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. 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 | Comments | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Monohedral prototiles | all | 1 | Shapes such that congruent copies can form a tiling of space | |||||||||
2 | Proof attributed to Thue[4] | |||||||||||
2 |
| Thomas Hales and Wöden Kusner[5] | ||||||||||
2 | ηso=
}{2\sqrt{2}-1} ≈ 0.902414 | Reinhardt[6] | ||||||||||
All 2-fold symmetric convex polygons | 2 | Linear-time (in number of vertices) algorithm given by Mount and Ruth Silverman[7] | ||||||||||
Sphere | 3 | See Kepler conjecture | ||||||||||
Bi-infinite cylinder | 3 | Bezdek and Kuperberg[8] | ||||||||||
Half-infinite cylinder | 3 | Wöden Kusner[9] | ||||||||||
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 |
≈ 0.2536695 | See Hypersphere packing[10] | |||||||||
Hypersphere | 24 |
≈ 0.000000471087 | See Hypersphere packing |