Let G be a finite group. We show that if |G| = pqrs, where p, q, r, and s are distinct odd primes, then every connected Cayley graph on G has a hamiltonian cycle.
We obtain a characterization of the Hamilton paths in the cartesian product Z x Zb of two directed cycles. This provides a correspondence between the collection of Hamilton paths in Za x Zb and the set of visible lattice points in the triangle with vertices (0,O) , (0,a) , and (b,0) . We use this correspondence to show there is a Hamilton circuit in the cartesian product of any three or more nontrivial directed cycles. Our methods are a synthesis of the theory of torus knots and the study of Hamilton paths in Cayley digraphs of abelian groups.
A process f o r c r e a t i n g r e p e a t i n g p a t t e r n s of t h e h y p e r b o l i c plane i s d e s c r i b e d .Unlike t h e Euclidean p l a n e , t h e h y p e r b o l i c p l a n e has i n  i n i t e l y many d i f f e r e n t kinds of r e p e a t i n g p a t t e r n s .The Poincare c i r c l e model of h y p e r b o l i c geometry has been used by t h e a r t i s t M. C . Escher t o d i s p l a y i n t e r l o c k i n g , r e p e a t i n g , h y p e r b o l i c p a t t e r n s . A program has been designed which w i l l do t h i s a u t o m a t i c a l l y . The u s e r e n t e r s a m o t i f , o r b a s i c s u b p a t t e r n , which could t h e o r e t i c a l l y be r e p l i c a t e d t o f i l l t h e h y p e r b o l i c p l a n e . I n p r a c t i c e , t h e r e p l i c a t i o n process can be i t e r a t e d s u f f i c i e n t l y o f t e n t o appear t o f i l l t h e c i r c l e model. There i s an i n t e r a c t i v e "boundary procedure" which allows t h e u s e r t o d e s i g n a motif which w i l l b e r e p l i c a t e d i n t o a completely i n t e r l o c k i n g p a t t e r n . D u p l i c a t i o n of two of Escher's p a t t e r n s and some e n t i r e l y new p a t t e r n s a r e included i n t h e paper.K N WORDS A m P W S E S : h y p e r b o l i c geometry, --Poincare c i r c l e model, t e s s e l l a t i o n s , symmetry groups, i n t e r a c t i v e g r a p h i c s , m o t i f , r e p e a t i n g p a t t e r n , M. C. Escher, computer a r t . which r e p e a t i n g p a t t e r n s of t h e h y p e r b o l i c p l a n e may be generated. A r e p e a t i n g p a t t e r n i s d e f i n e d t o be a p a t t e r n which remains i n v a r i a n t under c e r t a i n t r a n s f o r m a t i o n s of t h e h y p e r b o l i c plane.The P o i n c a r e c i r c l e model of h y p e r b o l i c geometry g i v e s a c o n c r e t e r e a l i z a t i o n of t h e h y p e r b o l i c p l a n e [Coxeter, 1961 1. The p o i n t s of t h i s model a r e t h e i n t e r i o r p o i n t s of a c i r c l e , c a l l e d t h e bounding, c i r c l e ; t h e h y p e r b o l i c l i n e s a r e r e p r e s e n t e d by t h e diameters o f t h e bounding c i r c l e and c i r c u l a r a r c s orthogonal t o t h e bounding c i r c l e .Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission.01981 ACM 0-8971-045-1/81-0800-0215 $00.75Theh y p e r b o l i c t r a n s f o r m a t i o n s of most i n t e r e s t t o us a r e r e f l e c t i o n s a c r o s s h y p e r b o l i c l i n e s and r o t a t i o n s about p o i n t s . Hyperbolic r e f l e c t i o n s c o n s i s t of o r d i n a r y Euclidean r e f l e c t i o n s a c r o s s di...
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