Jump to content

List of shapes with known packing constant

Add links
From Wikipedia, the free encyclopedia

This is an old revision of this page, as edited by Citation bot (talk | contribs) at 03:33, 20 May 2020 (Alter: template type. Add: arxiv, doi, pages, issue, volume, journal. Removed parameters. Formatted dashes. Some additions/deletions were actually parameter name changes. | You can use this bot yourself. Report bugs here. | Activated by Amigao | Category:Mathematics-related lists | via #UCB_Category). The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.

The packing constant of a geometric body is the largest average density achieved by packing arrangements of congruent copies of the body. For most bodies the value of the packing constant is unknown.[1] The following is a list of bodies in Euclidean spaces whose packing constant is known.[1] Fejes Tóth proved that in the plane, a point symmetric body has a packing constant that is equal to its translative packing constant and its lattice packing constant.[2] Therefore, any such body for which the lattice packing constant was previously known, such as any ellipse, consequently has a known packing constant. In addition to these bodies, the packing constants of hyperspheres in 8 and 24 dimensions are almost exactly known.[3]

Image Description Dimension Packing constant Comments
All shapes that tile space all 1 By definition
Circle, Ellipse 2 π/√12 ≈ 0.906900 Proof attributed to Thue[4]
Smoothed octagon 2 Reinhardt[5]
All 2-fold symmetric convex polygons 2 Linear-time (in number of vertices) algorithm given by Mount and Ruth Silverman[6]
Sphere 3 π/√18 ≈ 0.7404805 See Kepler conjecture
Bi-infinite cylinder 3 π/√12 ≈ 0.906900 Bezdek and Kuperberg[7]
All shapes contained in a rhombic dodecahedron whose inscribed sphere is contained in the shape 3 Fraction of the volume of the rhombic dodecahedron filled by the shape Corollary of Kepler conjecture. Examples pictured: rhombicuboctahedron and rhombic enneacontahedron.
Hypersphere 8 See Hypersphere packing[8][9]
Hypersphere 24 See Hypersphere packing

References

  1. ^ a b Bezdek, András; Kuperberg, Włodzimierz (2010). "Dense packing of space with various convex solids". arXiv:1008.2398v1 [math.MG].
  2. ^ Fejes Tóth, László (1950). "Some packing and covering theorems". Acta Sci. Math. Szeged. 12.
  3. ^ Cohn, Henry; Kumar, Abhinav (2009). "Optimality and uniqueness of the Leech lattice among lattices". Annals of Mathematics. 170 (3): 1003–1050. arXiv:math.MG/0403263. doi:10.4007/annals.2009.170.1003.
  4. ^ Chang, Hai-Chau; Wang, Lih-Chung (2010). "A Simple Proof of Thue's Theorem on Circle Packing". arXiv:1009.4322v1 [math.MG].
  5. ^ Reinhardt, Karl (1934). "Über die dichteste gitterförmige Lagerung kongruente Bereiche in der Ebene und eine besondere Art konvexer Kurven". Abh. Math. Sem. Univ. Hamburg. 10: 216–230. doi:10.1007/bf02940676.
  6. ^ Mount, David M.; Silverman, Ruth (1990). "Packing and covering the plane with translates of a convex polygon". Journal of Algorithms. 11 (4): 564–580. doi:10.1016/0196-6774(90)90010-C.
  7. ^ Bezdek, András; Kuperberg, Włodzimierz (1990). "Maximum density space packing with congruent circular cylinders of infinite length". Mathematika. 37: 74–80. doi:10.1112/s0025579300012808.
  8. ^ Klarreich, Erica (March 30, 2016), "Sphere Packing Solved in Higher Dimensions", Quanta Magazine
  9. ^ Viazovska, Maryna (2016). "The sphere packing problem in dimension 8". Annals of Mathematics. 185 (3): 991–1015. arXiv:1603.04246. doi:10.4007/annals.2017.185.3.7.
Retrieved from "https://en.wikipedia.org/w/index.php?title=List_of_shapes_with_known_packing_constant&oldid=957698883"