List of shapes with known packing constant
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 |
.svg.png) | 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 Silverman[6] | |
.jpg) | 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. |
References
- 1 2 Bezdek, András; Kuperberg, Włodzimierz (2010). "Dense packing of space with various convex solids". arXiv:1008.2398v1
[math.MG]. - ↑ Fejes Tóth, László (1950). "Some packing and covering theorems". Acta Sci. Math. Szeged. 12.
- ↑ Cohn, Henry; Kumar, Abhinav (2009). "Optimality and uniqueness of the Leech lattice among lattices". Annals of Mathematics. 170 (3): 1003–1050. doi:10.4007/annals.2009.170.1003.
- ↑ Chang, Hai-Chau; Wang, Lih-Chung (2010). "A Simple Proof of Thue's Theorem on Circle Packing". arXiv:1009.4322v1 [math.MG].
- ↑ 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.
- ↑ 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.
- ↑ Bezdek, András; Kuperberg, Włodzimierz (1990). "Maximum density space packing with congruent circular cylinders of infinite length". Mathematika. 37: 74–80. doi:10.1112/s0025579300012808.
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