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Packing, tiling, and covering with tetrahedra
1Department of Mathematics, Program in Applied and Computational Mathematics, Princeton Institute for the Science and Technology of Materials (PRISM), and Princeton Center for Theoretical Physics, Princeton University, NJ 08544, USA.
Summary
Regular tetrahedra cannot perfectly tile 3D space. This study explores their packing, tiling, and covering limits, suggesting they may have the lowest packing density among convex bodies, challenging existing conjectures.
Area of Science:
- Geometry
- Convex Geometry
- Spatial Packing
Background:
- Regular tetrahedra cannot tile three-dimensional Euclidean space.
- Understanding the limits of packing, tiling, and covering with regular tetrahedra is an open problem.
Purpose of the Study:
- To investigate how well regular tetrahedra can pack, tile, and cover three-dimensional Euclidean space.
- To explore the packing density of regular tetrahedra.
Main Methods:
- Construction of several spatial arrangements.
- Analysis of packing, tiling, and covering properties of tetrahedra.
Main Results:
- Provided constructions for packing, tiling, and covering problems involving regular tetrahedra.
- Results indicate that regular tetrahedra may not achieve the packing density of spheres.
- The regular tetrahedron might possess the minimal packing density among all convex bodies.
Conclusions:
- The study offers solutions to packing, tiling, and covering problems for regular tetrahedra.
- Findings challenge Ulam's conjecture regarding packing density.
- Regular tetrahedra may represent the convex body with the lowest possible packing density.
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