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Discrete geometry'''Discrete geometry''' or '''combinatorial geometry''' may be loosely defined as study of geometrical objects and properties that are discrete space|discrete or combinatorial, either by their nature or by their representation; the study that does not essentially rely on the notion of continuum|continuity.
Parts of its domain of research is often attributed to other kinds of geometry: digital geometry, computational geometry, finite geometry, and toric geometry.
It also overlaps with convex geometry, and combinatorial topology.
Although polyhedra and tessellations have been studied for many years by people such as Johannes Kepler| Kepler, and Augustin-Louis Cauchy | Cauchy, modern discrete geometry has its origins in the late 19th century. Early topics studied were: the density of circle packings by Axel Thue | Thue, Projective configurations by Reye and Ernst Steinitz| Steinitz, the Geometry of numbers by Minkowski, and Four colour theorem | map colourings by Tait, Heawood, and Hadwiger.
(The term ''combinatorial geometry'' has also been used as a synonym for ''simple matroid'', but that is no longer popular.)
Topics in discrete geometry
- Polytopes and polyhedral combinatorics
- Convex lattice polytope|Lattice polytopes
- Ehrhart polynomials
- Pick's theorem
- Hirsch conjecture
- Packing, covering and tessellation|tiling
- Kepler conjecture
- Simplicial complexes
- Triangulation
- Topological combinatorics
- Sperner's lemma
- Incidence structures
- Configuration (geometry) | Configurations
- Arrangement of hyperplanes
- Building (mathematics)| Buildings
- Discrete groups
- Coxeter groups
- Triangle groups
- Lattice (discrete subgroup) | Lattices
- Regular map (graph theory)| Regular maps
- Discrete differential geometry
- Geometric set partitioning
- Geometric set transversals
See also
- Discrete mathematics
- Oriented matroid
- Paul Erdős
References
- A.L. Cauchy, "Recherche sur les polyèdres - premier mémoire", ''Journal de l'Ecole Polytechnique'' '''9''' (1813), 66–86.
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Category:Discrete geometry|*
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