156 KANSAS UNIVERSITY QUARTERLY. 



of a plane or a cylinder and the bifacial surface is always possible, 

 though a physical deformation of the one into the other cannot be 

 done. But the surface can be put entirely in the plane, if a number 

 of foldings are made. Each folding is then ecpiivalent to a twist. 

 Let III be the number of required foldings, so that /// — 2(2;; — i) 

 foldings are left which correspond to zero twists. Now two twists of 

 opposite sense cancel each other; two folds as given in Fig. 2, Plate 

 XIII, illustrate this process, while the folds as in Fig. i represent 

 twists of the same sense. From this it follows that /// — ^(2// — 1)=/6 

 must be an even number, therefoJ'e m must be an even number; i. e. 



Only an even number of foldings can transform a bifacial surface 

 into a plane figure. See Figs. 4 and 7. 



Also ;// — (2;/ — 1)=^/' must be an even number; but this is true 

 only when ni is odd. Therefore:-- 



An original unifacial surface can be transformed into a plane 

 figure only bv an odd number oj folds. 



We have also the converse theorem: 



If such a plane figure has an odd number of foldings, it repre- 

 sents a unifacial surface. Figs. 3, 5 and 6 illustrate these cases. 



It will be noticed that certain parts of the plane figures cover 

 each other either twice or several times. 



There are a great number of representations possible according to 

 the value of /'. The case k=^o, or ;//:=:2(2;/— i ), is the most inter- 

 esting, because it gives the best conception of the twists and the 

 knots. Figs. 8 and 9 on Plate XIII show the cases n=\ and //=3. 

 For 2/z=:i there is no knot, but a bifacial surface of two twists 

 cannot be represented by a plane figure of only two folds. A 

 bifacial plane figure has therefore at least four folds. In this case 

 k=2, and it is illustrated in Fig. 7. Fig. 3 represents a plane 

 unifacial surface with three twists. 



In Figs. 8 and 9 the character of the knot is apparent at the 

 center. Each part of the surface crossing the center covers the 

 foregoing part. All the parts together make up a geometrical group* 

 as is evident from the standpoint of Klein's definition of a group in 

 the most general sense. 



From the generation of unifacial and bifacial surfaces as treated in 

 this paper, it is obvious: A unifacial surface can never have a knot. 



How loup-cuts affect a bifacial surface, I have mentioned already. 

 Fig. 10 illustrates such a case of three loup-cuts being made in a 

 unifacial surface with three twists. Another case is added, Fig. 11, 

 where four loup-cuts were make in a bifacial surface with four twists. 



*3ee Klein's Eiuleiiung in die lioehere Geuinetrie, Vol. II. 



