3‐VALENT 3‐POLYTOPES WITH FACES HAVING FEWER THAN 7 EDGES

It i s conven ien t t o have a p i c t o r i a l way of v i s u a l i z i n g a c y c l e L i n a p l a n a r g raph . To do t h i s , we c o n s t r u c t a s h e l l diagram L* f o r L a s f o l l o w s : D r a w a c y c l e w i t h t h e same number of v e r t i c e s as t h e r e a r e on L . P i ck a v e r t e x v on L a and c o n s i d e r t h e co r re spond ing v e r t e x v* of L*. For each edge a t v , n o t a n edge of L a draw a s h o r t " sp ike" i n t h e i n t e r i o r o r e x t e r i o r of t h e c y c l e of L* a t v*, acc o r d i n g as t h e edge E i s i n t h e bounded or unbounded component of L. Given a s h e l l diagram L* t h e i n t e r i o r s p i k e s can b e used t o c o n s t r u c t a graph i n t h e bounded component formed by t h e c y c l e o f L* . Such a graph i s c a l l e d a f i l l i n g of L*. S i m i l a r l y , t h e e x t e r i o r s p i k e s of L* can be used t o c o n s t r u c t an i c i n o f L*, i n t h e unbounded component of t h e c y c l e of L d i v e n a s h e l l diagram L * , i f we i n v e r t L* i n a p o i n t i n i t s i n t e r i o r , w e o b t a i n a new diagram c a l l e d t h e i n v e r t of L*. I f t h e i n v e r t of L* h a s t h e same s p i k e p a t t e r n as L*, up t o c y c l i c o r d e r , L* i s c a l l e d s e l f i n v e r t e d . For s e l f i n v e r t e d diagrams, f i l l i n g s can be used as i c i n g s and c o n v e r s e l y . A s h e l l diagram L* i s s a i d t o be r e p l i c a b l e by k hexagons, i f L* can be i c e d by k hexagons t o o b t a i n a s h e l l diagram M* w i t h t h e same s p i k e p a t t e r n a s L*. F i g u r e l ( a ) shows a s e l f i n v e r t e d s h e l l diagram r e p l i c a b l e w i t h 1 hexagon. F i g u r e l ( b ) how t h i s r e p l i c a t i o n i s accomplished. F i g u r e l ( c ) shows an i c i n g w i t h p '=l ,p4 '=1 , p 5 ' = 1 , and a f i l l i n g w i t h