
Polygon with an infinite number of sides
M. (1937). "Regular Skew Polyhedra in Three and Four Dimensions". Proc. London Math. Soc. 43: 33–62. Look up apeirogon in Wiktionary, the free dictionary
Apeirogon
Overview of and topical guide to geometry
Heronian tetrahedron Platonic solid Archimedean solid Kepler-Poinsot polyhedra Johnson solid Uniform polyhedron Polyhedral compound Hilbert's third problem
Outline_of_geometry
Polygon with four crossed edges of two lengths
Antiparallelograms occur as the vertex figures of certain nonconvex uniform polyhedra. In the theory of four-bar linkages, the linkages with the form of an antiparallelogram
Antiparallelogram
Shape in hyperbolic geometry
In three-dimensional hyperbolic geometry, an ideal polyhedron is a convex polyhedron all of whose vertices are ideal points, points "at infinity" rather
Ideal_polyhedron
Arrangement of points on a sphere
except in the cases N = 2, 3, 4, 6, 12, and the geodesic polyhedra, the convex hull is only topologically equivalent to the figure listed in the last
Thomson_problem
In geometry a line segment joining two nonconsecutive vertices of a polygon or polyhedron
On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Poonen, Bjorn; Rubinstein, Michael. "The number of intersection points made by the diagonals
Diagonal
Graph that can be embedded in the plane
to a convex polyhedron in this way: the trees do not, for example. Steinitz's theorem says that the polyhedral graphs formed from convex polyhedra are
Planar_graph
Software for the algorithmic treatment of convex polyhedra
algorithmic treatment of convex polyhedra. Albeit primarily a tool to study the combinatorics and the geometry of convex polytopes and polyhedra, it is also capable
Polymake
Number, approximately 1.618
Vinci's illustrations of polyhedra in Pacioli's Divina proportione have led some to speculate that he incorporated the golden ratio in his paintings. But the
Golden_ratio
In mathematics, the order polytope of a finite partially ordered set is a convex polytope defined from the set. The points of the order polytope are the
Order_polytope
Uniform 6-dimensional polytope
(similar to the nonconvex uniform polyhedra) Ongoing: Jonathan Bowers and other researchers search for other non-convex uniform 6-polytopes, with a current
Uniform_6-polytope
Polyhedron related to sphere packing
convex hull of the sphere centers. Cubic Close(st) Packed spheres with radius √24 Corresponding Waterman polyhedron W24 Origin 1 Waterman polyhedra form
Waterman_polyhedron
Tiling of hyperbolic 3-space by uniform polyhedra
3-dimensional hyperbolic space there are nine Coxeter group families of compact convex uniform honeycombs, generated as Wythoff constructions, and represented
Uniform honeycombs in hyperbolic space
Uniform_honeycombs_in_hyperbolic_space
Problems which attempt to find the most efficient way to pack objects into containers
are allowed to overlap. In a bin packing problem, people are given: A container, usually a two- or three-dimensional convex region, possibly of infinite
Packing_problems
Approach to static program analysis
relational numerical abstract domains are: congruence relations on integers convex polyhedra (cf. left picture) – with some high computational costs difference-bound
Abstract_interpretation
of convex polyhedra), convex geometry (the study of convex sets, in particular combinatorics of their intersections), and discrete geometry, which in turn
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Covering by shapes without overlaps or gaps
placed points can be used to construct random tilings of the plane. Tessellation can be extended to three dimensions. Certain polyhedra can be stacked in a
Tessellation
Mathematical system of orderings or sets
{\displaystyle U} of points in the Euclidean plane or a higher-dimensional Euclidean space is formed by repeatedly removing vertices of the convex hull. The feasible
Antimatroid
General concept and operation in mathematics
self-dual. The dual polyhedron of any of these polyhedra may be formed as the convex hull of the center points of each face of the primal polyhedron, so the
Duality_(mathematics)
Complexity class of problems
two triangulations of the same convex polygon is below a given threshold The turnpike problem of reconstructing points on line from their distance multiset
NP-intermediate
Tiling of the plane with 60° rhombi
into a subset of a three-dimensional integer lattice, consisting of the points (x,y,z) with |x + y + z| ≤ 1, in such a way that two vertices are adjacent
Rhombille_tiling
Problem of finding obscured edges in a wire-frame 3D model
In 3D computer graphics, solid objects are usually modeled by polyhedra. A face of a polyhedron is a planar polygon bounded by straight line segments
Hidden-line_removal
Graph formed by subdivision of triangles
Steinitz's theorem, can always be represented as the graphs of convex polyhedra. The convex polyhedron representing an Apollonian network is a 3-dimensional
Apollonian_network
and mathematical objects such as polyhedra and the Möbius strip. Magnus Wenninger creates colourful stellated polyhedra, originally as models for teaching
Mathematics_and_art
2007 mathematics textbook
Discretely: Integer-Point Enumeration in Polyhedra is an undergraduate-level textbook in geometry, on the interplay between the volume of convex polytopes
Computing the Continuous Discretely
Computing_the_Continuous_Discretely
Convex polygon which can tile the plane by itself
In geometry, a planigon is a convex polygon that can fill the plane with only copies of itself (isotopic to the fundamental units of monohedral tessellations)
Planigon
Branch of discrete mathematics
of convex polyhedra), convex geometry (the study of convex sets, in particular combinatorics of their intersections), and discrete geometry, which in turn
Combinatorics
On tangency patterns of circles
1007/978-3-662-45803-7_11, ISBN 978-3-319-12567-1 Andreev, E. M. (1970a), "Convex polyhedra in Lobačevskiĭ spaces", Matematicheskii Sbornik, New Series, 81 (123):
Circle_packing_theorem
Polygon associated with a compact Riemann surface
Graduate Texts in Mathematics, vol. 109, Springer-Verlag, ISBN 978-0-387-96310-5 Lyusternik, L. A. (1966), Convex figures and polyhedra, translated by
Fundamental_polygon
Set of points equidistant from a center
various properties of the sphere in book XII, and shows how to inscribe the five regular polyhedra within a sphere in book XIII. Euclid does not include
Sphere
Five-dimensional geometric shape
both. The 5-cube family of 5-polytopes are given by the convex hulls of the base points listed in the following table, with all permutations of coordinates
Uniform_5-polytope
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
finite point set on the moment curve is in affine general position. The convex hull of any finite set of points on the moment curve is a cyclic polytope
Moment_curve
requires software to represent the objects of this framework (sets of integer-valued points in regions of various spaces) and perform operations upon them (e
Frameworks supporting the polyhedral model
Frameworks_supporting_the_polyhedral_model
Groups of point isometries in 3 dimensions
other polyhedra with the same symmetry. The polyhedron is convex if the surface fits to its copies and the radial line perpendicular to the plane is in the
Point groups in three dimensions
Point_groups_in_three_dimensions
Shape in the geometry of numbers
1)\}} give rise to two Klein polyhedra, each of which is bounded by a sequence of adjoining line segments. Define the integer length of a line segment to
Klein_polyhedron
Theorem that any three objects in space can be simultaneously bisected by a plane
In mathematical measure theory, for every positive integer n the ham sandwich theorem states that given n measurable "objects" in n-dimensional Euclidean
Ham_sandwich_theorem
Group of geometric symmetries with at least one fixed point
3 mirror planes, can also be given by their Coxeter group and related polyhedra. The [3,3] group can be doubled, written as [[3,3]], mapping the first
Point_group
Special type of lattice
Young diagrams representing integer partitions is a distributive lattice. The points of a distributive polytope (a convex polytope closed under coordinatewise
Distributive_lattice
In mathematics, economics, and computer science, the stable matching polytope or stable marriage polytope is a convex polytope derived from the solutions
Stable_matching_polytope
Spherical triangle that can be used to tile a sphere
List of uniform polyhedra by Schwarz triangle Nonconvex uniform polyhedron Regular hyperbolic tiling Uniform polyhedron Uniform tilings in hyperbolic plane
Schwarz_triangle
Generalization of a polytope in real space
has p (p ≥ 2) vertex points arranged to form a convex regular polygon {p} in the Argand plane. Unlike points on the real line, points on the complex line
Complex_polytope
Polytope whose vertices represent permutations
hdl:1721.1/105344, MR 3061550 Bowman, V. Joseph (1972), "Permutation polyhedra", SIAM Journal on Applied Mathematics, 22 (4): 580–589, doi:10.1137/0122054
Permutohedron
Generalization of a rectangle for higher dimensions
hypercube. A hyperrectangle is a special case of a parallelotope. For every integer i {\displaystyle i} from 1 {\displaystyle 1} to k {\displaystyle k} , let
Hyperrectangle
Pictorial representation of symmetry
diagram: . This is also called octahedral symmetry. There are 7 convex uniform polyhedra that can be constructed from this symmetry group and 3 from its
Coxeter–Dynkin_diagram
Form of differential geometry
In mathematics, systolic geometry is the study of systolic invariants of manifolds and polyhedra, as initially conceived by Charles Loewner and developed
Systolic_geometry
Compact Riemann surface of genus 3
embedded. Such polyhedra may have various convex hulls, including the truncated cube, the snub cube, or the rhombicuboctahedron, as in the small cubicuboctahedron
Klein_quartic
Mathematical space with a notion of distance
In mathematics, a metric space is a set together with a notion of distance between its points. The distance is measured by a function called a metric
Metric_space
Method for mathematical optimization
(December 1992). "A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra". Discrete and Computational Geometry. 8 (ACM
Criss-cross_algorithm
Category of routing problem minimizing total distance and time
with the edges being the points of the hull. The convex hull problem can be solved through linear programming or through convex hull algorithms, but the
Arc_routing
Triangle with at least two sides congruent
used in mathematics to show that the area of a smooth surface cannot always be accurately approximated by polyhedra converging to the surface. In graphic
Isosceles_triangle
Empirical study of systems in transformation
of low order primes (some integer). He then related the "multiplicative 2" and "additive 2" in this formula to the convex versus concave aspects of shapes
Synergetics_(Fuller)
English mathematician (1937–2020)
antiprism in the process, the only non-Wythoffian uniform polychoron. Conway also suggested a system of notation dedicated to describing polyhedra called
John_Horton_Conway
Hexahedron with parallelogram faces
perfect parallelepiped is a parallelepiped with integer-length edges, face diagonals, and space diagonals. In 2009, dozens of perfect parallelepipeds were
Parallelepiped
British-Lebanese mathematician (1929–2019)
(Clifford algebras), L. Smith (homotopy groups of spheres), Paul Sutcliffe (polyhedra), David O. Tall (lambda rings), John A. Todd (Stiefel manifolds), Cumrun
Michael_Atiyah
Four-dimensional analog of the dodecahedron
1-polytope occurs in only 15 distinct lengths in any of the component polytopes of the 120-cell. By Alexandrov's uniqueness theorem, convex polyhedra with shapes
120-cell
Three linked but pairwise separated rings
Center. Hyperbolic manifolds can be decomposed in a canonical way into gluings of hyperbolic polyhedra (the Epstein–Penner decomposition) and for the
Borromean_rings
Series of books published by Springer-Verlag
Robins, Sinai (2015). Computing the Continuous Discretely: Integer-point Enumeration in Polyhedra (2nd ed.). doi:10.1007/978-1-4939-2969-6. ISBN 978-1-4939-2968-9
Undergraduate Texts in Mathematics
Undergraduate_Texts_in_Mathematics
Abstraction of linear independence of vectors
Jack (5–9 March 2001). "Submodular functions, matroids, and certain polyhedra". In Jünger, Michael; Reinelt, Gerhard; Rinaldi, Giovanni (eds.). Combinatorial
Matroid
regular polygons and regular polyhedra to higher dimensions. Originating with an essay entitled Dimensional Analogy written in 1923, the first edition of
List of publications in mathematics
List_of_publications_in_mathematics
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INTEGER POINTS-IN-CONVEX-POLYHEDRA
INTEGER POINTS-IN-CONVEX-POLYHEDRA
INTEGER POINTS-IN-CONVEX-POLYHEDRA
INTEGER POINTS-IN-CONVEX-POLYHEDRA
INTEGER POINTS-IN-CONVEX-POLYHEDRA
INTEGER POINTS-IN-CONVEX-POLYHEDRA
INTEGER POINTS-IN-CONVEX-POLYHEDRA
INTEGER POINTS-IN-CONVEX-POLYHEDRA
INTEGER POINTS-IN-CONVEX-POLYHEDRA
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