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Because matroids and oriented matroids are abstractions of other mathematical abstractions, nearly all the relevant books are written for mathematical scientists rather than for the general public. For learning about oriented matroids, a good preparation is to study the textbook on
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534:) as can be seen from their axiomatic definitions. Incidence structures also generalize the higher-dimensional analogs and the finite structures are sometimes called
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and this theorem retains a prominent role in this new field. This theorem has many equivalent versions and analogs and has been used in the study of
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Packings, coverings, and tilings are all ways of arranging uniform objects (typically circles, spheres, or tiles) in a regular way on a surface or
1147:, and both the algebraic structure of lattices and the geometry of the totality of all lattices are relatively well understood. Deep results of
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by Nering and Tucker, which is infused with oriented-matroid ideas, and then to proceed to
Ziegler's lectures on polytopes.
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appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a simplicial complex is an
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within a containing space. The spheres considered are usually all of identical size, and the space is usually three-
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obtained from the 1950s through the 1970s provided examples and generalized much of the theory to the setting of
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177:, modern discrete geometry has its origins in the late 19th century. Early topics studied were: the density of
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in four dimensions). Some theories further generalize the idea to include such objects as unbounded polytopes (
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drawn as a square can be tilted over by the blue force into a parallelogram, so it is a flexible graph. K
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30:"Combinatorial geometry" redirects here. The term combinatorial geometry is also used in the theory of
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In 1978, the situation was reversed – methods from algebraic topology were used to solve a problem in
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949:(see illustration). Simplicial complexes should not be confused with the more abstract notion of a
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using one or more geometric shapes, called tiles, with no overlaps and no gaps. In
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The following are some of the aspects of polytopes studied in discrete geometry:
27:
Branch of geometry that studies combinatorial properties and constructive methods
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Li Chen, Digital and discrete geometry: Theory and
Algorithms, Springer, 2014.
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1969. Björner et alia, Chapters 1-3. Bokowski, Chapter 1. Ziegler, Chapter 7.
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The discipline of combinatorial topology used combinatorial concepts in
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one another, or how they may be arranged to cover a larger object.
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40:
1445:(2010), "Discrete and convex geometry", in Horváth, János (ed.),
1309:
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geometric objects. Most questions in discrete geometry involve
1774:
1659:
Handbook of
Discrete and Computational Geometry, Second Edition
1447:
A Panorama of
Hungarian Mathematics in the Twentieth Century, I
258:
in three dimensions, and so on in higher dimensions (such as a
1566:
Lectures on Sphere
Arrangements - the Discrete Geometric Side
979:
and in the early 20th century this turned into the field of
1522:
Discrete geometry: in honor of W. Kuperberg's 60th birthday
446:
Graphs are drawn as rods connected by rotating hinges. The
1423:
Katona, G. O. H. (2005), "Laszlo Fejes Toth – Obituary",
388:, tessellations can be generalized to higher dimensions.
357:-dimensional Euclidean space (where the problem becomes
1480:
Björner et alia, Chapters 1-3. Bokowski, Chapters 1-4.
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for predicting the flexibility of ensembles formed by
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is the study of discrete counterparts of notions in
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display of a computer, or in newspapers are in fact
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of a certain kind, constructed by "gluing together"
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2141:
2103:
2070:
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1994:
1941:
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1304:. Instead of smooth curves and surfaces, there are
801:). In comparison, an ordinary (i.e., non-oriented)
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1143:, this amounts to the usual geometric notion of a
859:objects. Examples include Euclidean graphs, the 1-
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703:
655:
631:
588:
169:had been studied for many years by people such as
1409:Intuitive Geometry, in Memoriam László Fejes Tóth
522:Incidence structures generalize planes (such as
514:Seven points are elements of seven lines in the
353:can be generalised to consider unequal spheres,
1786:
125:, and is closely related to subjects such as
8:
1762:: CS1 maint: multiple names: authors list (
1681:: CS1 maint: multiple names: authors list (
1425:Studia Scientiarum Mathematicarum Hungarica
117:Discrete geometry has a large overlap with
1793:
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1202:, which remains an active research area.
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1449:, New York: Springer, pp. 431–441,
441:
1412:, Alfréd Rényi Institute of Mathematics
1398:
518:, an example of an incidence structure.
71:properties and constructive methods of
1755:
1740:Excursions into Combinatorial Geometry
1674:
1661:. Boca Raton: Chapman & Hall/CRC.
1611:Research problems in discrete geometry
1139:. In the special case of subgroups of
1085:is the discrete one. For example, the
817:, and to arrangements of vectors over
391:Specific topics in this area include:
1544:Classical Topics in Discrete Geometry
380:of a flat surface is the tiling of a
338:is an arrangement of non-overlapping
7:
2355:
789:and of arrangements of vectors in a
632:{\displaystyle I\subseteq P\times L}
365:packing in higher dimensions) or to
86:of basic geometric objects, such as
2367:
1586:; Deza, Antoine; Ye, Yinyu (2013).
1375:Discrete and Computational Geometry
809:properties that are common both to
432:Structural rigidity and flexibility
254:is a polytope in two dimensions, a
1588:Discrete Geometry and Optimization
1093:, form a discrete subgroup of the
998:, thus beginning the new study of
25:
1406:Pach, János; et al. (2008),
785:that abstracts the properties of
2366:
2354:
2343:
2342:
2330:
1639:. New York: Wiley-Interscience.
1524:. New York, N.Y: Marcel Dekker.
2251:Computational complexity theory
1275:Its main application areas are
799:partially ordered vector spaces
313:Packings, coverings and tilings
1605:Brass, Peter; Moser, William;
1298:Discrete differential geometry
1293:Discrete differential geometry
1287:Discrete differential geometry
689:
677:
579:
561:
139:discrete differential geometry
1:
1714:Lectures on discrete geometry
1657:and O'Rourke, Joseph (2004).
1631:; Agarwal, Pankaj K. (1995).
1327:Topics in this area include:
1316:. It is used in the study of
1205:Topics in this area include:
1013:Topics in this area include:
882:Topics in this area include:
741:Topics in this area include:
484:Topics in this area include:
1695:Convex and Discrete Geometry
1029:Lattices and discrete groups
821:, which are not necessarily
813:, which are not necessarily
589:{\displaystyle C=(P,L,I).\,}
38:, especially in older texts.
1590:. New York, N.Y: Springer.
1568:. New York, N.Y: Springer.
1546:. New York, N.Y: Springer.
1253:of objects of the 2D or 3D
1184:semisimple algebraic groups
1131:with the property that the
955:abstract simplicial complex
704:{\displaystyle (p,l)\in I,}
2411:
2301:Films about mathematicians
1738:, Petru S. Soltan (1997).
1337:Discrete exterior calculus
1290:
1224:
1032:
1002:. Lovász's proof used the
968:
959:random geometric complexes
915:
832:
770:
643:relation. The elements of
503:
435:
316:
235:
131:combinatorial optimization
29:
2324:
1870:Philosophy of mathematics
1810:
1352:Topological combinatorics
1332:Discrete Laplace operator
1322:topological combinatorics
1000:topological combinatorics
971:Topological combinatorics
965:Topological combinatorics
947:-dimensional counterparts
187:projective configurations
2306:Recreational mathematics
607:is a set of "lines" and
426:Finite subdivision rules
282:Polyhedral combinatorics
219:laid the foundations of
2191:Mathematical statistics
2181:Mathematical psychology
2151:Engineering mathematics
2085:Algebraic number theory
1712:Matoušek, Jiří (2002).
1520:Bezdek, András (2003).
1357:Spectral shape analysis
1242:sets) considered to be
1238:sets (usually discrete
1198:initiated the study of
1070:of a topological group
906:Delaunay triangulations
897:Random geometric graphs
756:Hyperplane arrangements
232:Polyhedra and polytopes
2337:Mathematics portal
2186:Mathematical sociology
2166:Mathematical economics
2161:Mathematical chemistry
2090:Analytic number theory
1971:Differential equations
1635:Combinatorial geometry
1058:. With this topology,
835:Geometric graph theory
829:Geometric graph theory
783:mathematical structure
732:
705:
657:
633:
603:is a set of "points",
590:
519:
473:connected by flexible
459:
361:in two dimensions, or
201:by Tait, Heawood, and
151:combinatorial topology
143:geometric graph theory
123:computational geometry
61:combinatorial geometry
53:
49:and the corresponding
18:Combinatorial geometry
2316:Mathematics education
2246:Theory of computation
1966:Hypercomplex analysis
1347:Discrete Morse theory
1302:differential geometry
733:
706:
658:
634:
591:
513:
445:
44:
2296:Informal mathematics
2176:Mathematical physics
2171:Mathematical finance
2156:Mathematical biology
2095:Diophantine geometry
1716:. Berlin: Springer.
1697:. Berlin: Springer.
1613:. Berlin: Springer.
1382:Discrete mathematics
1362:Analysis on fractals
1314:simplicial complexes
912:Simplicial complexes
855:are associated with
722:
674:
647:
611:
552:
500:Incidence structures
467:combinatorial theory
2311:Mathematics and art
2221:Operations research
1976:Functional analysis
1732:Vladimir Boltyanski
1492:linear optimization
1101:(with the standard
1004:Borsuk-Ulam theorem
543:incidence structure
506:Incidence structure
463:Structural rigidity
438:Structural rigidity
292:Ehrhart polynomials
195:geometry of numbers
2256:Numerical analysis
1865:Mathematical logic
1860:Information theory
1054:equipped with the
981:algebraic topology
924:simplicial complex
918:Simplicial complex
797:(particularly for
728:
714:we say that point
701:
653:
629:
586:
520:
494:Flexible polyhedra
460:
349:. However, sphere
272:abstract polytopes
197:by Minkowski, and
54:
2395:Discrete geometry
2382:
2381:
1981:Harmonic analysis
1704:978-3-540-71132-2
1655:Goodman, Jacob E.
1597:978-3-319-00200-2
1575:978-1-4614-8117-1
1553:978-1-4419-0599-4
1342:Discrete calculus
1318:computer graphics
1277:computer graphics
1210:Reflection groups
1165:M. S. Raghunathan
1137:invariant measure
1129:discrete subgroup
1125:topological group
1083:relative topology
1068:discrete subgroup
1064:topological group
1056:discrete topology
996:Kneser conjecture
928:topological space
892:Polyhedral graphs
877:visibility graphs
767:Oriented matroids
751:Line arrangements
731:{\displaystyle l}
656:{\displaystyle I}
536:finite geometries
416:Aperiodic tilings
406:Kepler conjecture
302:Hirsch conjecture
287:Lattice polytopes
221:discrete geometry
209:László Fejes Tóth
57:Discrete geometry
16:(Redirected from
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2266:Computer algebra
2241:Computer science
1961:Complex analysis
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1691:Gruber, Peter M.
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1232:Digital geometry
1227:Digital geometry
1221:Digital geometry
1190:. In the 1990s,
1107:rational numbers
902:Voronoi diagrams
873:unit disk graphs
779:oriented matroid
773:Oriented matroid
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489:Cauchy's theorem
371:hyperbolic space
351:packing problems
135:digital geometry
63:are branches of
45:A collection of
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1122:locally compact
1103:metric topology
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1035:Lattice (group)
1033:Main articles:
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1018:Sperner's lemma
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841:geometric graph
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401:Sphere packings
396:Circle packings
369:spaces such as
347:Euclidean space
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317:Main articles:
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236:Main articles:
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179:circle packings
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127:finite geometry
119:convex geometry
51:unit disk graph
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1291:Main article:
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1281:image analysis
1225:Main article:
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1153:Harish-Chandra
1133:quotient space
1045:discrete group
1039:discrete group
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969:Main article:
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951:simplicial set
916:Main article:
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833:Main article:
830:
827:
805:abstracts the
771:Main article:
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746:Configurations
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436:Main article:
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421:Periodic graph
418:
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359:circle packing
336:sphere packing
319:circle packing
314:
311:
310:
309:
304:
299:
297:Pick's theorem
294:
289:
284:
233:
230:
228:
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213:H.S.M. Coxeter
199:map colourings
158:
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147:toric geometry
36:simple matroid
34:to refer to a
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4:
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847:in which the
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541:Formally, an
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411:Quasicrystals
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367:non-Euclidean
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167:tessellations
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69:combinatorial
66:
62:
58:
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2261:Optimization
2123:Differential
2051:
2047:Differential
2014:Order theory
2009:Graph theory
1913:Group theory
1742:. Springer.
1739:
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1261:
1260:Simply put,
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1059:
1051:
1044:
1042:
1023:Regular maps
1012:
999:
985:
974:
944:
942:, and their
923:
921:
881:
840:
838:
822:
814:
791:vector space
778:
776:
740:
715:
713:
664:
604:
600:
598:
545:is a triple
542:
540:
521:
483:
471:rigid bodies
462:
461:
390:
378:tessellation
377:
375:
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326:
323:tessellation
276:
247:
245:
220:
207:
189:by Reye and
160:
116:
60:
56:
55:
2373:WikiProject
2216:Game theory
2196:Probability
1933:Homological
1923:Multilinear
1903:Commutative
1880:Type theory
1847:Foundations
1803:mathematics
1629:Pach, János
1607:Pach, János
1469:Rockafellar
1234:deals with
1188:local field
1135:has finite
1105:), but the
994:proved the
957:. See also
663:are called
448:cycle graph
386:mathematics
363:hypersphere
344:dimensional
264:apeirotopes
67:that study
2201:Statistics
2080:Arithmetic
2042:Arithmetic
1908:Elementary
1875:Set theory
1514:References
1387:Paul Erdős
1262:digitizing
1180:Lie groups
1113:, do not.
1062:becomes a
1010:problems.
865:polyhedron
807:dependence
528:projective
516:Fano plane
307:Opaque set
260:4-polytope
256:polyhedron
238:Polyhedron
217:Paul Erdős
2128:Geometric
2118:Algebraic
2057:Euclidean
2032:Algebraic
1928:Universal
1758:cite book
1677:cite book
1378:(journal)
1244:digitized
1177:nilpotent
940:triangles
857:geometric
761:Buildings
693:∈
641:incidence
624:×
618:⊆
163:polyhedra
161:Although
112:intersect
2389:Category
2349:Category
2105:Topology
2052:Discrete
2037:Analytic
2024:Geometry
1996:Discrete
1951:Calculus
1943:Analysis
1898:Abstract
1837:Glossary
1820:Timeline
1693:(2007).
1609:(2005).
1564:(2013).
1542:(2010).
1431:(2): 113
1368:See also
1306:polygons
1272:images.
1236:discrete
1196:Lubotzky
1169:Margulis
1161:Tamagawa
1087:integers
1076:subgroup
977:topology
869:polytope
861:skeleton
849:vertices
815:directed
793:over an
475:linkages
329:manifold
248:polytope
242:Polytope
203:Hadwiger
191:Steinitz
108:polygons
81:discrete
73:discrete
65:geometry
32:matroids
2361:Commons
2143:Applied
2113:General
1890:Algebra
1815:History
1270:digital
1186:over a
1145:lattice
1118:lattice
990:– when
823:ordered
803:matroid
639:is the
340:spheres
270:), and
252:polygon
157:History
104:spheres
100:circles
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2062:Finite
1918:Linear
1825:Future
1801:Major
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1312:, and
1310:meshes
1266:raster
1251:images
1247:models
1173:Zimmer
1157:Mostow
1081:whose
932:points
875:, and
819:fields
811:graphs
665:flags.
599:where
530:, and
524:affine
479:hinges
227:Topics
215:, and
193:, the
175:Cauchy
171:Kepler
149:, and
96:planes
88:points
77:finite
2289:lists
1832:Lists
1805:areas
1393:Notes
1240:point
1149:Borel
1127:is a
1120:in a
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