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. 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. 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.
Image Description Dimension Packing constant Comments Monohedral prototiles all 1 Shapes such that congruent copies can form a tiling of space Circle, Ellipse 2 Proof attributed to Thue Regular pentagon 2 Thomas Hales and Wöden Kusner Smoothed octagon 2 Reinhardt All 2-fold symmetric convex polygons 2 Linear-time (in number of vertices) algorithm given by Mount and Ruth Silverman Sphere 3 See Kepler conjecture Bi-infinite cylinder 3 Bezdek and Kuperberg Half-infinite cylinder 3 Wöden Kusner 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 Hypersphere 24 See Hypersphere packing
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