Problem 35. Abutting, Nonoverlapping Unit Squares

1979 ◽  
pp. 82-89
Keyword(s):  
Nonoverlapping Unit

scholarly journals Uniqueness of Certain Spherical Codes

1981 ◽  
Vol 33 (2) ◽  
pp. 437-449 ◽  
Author(s):  
Eiichi Bannai ◽  
N. J. A. Sloane
Keyword(s):  
Unit Sphere ◽  
Spherical Codes ◽  
Nonoverlapping Unit

In this paper we show that there is essentially only one way of arranging 240 (resp. 196560) nonoverlapping unit spheres in R8 (resp. R24) so that they all touch another unit sphere, and only one way of arranging 56 (resp. 4600) spheres in R8 (resp. R24) so that they all touch two further, touching spheres. The following tight spherical t-designs are unique: the 5-design in Ω7, the 7-designs in Ω8 and Ω23, and the 11-design in Ω24. It was shown in [20] that the maximum number of nonoverlapping unit spheres in R8 (resp. R24) that can touch another unit sphere is 240 (resp. 196560). Arrangements of spheres meeting these bounds can be obtained from the E8 and Leech lattices, respectively. The present paper shows that these are the only arrangements meeting these bounds.

Isoperimetric Inequalities and the Dodecahedral Conjecture

1997 ◽  
Vol 08 (06) ◽  
pp. 759-780 ◽  
Author(s):  
Károly Bezdek
Keyword(s):  
Unit Sphere ◽  
Sphere Packing ◽  
Isoperimetric Inequalities ◽  
Sphere Packings ◽  
Voronoi Polyhedron ◽  
Regular Dodecahedron ◽  
Nonoverlapping Unit

The dodecahedrad conjecture, posed more than 50 years ago, says that the volume of any Voronoi polyhedron of a unit sphere packing in [Formula: see text] is at least as large as the volume of a regular dodecahedron of inradius 1. In this paper we show how the dodecahedral conjecture can be obtained from the distance conjecture of 14 and 15 nonoverlapping unit spheres and from the isoperimetric conjecture of Voronoi faces of unit sphere packings.

A NEW PROOF FOR ZASSENHAUS–GROEMER–OLER INEQUALITY

2012 ◽  
Vol 04 (02) ◽  
pp. 1250014 ◽  
Author(s):  
QINGHAI LIU ◽  
XIANG LI ◽  
LIDONG WU ◽  
HAI DU ◽  
ZHAO ZHANG ◽  
...  
Keyword(s):  
Sensor Networks ◽  
Upper Bound ◽  
Compact Convex ◽  
Convex Region ◽  
Nonoverlapping Unit

In this paper, we present a new proof for a well-known inequality, conjectured by Zassenhaus in 1947 and proved independently by Groemer in 1960 and Oler in 1961. The inequality gives an upper bound for the number of nonoverlapping unit discs whose centers can be packed into a compact convex region, and recently obtains a lot of applications in study of sensor networks.

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