Dodecahedral conjecture
Encyclopedia
The dodecahedral conjecture in geometry
is intimately related to sphere packing
.
László Fejes Tóth
, a 20th century Hungarian geometer, considered the Voronoi decomposition of any given packing of unit sphere
s. He conjectured in 1943 that the minimal volume of any cell in the resulting Voronoi decomposition was at least as large as the volume of a regular dodecahedron circumscribed to a unit sphere.
Thomas Callister Hales
and Sean McLaughlin proved the conjecture in 1998, following the same strategy that led Hales to his proof of the Kepler conjecture
. The proofs rely on extensive computations. McLaughlin was awarded the 1999 Morgan Prize
for his contribution to this proof.
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
is intimately related to sphere packing
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-dimensional Euclidean space...
.
László Fejes Tóth
László Fejes Tóth
László Fejes Tóth was a Hungarian mathematician who specialised in geometry. He proved that a honeycomb pattern is the most efficient way to pack centrally symmetric convex sets on the Euclidean plane . He also investigated packings on the sphere...
, a 20th century Hungarian geometer, considered the Voronoi decomposition of any given packing of unit sphere
Unit sphere
In mathematics, a unit sphere is the set of points of distance 1 from a fixed central point, where a generalized concept of distance may be used; a closed unit ball is the set of points of distance less than or equal to 1 from a fixed central point...
s. He conjectured in 1943 that the minimal volume of any cell in the resulting Voronoi decomposition was at least as large as the volume of a regular dodecahedron circumscribed to a unit sphere.
Thomas Callister Hales
Thomas Callister Hales
Thomas Callister Hales is an American mathematician. He is known for his 1998 computer-aided proof of the Kepler conjecture, a centuries-old problem in discrete geometry which states that the most space-efficient way to pack spheres is in a pyramid shape...
and Sean McLaughlin proved the conjecture in 1998, following the same strategy that led Hales to his proof of the Kepler conjecture
Kepler conjecture
The Kepler conjecture, named after the 17th-century German astronomer Johannes Kepler, is a mathematical conjecture about sphere packing in three-dimensional Euclidean space. It says that no arrangement of equally sized spheres filling space has a greater average density than that of the cubic...
. The proofs rely on extensive computations. McLaughlin was awarded the 1999 Morgan Prize
Morgan Prize
The Morgan Prize in Mathematics may refer to:* Frank and Brennie Morgan Prize for Outstanding Research in Mathematics by an Undergraduate Student awarded jointly by the American Mathematical Society, Mathematical Association of America and Society for Industrial and Applied Mathematics* De Morgan...
for his contribution to this proof.