Discrete geometry -- Please Click to Return to Front Page

Discrete geometry or combinatorial geometry may be loosely defined as study of geometrical objects and properties that are discrete or combinatorial, either by their nature or by their representation; the study that does not essentially rely on the notion of 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 Kepler, and Cauchy, modern discrete geometry has its origins in the late 19th century. Early topics studied were: the density of circle packings by Thue, Projective configurations by Reye and Steinitz, the Geometry of numbers by Minkowski, and 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
  • Lattice polytopes
  • Ehrhart polynomials
  • Pick's theorem
  • Hirsch conjecture
• Packing, covering and tiling
  • Kepler conjecture
• Simplicial complexes
  • Triangulation
• Topological combinatorics
  • Sperner's lemma
• Incidence structures
  • Configurations
  • Arrangement of hyperplanes
  • Buildings
• Discrete groups
  • Coxeter groups
  • Triangle groups
  • Lattices
  • 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.
• Minkowski, H. (1896). Geometrie der Zahlen. Leipzig: Teubner. 
• Steinitz (1894). Uber die Construction der Configurationen. Breslau. 

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This page was last modified on 22 January 2010 at 16:44.

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