List of shapes with known packing constant
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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]
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Image | Description | Dimension | Packing constant | Comments |
---|---|---|---|---|
Monohedral prototiles | all | 1 | Shapes such that congruent copies can form a tiling of space | |
Circle, Ellipse | 2 | π/√12 ≈ 0.906900 | Proof attributed to Thue[4] | |
Regular pentagon | 2 | Thomas Hales and Wöden Kusner[5] | ||
Smoothed octagon | 2 | 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 | π/√18 ≈ 0.7404805 | See Kepler conjecture | |
Bi-infinite cylinder | 3 | π/√12 ≈ 0.906900 | Bezdek and Kuperberg[8] | |
Half-infinite cylinder | 3 | π/√12 ≈ 0.906900 | 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 | See Hypersphere packing[10][11] | ||
Hypersphere | 24 | See Hypersphere packing |
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