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]

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

References

  1. ^
    arXiv:1008.2398v1 [math.MG
    ].
  2. ^ Fejes Tóth, László (1950). "Some packing and covering theorems". Acta Sci. Math. Szeged. 12.
  3. S2CID 10696627
    .
  4. arXiv:1009.4322v1 [math.MG
    ].
  5. arXiv:1602.07220 [math.MG
    ].
  6. S2CID 120336230
    .
  7. doi:10.1016/0196-6774(90)90010-C
    .
  8. doi:10.1112/s0025579300012808
    .
  9. S2CID 38234737
    .
  10. ^ Klarreich, Erica (March 30, 2016), "Sphere Packing Solved in Higher Dimensions", Quanta Magazine
  11. S2CID 119286185
    .
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