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Branch of geometry that studies combinatorial properties and constructive methods
Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric
Discrete_geometry
Study of discrete mathematical structures
numbers, calculus or Euclidean geometry. Discrete objects can often be enumerated by integers; more formally, discrete mathematics has been characterized
Discrete_mathematics
Area of mathematics
Discrete differential geometry is the study of discrete counterparts of notions in differential geometry. Instead of smooth curves and surfaces, there
Discrete differential geometry
Discrete_differential_geometry
Academic journal
Discrete & Computational Geometry is a peer-reviewed mathematics journal published quarterly by Springer. Founded in 1986 by Jacob E. Goodman and Richard
Discrete & Computational Geometry
Discrete_&_Computational_Geometry
Branch of discrete mathematics
examples for design theory. It should not be confused with discrete geometry (combinatorial geometry). Order theory is the study of partially ordered sets
Combinatorics
Branch of computer science
computational geometry are: Combinatorial computational geometry, also called algorithmic geometry, which deals with geometric objects as discrete entities
Computational_geometry
Branch of geometry
computational geometry, convex analysis, discrete geometry, functional analysis, geometry of numbers, integral geometry, linear programming, probability theory
Convex_geometry
algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, mathematical logic, number theory
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Types of numerical variables in mathematics
Continuous geometry Continuous modelling Continuous or discrete spectrum Continuous spectrum Count data Discrete-time stochastic process Discrete geometry Discrete
Continuous or discrete variable
Continuous_or_discrete_variable
Branch of mathematics
methods—differential geometry, algebraic geometry, computational geometry, algebraic topology, discrete geometry (also known as combinatorial geometry), etc.—or
Geometry
conjecture (discrete geometry) Kirchberger's theorem (discrete geometry) Krein–Milman theorem (mathematical analysis, discrete geometry) Minkowski's
List_of_theorems
Deals with digitized models or images of objects of the 2D or 3D Euclidean space
Digital geometry deals with discrete sets (usually discrete point sets) considered to be digitized models or images of objects of the 2D or 3D Euclidean
Digital_geometry
Problem in discrete geometry
In discrete geometry, the Erdős distinct distances problem states that every set of points in the plane has a nearly linear number of distinct distances
Erdős distinct distances problem
Erdős_distinct_distances_problem
Study of spaces with group actions
Combinatorics and Geometry. Universitext. Springer. Goodman, Jacob E.; O'Rourke, Joseph, eds. (2004-04-15). Handbook of Discrete and Computational Geometry, Second
Equivariant_topology
Combinatorial theory of mechanics and discrete geometry
In discrete geometry and mechanics, structural rigidity is a combinatorial theory for predicting the flexibility of ensembles formed by rigid bodies connected
Structural_rigidity
Line that intersects a curve at least twice
incidence geometry and discrete geometry. For instance, the Sylvester–Gallai theorem of incidence geometry states that if n points of Euclidean geometry are
Secant_line
Unsolved problem in combinatorial geometry
on Discrete and Computational Geometry: Proceedings of the 3rd AMS–IMS–SIAM Joint Summer Research Conference "Discrete and Computational Geometry—Twenty
Kobon_triangle_problem
Quickly growing function
cell-probe model of computational complexity. Certain problems in discrete geometry related to Davenport–Schinzel sequences have complexity bounds in
Ackermann_function
American mathematician
American mathematician working in the areas of representation theory, discrete geometry, and formal verification. In representation theory he is known for
Thomas_Callister_Hales
Overview of and topical guide to geometry
solid geometry Contact geometry Convex geometry Descriptive geometry Differential geometry Digital geometry Discrete geometry Distance geometry Elliptic
Outline_of_geometry
Levi–Hadwiger Covering Problem and Illumination". Research Problems in Discrete Geometry. Springer-Verlag. pp. 136–142. ISBN 978-0-387-23815-9.. Campos, Marcelo;
Hadwiger conjecture (combinatorial geometry)
Hadwiger_conjecture_(combinatorial_geometry)
North American undergraduate mathematics award
(Combinatorics, discrete geometry, and probability, Massachusetts Institute of Technology), Mehtaab Sawhney (Combinatorics, discrete geometry, and probability
Morgan_Prize
Alexander Grothendieck (1928–2014) – algebraic geometry Branko Grünbaum (1929–2018) – discrete geometry Michael Atiyah (1929–2019) Lev Semenovich Pontryagin
List_of_geometers
Field of mathematics which studies incidence structures
Improved Bounds for Incidences% and Complexity of Many Faces". Discrete & Computational Geometry. 28 (4): 475–490. doi:10.1007/s00454-001-0084-1. Aigner, Martin;
Incidence_geometry
Shape containing unit line segments in all directions
doi:10.1007/BF01171101. S2CID 121781065. Falconer, K. J. (1985). The Geometry of Fractal Sets. Cambridge University Press. pp. 96–99. ISBN 978-0-521-33705-2
Kakeya_set
On continuous motion of a simple polygon to convex
The carpenter's rule problem is a discrete geometry problem, which can be stated in the following manner: Can a simple planar polygon be moved continuously
Carpenter's_rule_problem
Branch of mathematics
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems
Algebraic_geometry
Hungarian-Canadian mathematician
mathematics and the director of the Centre for Computational and Discrete Geometry at the University of Calgary in Calgary, Alberta, Canada. Also he
Károly_Bezdek
Geometry problem on grid points
unsolved problems in mathematics The no-three-in-line problem in discrete geometry asks how many points can be placed in the n × n {\displaystyle n\times
No-three-in-line_problem
Study of geometry using a coordinate system
foundation of most modern fields of geometry, including algebraic, differential, discrete and computational geometry. Usually the Cartesian coordinate system
Analytic_geometry
Field of knowledge
methods, mainly homological algebra. Discrete geometry, the study of finite configurations in geometry. Convex geometry, the study of convex sets, which takes
Mathematics
In discrete geometry and computational geometry, the relative convex hull or geodesic convex hull is an analogue of the convex hull for the points inside
Relative_convex_hull
Points separated from others by a line
In discrete geometry, a k {\displaystyle k} -set of a finite point set S {\displaystyle S} in the Euclidean plane is a subset of k {\displaystyle k} elements
K-set_(geometry)
Type of geometry
formalism. There are many projective geometries, which may be divided into discrete and continuous: a discrete geometry comprises a set of points, which may
Projective_geometry
Philosophical argument
Weyl in 1949, is an argument against the notion that physical space is "discrete", as if composed of a number of finite sized units or tiles. The argument
Weyl's_tile_argument
geometry Discrete exterior calculus Discrete geometry a branch of geometry that studies combinatorial properties and constructive methods of discrete
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
German mathematician
work in discrete geometry, in particular on realization spaces of polytopes citing "his wide-ranging and deep contributions to discrete geometry using analytic
Karim_Adiprasito
Unsolved geometry problem about planar regions
"An improved upper bound for Leo Moser's worm problem", Discrete and Computational Geometry, 29 (3): 409–417, doi:10.1007/s00454-002-0774-3, MR 1961007
Moser's_worm_problem
Planar surface that forms part of the boundary of a solid object
face of C {\displaystyle C} . Face lattice Polyhedral combinatorics Discrete geometry Some other polygons, which are not faces, have also been considered
Face_(geometry)
Area of mathematics
in natural languages Computational algebraic geometry Computational group theory Computational geometry Computational number theory Computational topology
Computational_mathematics
Math theorem about sphere packing
Hales, Thomas C. (20 May 2002). "The Honeycomb Conjecture". Discrete & Computational Geometry. 25: 1–22. arXiv:math/9906042. doi:10.1007/s004540010071.
Kepler_conjecture
Five coplanar points have a subset forming a convex quadrilateral
Discrete and Computational Geometry, 19 (3): 367–371, doi:10.1007/PL00009353 Erdős, P.; Szekeres, G. (1935), "A combinatorial problem in geometry",
Happy_ending_problem
Can every bounded subset of Rn be partitioned into (n+1) smaller diameter sets?
The Borsuk problem in geometry, for historical reasons incorrectly called Borsuk's conjecture, is a question in discrete geometry. It is named after Karol
Borsuk's_conjecture
Geometry; how many 3-point lines can n points form
In discrete geometry, the original orchard-planting problem (or the tree-planting problem) asks for the maximum number of 3-point lines attainable by
Orchard-planting_problem
Branch of algebraic geometry
arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around
Arithmetic_geometry
Book on discrete geometry
Combinatorial Geometry in the Plane is a book in discrete geometry. It was translated from a German-language book, Kombinatorische Geometrie in der Ebene
Combinatorial Geometry in the Plane
Combinatorial_Geometry_in_the_Plane
Mathematical problem
4064/cm-9-2-233-240, MR 0143181. Pak, Igor (2010), Lectures on Discrete and Polyhedral Geometry, p. 39. Sikorski, R.; Zarankiewicz, K. (1955), "On uniformization
Mountain_climbing_problem
Hungarian mathematician (1915–2005)
Together with H.S.M. Coxeter and Paul Erdős, he laid the foundations of discrete geometry. As described in a 1999 interview with István Hargittai, Fejes Tóth's
László_Fejes_Tóth
Coordinate system in two dimensions
log-polar, coordinates. In order to solve a PDE numerically in a domain, a discrete coordinate system must be introduced in this domain. If the domain has
Log-polar_coordinates
Point in the convex hull of a set P in Rd, is the convex combination of d+1 points in P
Roman (2012-07-20). "Notes About the Carathéodory Number". Discrete & Computational Geometry. 48 (3): 783–792. arXiv:1112.5942. doi:10.1007/s00454-012-9439-z
Carathéodory's theorem (convex hull)
Carathéodory's_theorem_(convex_hull)
Notion in combinatorics
Vapnik and Chervonenkis was in statistics. It has also been applied in discrete geometry and graph theory. If F = { S 1 , S 2 , … } {\displaystyle \textstyle
Sauer–Shelah_lemma
Mathematics of varieties with integer coordinates
geometry. The extensive development of algebraic geometry in the 20th century produced powerful tools to study these equations. Diophantine geometry is
Diophantine_geometry
Sufficiently long sequences of numbers have long monotonic subsequences
Aldous, David; Diaconis, Persi; Spencer, Joel; Steele, J. Michael (eds.), Discrete Probability and Algorithms (PDF), IMA Volumes in Mathematics and its Applications
Erdős–Szekeres_theorem
Overview of and topical guide to discrete mathematics
discrete mathematics Finite mathematics – Syllabus in college and university mathematics Graph theory – Area of discrete mathematics Digital geometry –
Outline of discrete mathematics
Outline_of_discrete_mathematics
On point sets with no small-area triangles
chosen to maximize this area? More unsolved problems in mathematics In discrete geometry and discrepancy theory, the Heilbronn triangle problem asks how to
Heilbronn_triangle_problem
problems in mathematics The McMullen problem is an open problem in discrete geometry named after Peter McMullen. In 1972, David G. Larman wrote about the
McMullen_problem
Type of plane partition
sites are allowed in the definition (this setting has applications in geometry of numbers and crystallography), but again, in many cases only finitely
Voronoi_diagram
Fundamental object of geometry
In geometry, a point is an abstract idealization of an exact position, without size, in physical space, or its generalization to other kinds of mathematical
Point_(geometry)
Sequence with limited alternation of symbols
sequences and their length bounds have also become a standard tool in discrete geometry and in the analysis of geometric algorithms. A finite sequence U =
Davenport–Schinzel_sequence
On sets of points with integer distances
MR 0013512 Solymosi, József (2003), "Note on integral distances", Discrete & Computational Geometry, 30 (2): 337–342, doi:10.1007/s00454-003-0014-7, MR 2007970
Erdős–Anning_theorem
The Mojette transform is an application of discrete geometry. More specifically, it is a discrete and exact version of the Radon transform, thus a projection
Mojette_transform
Method in geometry for representing a polygon by a topological skeleton
of a data structure for finding pairwise interactions". Discrete and Computational Geometry. 22 (4): 569–592. doi:10.1007/PL00009479. MR 1721026. S2CID 12460625
Straight_skeleton
Question about single-shape aperiodic tiling
In plane discrete geometry, the einstein problem asks about the existence of a single prototile that by itself forms an aperiodic set of prototiles; that
Einstein_problem
Unsolved geometry question on moving a sofa through a 90° angle
subplot that revolves around such a problem. Moser's worm problem – Unsolved geometry problem about planar regions Square packing in a square – Two-dimensional
Moving_sofa_problem
Theorem about the intersections of d-dimensional convex sets
Helly's theorem is a basic result in discrete geometry on the intersection of convex sets. It was discovered by Eduard Helly in 1913, but not published
Helly's_theorem
In discrete geometry, a polytope is projectively unique (or projectively stable) if it has a unique convex realization up to projective transformations
Projectively_unique_polytope
Property in graph theory
types of sparse graph family. They arise in incidence problems in discrete geometry, and have also been used in parameterized complexity. According to
Biclique-free_graph
Ziegler, G. M. (2000-09-01). "Neighborly Cubical Polytopes". Discrete & Computational Geometry. 24 (2): 325–344. arXiv:math/9812033. doi:10.1007/s004540010039
Graph_of_a_polytope
Bound on the number of incidences between points and lines in the plane
Szemerédi–Trotter theorem is a mathematical result in the field of Discrete geometry. It asserts that given n points and m lines in the Euclidean plane
Szemerédi–Trotter_theorem
Geometric partition where pieces are connected by "hinged" points
Nakamura, Gisaku (2000). "Dudeney Dissection of Polygons". Discrete and Computational Geometry. Lecture Notes in Computer Science. Vol. 1763. pp. 14–29
Hinged_dissection
Hungarian mathematician (born 1948)
Hungarian mathematician known for her contributions to graph theory and discrete geometry. A student of Vera T. Sós and a co-author of Paul Erdős, she is an
Katalin_Vesztergombi
Branch of mathematics concerning probability
space is called an event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic
Probability_theory
Austrian mathematician
mathematician working in geometric number theory as well as in convex and discrete geometry. Gruber obtained his PhD at the University of Vienna in 1966, under
Peter_M._Gruber
Unsolved geometric problem
Bellman's lost-in-a-forest problem is an unsolved minimization problem in geometry, originating in 1955 by the American applied mathematician Richard E. Bellman
Bellman's lost-in-a-forest problem
Bellman's_lost-in-a-forest_problem
One of several theorems in different areas of mathematics
In discrete mathematics, Schur's theorem is any of several theorems of the mathematician Issai Schur. In differential geometry, Schur's theorem is a theorem
Schur's_theorem
Study of Lie groups, Lie algebras and differential equations
to be called Lie theory. For instance, the latter subject is Lie sphere geometry. This article addresses his approach to transformation groups, which is
Lie_theory
Mathematical problem
Gambini, Ian (1999). "A method for cutting squares into distinct squares". Discrete Applied Mathematics. 98 (1–2): 65–80. doi:10.1016/S0166-218X(99)00158-4
Squaring_the_square
Length of a line segment
p. 106, ISBN 978-3-527-63457-6 Matoušek, Jiří (2002), Lectures on Discrete Geometry, Graduate Texts in Mathematics, Springer, p. 349, ISBN 978-0-387-95373-1
Euclidean_distance
Theorem on polygon dissections
In geometry, the Wallace–Bolyai–Gerwien theorem, named after William Wallace, Farkas Bolyai and P. Gerwien, is a theorem related to dissections of polygons
Wallace–Bolyai–Gerwien theorem
Wallace–Bolyai–Gerwien_theorem
Gives a lower bound on the number of lines determined by n points in a projective plane
In incidence geometry, the De Bruijn–Erdős theorem, originally published by Nicolaas Govert de Bruijn and Paul Erdős in 1948, states a lower bound on the
De Bruijn–Erdős theorem (incidence geometry)
De_Bruijn–Erdős_theorem_(incidence_geometry)
Decomposition into connected open cells of lower dimensions, by a finite set of objects
In discrete geometry, an arrangement is the decomposition of the d-dimensional linear, affine, or projective space into connected cells of different dimensions
Arrangement_(space_partition)
Solid with six equal square faces
p. 247. Grünbaum, Branko (1997). "Isogonal Prismatoids". Discrete & Computational Geometry. 18 (1): 13–52. doi:10.1007/PL00009307. Senechal, Marjorie
Cube
An integrally convex set is the discrete geometry analogue of the concept of convex set in geometry. A subset X of the integer grid Z n {\displaystyle
Integrally_convex_set
Geometry without using coordinates
Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is geometry without the use of coordinates. It relies on the axiomatic
Synthetic_geometry
Mathematics textbook
particular the Borsuk–Ulam theorem, to prove theorems in combinatorics and discrete geometry. It was written by Czech mathematician Jiří Matoušek, and published
Using_the_Borsuk–Ulam_Theorem
The theorem can be classified as belonging to convex geometry, discrete geometry, and the geometry of numbers. It is named after Belgian mathematician
Doignon's_theorem
Mathematical subject
polyhedra), convex geometry (the study of convex sets, in particular combinatorics of their intersections), and discrete geometry, which in turn has many
Geometric_combinatorics
Shape made from cubes joined together
satisfying Conway's criterion" (PDF), 19th Japan Conference on Discrete and Computational Geometry, Graphs, and Games (JCDCG^3 2016). Turney, Peter (1984),
Polycube
Rigidity theorem for convex polyhedra
Cauchy's theorem is a theorem in geometry, named after Augustin Cauchy. It states that convex polytopes in three dimensions with congruent corresponding
Cauchy's_theorem_(geometry)
On triangles in line arrangements
Roberts's triangle theorem, a result in discrete geometry, states that every arrangement of n {\displaystyle n} lines, with no parallel lines and no crossings
Roberts's_triangle_theorem
Mathematical theorem
Hales, Thomas C. (January 2001). "The Honeycomb Conjecture". Discrete and Computational Geometry. 25 (1): 1–22. arXiv:math/9906042. doi:10.1007/s004540010071
Honeycomb_theorem
Arrangement of hyperplanes
In mathematics, a supersolvable arrangement is a hyperplane arrangement that has a maximal flag consisting of modular elements. Equivalently, the intersection
Supersolvable_arrangement
Number associated with self-avoiding walks
this fact came from a program of applying tools from complex analysis to discrete probabilistic models that has also produced impressive results about the
Connective_constant
Three-dimensional packing problem
At the peaks of this curve lie the uniform structures. In-between these discrete diameter ratios are the line slips at a lower packing density. Their packing
Sphere_packing_in_a_cylinder
Geometric system with a finite number of points
projective plane. Discrete space Finite space Generalized polygon Incidence geometry Linear space (geometry) Near polygon Partial geometry Polar space Laywine
Finite_geometry
Geometric concept
Peter; Moser, W. O. J.; Pach, János (2005). Research problems in discrete geometry. Springer. p. 93. ISBN 978-0-387-23815-9. Mittelmann, Hans D.; Vallentin
Kissing_number
Subdivision of the plane by lines
complexity of other features of arrangements have been studied in discrete geometry; these include zones, the cells touching a single line, and levels
Arrangement_of_lines
Kirchberger's theorem is a theorem in discrete geometry, on linear separability. The two-dimensional version of the theorem states that, if a finite set
Kirchberger's_theorem
Study of abstract machines and automata
considered a branch of mathematical systems theory, studying the behavior of discrete-parameter systems. Early work in automata theory differed from previous
Automata_theory
One can't dissect a square into an odd number of triangles of equal area
In geometry, Monsky's theorem states that it is not possible to dissect a square into an odd number of triangles of equal area. In other words, a square
Monsky's_theorem
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DISCRETE GEOMETRY