Coordination sequences for hyperbolic tilings

✍ Scribed by O'Keeffe, M.

• Book ID: 120507901
• Publisher: Oldenbourg Wissenschaftsverlag
• Year: 1998
• Tongue: English
• Weight: 815 KB
• Volume: 213
• Category: Article
• ISSN: 2194-4946
• DOI: 10.1524/zkri.1998.213.3.135

✦ Synopsis

Abstract

Coordination sequences have been determined for tilings {p,q} of the hyperbolic plane in which__q p__-gons meet at each vertex. The number of vertices in successive shells is related by a recurrence to the numbers in earlier shells and grows exponentially. General expressions for the recurrences are given and numerical expressions for the growth rate are determined. For__p__= 3 – 10, 12, 14, 16 and 18 exact expressions, that require finding roots of polynomials of degree ≤4, are given for the coordination sequence. Some differences between hyperbolic tilings and 3-dimensional nets are noted.