The Optimal Packing of Eight Points in the Real Projective Plane
| Autor: | Hans Parshall, Dustin G. Mixon |
|---|---|
| Rok vydání: | 2019 |
| Předmět: |
FOS: Computer and information sciences
Euclidean space Computer Science - Information Theory Information Theory (cs.IT) General Mathematics 010102 general mathematics Metric Geometry (math.MG) 0102 computer and information sciences 01 natural sciences Combinatorics Mathematics - Metric Geometry 010201 computation theory & mathematics Real projective plane FOS: Mathematics Mathematics - Combinatorics Combinatorics (math.CO) 0101 mathematics Internal and external angle Mathematics |
| Zdroj: | Experimental Mathematics. 31:474-485 |
| ISSN: | 1944-950X 1058-6458 |
| Popis: | How can we arrange $n$ lines through the origin in three-dimensional Euclidean space in a way that maximizes the minimum interior angle between pairs of lines? Conway, Hardin and Sloane (1996) produced line packings for $n \leq 55$ that they conjectured to be within numerical precision of optimal in this sense, but until now only the cases $n \leq 7$ have been solved. In this paper, we resolve the case $n = 8$. Drawing inspiration from recent work on the Tammes problem, we enumerate contact graph candidates for an optimal configuration and eliminate those that violate various combinatorial and geometric necessary conditions. The contact graph of the putatively optimal numerical packing of Conway, Hardin and Sloane is the only graph that survives, and we recover from this graph an exact expression for the minimum distance of eight optimally packed points in the real projective plane. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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