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. 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.
ImageDescriptionDimensionPacking constantComments
All shapes that tile spaceall1By definition
Circle, Ellipse2Proof attributed to Thue
Smoothed octagon2Reinhardt
All 2-fold symmetric convex polygons2Linear-time algorithm given by Mount and Ruth Silverman
Sphere3See Kepler conjecture
Bi-infinite cylinder3Bezdek and Kuperberg
All shapes contained in a rhombic dodecahedron whose inscribed sphere is contained in the shape3Fraction of the volume of the rhombic dodecahedron filled by the shapeCorollary of Kepler conjecture. Examples pictured: rhombicuboctahedron and rhombic enneacontahedron.
Hypersphere8See Hypersphere packing
Hypersphere24See Hypersphere packing