Kissing number
In geometry, the kissing number of a mathematical space is defined as the greatest number of non-overlapping unit spheres that can be arranged in that space such that they each touch a common unit sph
Erdős distinct distances problem
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. It was posed by Paul Erdős in 1946 and almost
Heilbronn triangle problem
In discrete geometry and discrepancy theory, the Heilbronn triangle problem is a problem of placing points in the plane, avoiding triangles of small area. It is named after Hans Heilbronn, who conject
Lebesgue's universal covering problem
Lebesgue's universal covering problem is an unsolved problem in geometry that asks for the convex shape of smallest area that can cover every planar set of diameter one. The diameter of a set by defin
Napkin folding problem
The napkin folding problem is a problem in geometry and the mathematics of paper folding that explores whether folding a square or a rectangular napkin can increase its perimeter. The problem is known
Moser's worm problem
Moser's worm problem (also known as mother worm's blanket problem) is an unsolved problem in geometry formulated by the Austrian-Canadian mathematician Leo Moser in 1966. The problem asks for the regi
Straight skeleton
In geometry, a straight skeleton is a method of representing a polygon by a topological skeleton. It is similar in some ways to the medial axis but differs in that the skeleton is composed of straight
Penrose tiling
A Penrose tiling is an example of an aperiodic tiling. Here, a tiling is a covering of the plane by non-overlapping polygons or other shapes, and aperiodic means that shifting any tiling with these sh
Kakeya set
In mathematics, a Kakeya set, or Besicovitch set, is a set of points in Euclidean space which contains a unit line segment in every direction. For instance, a disk of radius 1/2 in the Euclidean plane
Sphere packing
In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three-dimension
Bellman's lost in a forest 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. The problem is often stated as follows
Disk covering problem
The disk covering problem asks for the smallest real number such that disks of radius can be arranged in such a way as to cover the unit disk. Dually, for a given radius ε, one wishes to find the smal
Sphere packing in a cylinder
Sphere packing in a cylinder is a three-dimensional packing problem with the objective of packing a given number of identical spheres inside a cylinder of specified diameter and length. For cylinders
Weighted Voronoi diagram
In mathematics, a weighted Voronoi diagram in n dimensions is a generalization of a Voronoi diagram. The Voronoi cells in a weighted Voronoi diagram are defined in terms of a distance function. The di
Weyl's tile argument
In philosophy, the Weyl's tile argument, introduced by Hermann 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 ti
Geometric combinatorics
Geometric combinatorics is a branch of mathematics in general and combinatorics in particular. It includes a number of subareas such as polyhedral combinatorics (the study of faces of convex polyhedra
Roberts's triangle theorem
Roberts's triangle theorem, a result in discrete geometry, states that every simple arrangement of lines has at least triangular faces. Thus, three lines form a triangle, four lines form at least two
Combinatorial Geometry in the Plane
Combinatorial Geometry in the Plane is a book in discrete geometry. It was translated from a German-language book, Kombinatorische Geometrie in der Ebene, which its authors Hugo Hadwiger and Hans Debr
Connective constant
In mathematics, the connective constant is a numerical quantity associated with self-avoiding walks on a lattice. It is studied in connection with the notion of universality in two-dimensional statist
K-set (geometry)
In discrete geometry, a -set of a finite point set in the Euclidean plane is a subset of elements of that can be strictly separated from the remaining points by a line. More generally, in Euclidean sp
Polycube
A polycube is a solid figure formed by joining one or more equal cubes face to face. Polycubes are the three-dimensional analogues of the planar polyominoes. The Soma cube, the Bedlam cube, the Diabol
Carpenter's rule problem
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 to a position where all its vertices are in
Happy ending problem
In mathematics, the "happy ending problem" (so named by Paul Erdős because it led to the marriage of George Szekeres and Esther Klein) is the following statement: Theorem — any set of five points in t
Hadwiger conjecture (combinatorial geometry)
In combinatorial geometry, the Hadwiger conjecture states that any convex body in n-dimensional Euclidean space can be covered by 2n or fewer smaller bodies homothetic with the original body, and that
Voronoi diagram
In mathematics, a Voronoi diagram is a partition of a plane into regions close to each of a given set of objects. In the simplest case, these objects are just finitely many points in the plane (called
Integrally-convex set
An integrally-convex set is the discrete geometry analogue of the concept of convex set in geometry. A subset X of the integer grid is integrally convex if any point y in the convex hull of X can be e
Integer triangle
An integer triangle or integral triangle is a triangle all of whose sides have lengths that are integers. A rational triangle can be defined as one having all sides with rational length; any such rati
Isosceles set
In discrete geometry, an isosceles set is a set of points with the property that every three of them form an isosceles triangle. More precisely, each three points should determine at most two distance
Hinged dissection
In geometry, a hinged dissection, also known as a swing-hinged dissection or Dudeney dissection, is a kind of geometric dissection in which all of the pieces are connected into a chain by "hinged" poi
List of shapes with known packing constant
The packing constant of a geometric body is the largest average density achieved by packing arrangements of congruent copies of the body. For most bodies the value of the packing constant is unknown.
Davenport–Schinzel sequence
In combinatorics, a Davenport–Schinzel sequence is a sequence of symbols in which the number of times any two symbols may appear in alternation is limited. The maximum possible length of a Davenport–S
Helly family
In combinatorics, a Helly family of order k is a family of sets in which every minimal subfamily with an empty intersection has k or fewer sets in it. Equivalently, every finite subfamily such that ev
Covering problem of Rado
The covering problem of Rado is an unsolved problem in geometry concerning covering planar sets by squares. It was formulated in 1928 by Tibor Radó and has been generalized to more general shapes and
Guillotine cutting
Guillotine cutting is the process of producing small rectangular items of fixed dimensions from a given large rectangular sheet, using only guillotine-cuts. A guillotine-cut (also called an edge-to-ed
Squaring the square
Squaring the square is the problem of tiling an integral square using only other integral squares. (An integral square is a square whose sides have integer length.) The name was coined in a humorous a
Vertex enumeration problem
In mathematics, the vertex enumeration problem for a polytope, a polyhedral cell complex, a hyperplane arrangement, or some other object of discrete geometry, is the problem of determination of the ob
Self-avoiding walk
In mathematics, a self-avoiding walk (SAW) is a sequence of moves on a lattice (a lattice path) that does not visit the same point more than once. This is a special case of the graph theoretical notio
Packing density
A packing density or packing fraction of a packing in some space is the fraction of the space filled by the figures making up the packing. In simplest terms, this is the ratio of the volume of bodies
Pinwheel tiling
In geometry, pinwheel tilings are non-periodic tilings defined by Charles Radin and based on a construction due to John Conway.They are the first known non-periodic tilings to each have the property t
Supersolvable arrangement
In mathematics, a supersolvable arrangement is a hyperplane arrangement which has a maximal flag with only modular elements. Equivalently, the intersection semilattice of the arrangement is a supersol
Guillotine partition
Guillotine partition is the process of partitioning a rectilinear polygon, possibly containing some holes, into rectangles, using only guillotine-cuts. A guillotine-cut (also called an edge-to-edge cu
Quaquaversal tiling
The quaquaversal tiling is a nonperiodic tiling of the euclidean 3-space introduced by John Conway and Charles Radin. The basic solid tiles are half prisms arranged in a pattern that relies essentiall
Dissection problem
In geometry, a dissection problem is the problem of partitioning a geometric figure (such as a polytope or ball) into smaller pieces that may be rearranged into a new figure of equal content. In this
Honeycomb conjecture
The honeycomb conjecture states that a regular hexagonal grid or honeycomb has the least total perimeter of any subdivision of the plane into regions of equal area. The conjecture was proven in 1999 b
Equidissection
In geometry, an equidissection is a partition of a polygon into triangles of equal area. The study of equidissections began in the late 1960s with Monsky's theorem, which states that a square cannot b
Kepler conjecture
The Kepler conjecture, named after the 17th-century mathematician and astronomer Johannes Kepler, is a mathematical theorem about sphere packing in three-dimensional Euclidean space. It states that no
Close-packing of equal spheres
In geometry, close-packing of equal spheres is a dense arrangement of congruent spheres in an infinite, regular arrangement (or lattice). Carl Friedrich Gauss proved that the highest average density –
Erdős–Diophantine graph
An Erdős–Diophantine graph is an object in the mathematical subject of Diophantine equations consisting of a set of integer points at integer distances in the plane that cannot be extended by any addi
Centroidal Voronoi tessellation
In geometry, a centroidal Voronoi tessellation (CVT) is a special type of Voronoi tessellation in which the generating point of each Voronoi cell is also its centroid (center of mass). It can be viewe
Discrete & Computational Geometry
Discrete & Computational Geometry is a peer-reviewed mathematics journal published quarterly by Springer. Founded in 1986 by Jacob E. Goodman and Richard M. Pollack, the journal publishes articles on
Tarski's circle-squaring problem
Tarski's circle-squaring problem is the challenge, posed by Alfred Tarski in 1925, to take a disc in the plane, cut it into finitely many pieces, and reassemble the pieces so as to get a square of equ
Tic-tac-toe
Tic-tac-toe (American English), noughts and crosses (Commonwealth English), or Xs and Os (Canadian or Irish English) is a paper-and-pencil game for two players who take turns marking the spaces in a t
Hilbert basis (linear programming)
The Hilbert basis of a convex cone C is a minimal set of integer vectors such that every integer vector in C is a conical combination of the vectors in the Hilbert basis with integer coefficients.
Borsuk's conjecture
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.
Discrete geometry
Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry in
Moving sofa problem
In mathematics, the moving sofa problem or sofa problem is a two-dimensional idealisation of real-life furniture-moving problems and asks for the rigid two-dimensional shape of largest area A that can
Nearest neighbor search
Nearest neighbor search (NNS), as a form of proximity search, is the optimization problem of finding the point in a given set that is closest (or most similar) to a given point. Closeness is typically
Kobon triangle problem
The Kobon triangle problem is an unsolved problem in combinatorial geometry first stated by Kobon Fujimura (1903-1983). The problem asks for the largest number N(k) of nonoverlapping triangles whose s
Necklace splitting problem
Necklace splitting is a picturesque name given to several related problems in combinatorics and measure theory. Its name and solutions are due to mathematicians Noga Alon and Douglas B. West. The basi
Arrangement of hyperplanes
In geometry and combinatorics, an arrangement of hyperplanes is an arrangement of a finite set A of hyperplanes in a linear, affine, or projective space S. Questions about a hyperplane arrangement A g
Opaque set
In discrete geometry, an opaque set is a system of curves or other set in the plane that blocks all lines of sight across a polygon, circle, or other shape. Opaque sets have also been called barriers,
Regular map (graph theory)
In mathematics, a regular map is a symmetric tessellation of a closed surface. More precisely, a regular map is a decomposition of a two-dimensional manifold (such as a sphere, torus, or real projecti
Arrangement of lines
In geometry an arrangement of lines is the subdivision of the plane formed by a collection of lines. Bounds on the complexity of arrangements have been studied in discrete geometry, and computational
Orchard-planting problem
In discrete geometry, the original orchard-planting problem asks for the maximum number of 3-point lines attainable by a configuration of a specific number of points in the plane. It is also called th
McMullen problem
The McMullen problem is an open problem in discrete geometry named after Peter McMullen.
Arrangement (space partition)
In discrete geometry, an arrangement is the decomposition of the d-dimensional linear, affine, or projective space into connected cells of different dimensions, induced by a finite collection of geome
Mountain climbing problem
In mathematics, the mountain climbing problem is a problem of finding the conditions that two functions forming profiles of a two-dimensional mountain must satisfy, so that two climbers can start on t