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Geometry and arithmetic of crystallographic sphere packings
Alex Kontorovich1,2, Kei Nakamura3
1Department of Mathematics, Rutgers University, New Brunswick, NJ 08854; alex.kontorovich@rutgers.edu.
Summary
We introduce crystallographic sphere packings, which have limit sets from hyperbolic reflection groups. An infinite family of such packings with integer reciprocal radii is shown, with superintegral packings limited to 20 dimensions.
Area of Science:
- Hyperbolic geometry
- Discrete groups
- Sphere packing
Background:
- Sphere packing problems are fundamental in geometry.
- Hyperbolic reflection groups play a key role in understanding geometric structures.
Purpose of the Study:
- To define and explore the properties of "crystallographic sphere packings."
- To investigate the existence and classification of such packings, particularly those with integer reciprocal radii and "superintegral" properties.
Main Methods:
- Defining crystallographic sphere packings based on limit sets of hyperbolic reflection groups.
- Constructing an infinite family of conformally inequivalent packings.
- Proving the finite existence of superintegral packings within specific commensurability classes and dimensional limits.
Main Results:
- An infinite family of conformally inequivalent crystallographic sphere packings with integer reciprocal radii was exhibited.
- It was proven that superintegral crystallographic sphere packings exist only in finitely many commensurability classes.
- These superintegral packings are restricted to at most 20 dimensions.
Conclusions:
- Crystallographic sphere packings offer a rich area for geometric study.
- The findings provide constraints on the types and dimensions of specific crystallographic sphere packings, particularly superintegral ones.
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