Regular Maps

Abstract

The branch of topology that deals with regular maps on multiply-connected surfaces may be said to have begun when Kepler (1619, p. 122) stellated the regular dodecahedron {5, 3} to obtain the star-polyhedron {⁵/₂, 5} (Coxeter 1963a, pp. 49, 95–105) which is essentially a map of twelve pentagons on a surface of genus 4. The modern work on this subject received its impetus from two different sources: automorphic functions and the four-colour problem. From the former came two maps on a surface of genus 3: one consisting of 24 heptagons (Klein 1879b, pp. 461, 560) and one of 12 octagons (Dyck 1880, pp. 188, 510). From the latter came two maps on a torus: one of seven hexagons (Heawood 1890, p. 334 and Fig. 16) and one of five quadrangles (Heffter 1891, p. 491, Fig. 2). The systematic enumeration of regular maps on a surface of genus 2 was begun by ERRéRA (1922) and completed by THRELFALL (1932 a, p. 44). Concerning the torus, it was remarked by Brahana (1926, p. 238) that “There is no regular map of 8, 10 or 11 hexagons, no map of 14 hexagons although there are maps of 7, 21 and 28 hexagons”. The expression that he sought is our 8.42 (Burnside 1911, p. 418). The first mention of maps on non-orientable surfaces seems to have been by Tietze (1910).