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]

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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	${\displaystyle {\frac {5-{\sqrt {5}}}{3}}\approx 0.92131}$	Thomas Hales and Wöden Kusner[5]
	Smoothed octagon	2	${\displaystyle \eta _{so}={\frac {8-4{\sqrt {2}}-\ln {2}}{2{\sqrt {2}}-1}}\approx 0.902414}$	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	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	${\displaystyle {\frac {({\frac {\pi }{2}})^{4}}{4!}}\approx 0.2536695}$	See Hypersphere packing[10][11]
	Hypersphere	24	${\displaystyle {\frac {({\frac {\pi }{2}})^{12}}{12!}}\approx 0.000000471087}$	See Hypersphere packing

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