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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.{{cite arXiv |first1=András | last1=Bezdek | first2=Włodzimierz | last2=Kuperberg |eprint=1008.2398v1 |title=Dense packing of space with various convex solids |class=math.MG |year=2010}} 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.{{cite journal | last=Fejes Tóth | first=László | title=Some packing and covering theorems | journal=Acta Sci. Math. Szeged | volume=12 | year=1950}} 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.{{cite journal | title=Optimality and uniqueness of the Leech lattice among lattices | last1=Cohn | first1=Henry | last2=Kumar | first2=Abhinav | pages=1003–1050 | volume=170 | year=2009 | issue=3 | journal = Annals of Mathematics | doi=10.4007/annals.2009.170.1003| arxiv=math/0403263 | s2cid=10696627 }}

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ImageDescriptionDimensionPacking constant || Comments
File:Rhombic dodecahedra.pngMonohedral prototilesall1Shapes such that congruent copies can form a tiling of space
File:Circle packing (hexagonal).svgCircle, Ellipse2{{math|π/{{sqrt|12}} ≈ 0.906900}}Proof attributed to Thue{{cite arXiv |last1=Chang|first1=Hai-Chau |last2=Wang|first2=Lih-Chung |authorlink= |eprint=1009.4322v1 |title=A Simple Proof of Thue's Theorem on Circle Packing |class=math.MG |year=2010}}
File:2-d_pentagon_packing_dual.svgRegular pentagon2 \frac{5 - \sqrt{5}}{3} \approx 0.92131 Thomas Hales and Wöden Kusner{{cite arXiv | last1=Hales |last2=Kusner | first1=Thomas |first2 = Wöden| title=Packings of regular pentagons in the plane |year=2016 |eprint = 1602.07220 | class=math.MG }}
File:Smoothed Octagon Packed.svgSmoothed octagon2\eta_{so} = \frac{ 8-4\sqrt{2}-\ln{2} }{2\sqrt{2}-1} \approx 0.902414 Reinhardt{{cite journal | last=Reinhardt | first=Karl | title=Über die dichteste gitterförmige Lagerung kongruente Bereiche in der Ebene und eine besondere Art konvexer Kurven | journal=Abh. Math. Sem. Univ. Hamburg | volume=10 | pages=216–230 | year=1934 | doi=10.1007/bf02940676 | doi-access= | s2cid=120336230 }}
File:Regular decagon.svgAll 2-fold symmetric convex polygons2Linear-time (in number of vertices) algorithm given by Mount and Ruth Silverman{{cite journal | title=Packing and covering the plane with translates of a convex polygon | last1=Mount | last2=Silverman | first1=David M. | first2 = Ruth | doi=10.1016/0196-6774(90)90010-C | journal=Journal of Algorithms | volume=11 | issue=4 | year=1990 | pages=564–580}}
File:FCC closed packing tetrahedron (20).jpgSphere3{{math|π/{{sqrt|18}} ≈ 0.7404805}}See Kepler conjecture
File:Red cylinder.svgBi-infinite cylinder3{{math|π/{{sqrt|12}} ≈ 0.906900}}Bezdek and Kuperberg{{cite journal | first1=András | last1=Bezdek | first2=Włodzimierz | last2=Kuperberg | title=Maximum density space packing with congruent circular cylinders of infinite length | journal=Mathematika | volume=37 | year=1990 | pages=74–80 | doi=10.1112/s0025579300012808}}
Half-infinite cylinder3{{math|π/{{sqrt|12}} ≈ 0.906900}}Wöden Kusner{{cite journal | first1=Wöden | last1=Kusner | title=Upper bounds on packing density for circular cylinders with high aspect ratio | journal=Discrete & Computational Geometry | volume=51| issue=4 | year=2014 | pages=964–978 | doi=10.1007/s00454-014-9593-6| doi-access=free | s2cid=38234737 | arxiv=1309.6996 }}
File:Small rhombicuboctahedron.png File:Rhombic enneacontahedron.pngAll 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.
Hypersphere8\frac{(\frac {\pi}{2})^4}{4!} \approx 0.2536695See Hypersphere packing{{citation|last1=Klarreich|first1=Erica|authorlink1=Erica Klarreich|title=Sphere Packing Solved in Higher Dimensions|url=https://www.quantamagazine.org/20160330-sphere-packing-solved-in-higher-dimensions|magazine=Quanta Magazine|date=March 30, 2016}}{{cite journal| first1 = Maryna | last1 = Viazovska |authorlink1 = Maryna Viazovska| year = 2016 | title = The sphere packing problem in dimension 8 | journal = Annals of Mathematics | volume = 185 | issue = 3 | pages = 991–1015 | doi = 10.4007/annals.2017.185.3.7 | arxiv = 1603.04246 | s2cid = 119286185 }}
Hypersphere24\frac{(\frac {\pi}{2})^{12}}{12!} \approx 0.000000471087See Hypersphere packing

References

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