- References
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 |
|---|---|---|---|---|
| All shapes that tile space | all | 1 | By definition | |
| Circle, Ellipse | 2 | π/{{sqrt|12}} ≈ 0.906900}} | Proof attributed to Thue[4] | |
| Smoothed octagon | 2 |  | Reinhardt[5] | |
| All 2-fold symmetric convex polygons | 2 | Linear-time (in number of vertices) algorithm given by Mount and Ruth Silverman[6] | ||
| Sphere | 3 | π/{{sqrt|18}} ≈ 0.7404805}} | See Kepler conjecture | |
| Bi-infinite cylinder | 3 | π/{{sqrt|12}} ≈ 0.906900}} | Bezdek and Kuperberg[7] | |
| 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[8][9] | |
| Hypersphere | 24 |  | See Hypersphere packing |
References
1. ^1 {{cite arXiv |first=András | last=Bezdek | first2=Włodzimierz | last2=Kuperberg |eprint=1008.2398v1 |title=Dense packing of space with various convex solids |class=math.MG |year=2010}}
2. ^{{cite journal | last=Fejes Tóth | first=László | title=Some packing and covering theorems | journal=Acta Sci. Math. Szeged | volume=12 | year=1950}}
3. ^{{cite journal | title=Optimality and uniqueness of the Leech lattice among lattices | last=Cohn | first=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.MG/0403263 }}
4. ^{{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}}
5. ^{{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}}
6. ^{{cite journal | title=Packing and covering the plane with translates of a convex polygon | last=Mount | last2=Silverman | first=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}}
7. ^{{cite journal | first=András | last=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}}
8. ^{{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}}
9. ^{{cite arXiv| first1 = Maryna | last1 = Viazovska |authorlink1 = Maryna Viazovska| year = 2016 | title = The sphere packing problem in dimension 8 | eprint = 1603.04246 | class = math.NT }}
2. ^{{cite journal | last=Fejes Tóth | first=László | title=Some packing and covering theorems | journal=Acta Sci. Math. Szeged | volume=12 | year=1950}}
3. ^{{cite journal | title=Optimality and uniqueness of the Leech lattice among lattices | last=Cohn | first=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.MG/0403263 }}
4. ^{{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}}
5. ^{{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}}
6. ^{{cite journal | title=Packing and covering the plane with translates of a convex polygon | last=Mount | last2=Silverman | first=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}}
7. ^{{cite journal | first=András | last=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}}
8. ^{{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}}
9. ^{{cite arXiv| first1 = Maryna | last1 = Viazovska |authorlink1 = Maryna Viazovska| year = 2016 | title = The sphere packing problem in dimension 8 | eprint = 1603.04246 | class = math.NT }}
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