Discrete geometry, also called combinatorial geometry, is the branch of geometry that studies the combinatorial properties and constructive methods of discrete geometric objects such as points, lines, polygons and packings. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
Central QuestionHow densely non-overlapping shapes, most famously equal spheres, can be packed into a containing space, and whether a proposed densest arrangement can be proved optimal rather than merely observed to be efficient. 1 Key DebateWhether the arrangement of equal spheres that Johannes Kepler conjectured in 1611 to be the densest possible, the same face-centered cubic packing seen in stacked cannonballs and fruit at a market, is truly optimal. Polyhedra and tessellations had been studied by people such as Kepler and Augustin-Louis Cauchy for centuries before discrete geometry took shape as its own subject in the late nineteenth century, and Kepler's own packing conjecture waited nearly four centuries for a computer-assisted proof. 1 Classification
Pure or Applied Connections
Associated With
Includes
Source Alexandrov's theorem on polyhedra (Wikipedia)
Source Beck's theorem (geometry) - Wikipedia
Source Wikipedia: Mahler volume
Source Caratheodory's theorem (convex hull) (Wikipedia)
Source Caratheodory's theorem (convex hull) (Wikipedia)
Source Fano Plane (Wikipedia)
Source Helly's theorem and related results (Wikipedia)
Source Hermann Minkowski (Wikipedia)
Source Hilbert Curve (Wikipedia)
Source Historisches Lexikon der Schweiz: Hadwiger, Hugo (Historisches Lexikon der Schweiz)
Primary research field per the source's own opening sentence: discrete, convex and combinatorial geometry, including finite sphere packings and the sausage conjecture.
Source Joerg Wills (Wikipedia, German)
Source Kawasaki's theorem (Wikipedia)
Additional Source Kepler Conjecture (Wikipedia)Introduction
Source Wikipedia: Mahler volume
Source Sierpiński Triangle (Wikipedia)
Proved the Kepler conjecture on sphere-packing density (1998, formally verified 2014) and the honeycomb and dodecahedral conjectures, all discrete geometry results.
Source Thomas Hales (Wikipedia)
Source Ulam's Packing Conjecture (Wikipedia)
Sources
1. Discrete Geometry (Wikipedia)
WikipediaLead section
Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects.
Packings, coverings and tilings
A sphere packing is an arrangement of non-overlapping spheres within a containing space.
History
Polyhedra and tessellations had been studied for many years by people such as Kepler and Cauchy, modern discrete geometry has its origins in the late 19th century.
View the Source Thomas Hales (Wikipedia)
Wikimedia FoundationIncludes: Thomas Hales, Infobox: known_forQuote, Includes: Thomas Hales, Infobox: known_for
Proof of the Kepler conjecture Proof of the honeycomb conjecture Proof of the dodecahedral conjecture
Joerg Wills (Wikipedia, German)
Wikimedia FoundationIncludes: Jorg Wills, opening paragraphQuote, Includes: Jorg Wills, opening paragraph
Joerg Michael Wills (born 5 March 1937 in Berlin) is a German mathematician working in discrete, convex and combinatorial geometry.
Kepler Conjecture (Wikipedia)
Wikimedia FoundationIncludes: Kepler Conjecture, IntroductionQuote, Includes: Kepler Conjecture, Introduction
is a mathematical theorem about sphere packing in three-dimensional Euclidean space.
Sierpiński Triangle (Wikipedia)
Hilbert Curve (Wikipedia)
Fano Plane (Wikipedia)
Historisches Lexikon der Schweiz: Hadwiger, Hugo (Historisches Lexikon der Schweiz)
Historisches Lexikon der SchweizIncludes: Hugo Hadwiger, Article textQuote, Includes: Hugo Hadwiger, Article text
Weiter trug er entscheidend zur Theorie der geometr. Wahrscheinlichkeiten und zur Entwicklung der kombinator. Geometrie zwischen Mathematik und Informatik bei.
Beck's theorem (geometry) - Wikipedia
Includes: Beck's Theorem (Geometry), Lead sentenceQuote, Includes: Beck's Theorem (Geometry), Lead sentence
In discrete geometry, Beck's theorem is any of several different results, two of which are given below.
Wikipedia: Mahler volume
WikipediaIncludes: Mahler Conjecture, Lead sentence
In convex geometry, the Mahler volume of a centrally symmetric convex body is a dimensionless quantity that is associated with the
Includes: Blaschke-Santalo Inequality, Lead sentence
In convex geometry, the Mahler volume of a centrally symmetric convex body is a dimensionless quantity that is associated with the
View the Source Helly's theorem and related results (Wikipedia)
Includes: Helly's Theorem, Lead sentenceQuote, Includes: Helly's Theorem, Lead sentence
Helly's theorem is a basic result in discrete geometry on the intersection of convex sets.
Caratheodory's theorem (convex hull) (Wikipedia)
- Includes: Caratheodory's Theorem (Convex Geometry), Lead sentence
- Includes: Caratheodory's Theorem (Convex Hull), Lead sentence
View the SourceHermann Minkowski (Wikipedia)
Includes: Hermann Minkowski, Lead paragraph [in-branch 2]Quote, Includes: Hermann Minkowski, Lead paragraph [in-branch 2]
elements of convex geometry
Alexandrov's theorem on polyhedra (Wikipedia)
Includes: Alexandrov's Theorem on Polyhedra, Lead sectionQuote, Includes: Alexandrov's Theorem on Polyhedra, Lead section
Alexandrov's theorem on polyhedra is a rigidity theorem in mathematics, describing
Ulam's Packing Conjecture (Wikipedia)
Wikimedia FoundationIncludes: Ulam's Packing Conjecture, Lead sectionQuote, Includes: Ulam's Packing Conjecture, Lead section
Stanisław Ulam, is a conjecture about the highest possible packing density of identical convex solids in three-dimensional Euclidean space. The conjecture
Kawasaki's theorem (Wikipedia)
Includes: Kawasaki's Theorem, Lead sectionQuote, Includes: Kawasaki's Theorem, Lead section
Kawasaki-Justin theorem is a theorem in the mathematics of paper folding that describes the crease patterns with a single
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