Name
Abstract | ||
|---|---|---|
We give a classification of all equivelar polyhedral maps on the torus. In particular, we classify all triangulations and quadrangulations of the torus admitting a vertex transitive automorphism group. These are precisely the ones which are quotients of the regular tessellations {3,6}, {6,3} or {4,4} by a pure translation group. An explicit formula for the number of combinatorial types of equivelar maps (polyhedral and non-polyhedral) with n vertices is obtained in terms of arithmetic functions in elementary number theory, such as the number of integer divisors of n. The asymptotic behaviour for n-~ is also discussed, and an example is given for n such that the number of distinct equivelar triangulations of the torus with n vertices is larger than n itself. The numbers of regular and chiral maps are determined separately, as well as the ones for all other kinds of symmetry. Furthermore, arithmetic properties of the integers of type p^2+pq+q^2 (or p^2+q^2, resp.) can be interpreted and visualized by the hierarchy of covering maps between regular and chiral equivelar maps or type {3,6} (or {4,4}, resp.). |
Year | DOI | Venue |
|---|---|---|
2008 | 10.1016/j.ejc.2008.01.010 | Eur. J. Comb. |
Keywords | Field | DocType |
arithmetic property,modular group,n vertex,binary quadratic form msc: primary 52b70,11a25.,05c30,equivelar map,polyhedral map,arithmetic function,equivelar,secondary 05c10,elementary number theory,equivelar polyhedral map,regular tessellation,dirichlet character,weakly regular,distinct equivelar triangulations,triangulated tori,chiral equivelar map,regular tessellations,chiral map,binary quadratic form,number theory | Integer,Discrete mathematics,Arithmetic function,Combinatorics,Vertex (geometry),Quotient,Torus,Tessellation,Divisor,Number theory,Mathematics | Journal |
Volume | Issue | ISSN |
29 | 8 | 0195-6698 |
Citations | PageRank | References |
6 | 0.87 | 3 |
Authors | ||
2 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Ulrich Brehm | 1 | 6 | 0.87 |
| Wolfgang Kühnel | 2 | 28 | 5.29 |