CH.IV] GEOMETRICAL RECREATIONS '81 



Take a strip of paper or piece of tape, say, for convenience, 

 an inch or two wide and at least nine or ten inches long, rule a 

 line in the middle down the length AB of the strip, gum one 

 end over the other end B, and we get a ring like a section of 

 a cylinder. If this ring is cut by a pair of scissors along the 

 ruled line we obtain two rings exactly like the first, except that 

 they are only half the width. Next suppose that the end A is 

 twisted through two right angles before it is gummed to B 

 (the result of which is that the back of the strip at A is gummed 

 over the front of the strip at B), then a cut along the line will 

 produce only one ring. Next suppose that the end A is twisted 

 once completely round (i.e. through four right angles) before it 

 is gummed to B, then a similar cut produces two interlaced 

 rings. If any of my readers think that these results could be 

 predicted off-hand, it may be interesting to them to see if they 

 can predict correctly the effect of again cutting the rings formed 

 in the second and third experiments down their middle lines in 

 a manner similar to that above described. 



The theory is due to J. B. Listing* who discussed the case 

 when the end A receives to half-twists, that is, is twisted through 

 «i7r, before it is gummed to B. 



If m is even we obtain a surface which has two sides and two 

 edges, which are termed paradromic. If the ring is cut along 

 a line midway between the edges, we obtain two rings, each 

 of which has m half-twists, and which are linked together ^m 

 times. 



If to is odd we obtain a surface having only one side and 

 one edge. If this ring is cut along its mid-line, we obtain only 

 one ring, but it has 2m half-twists, and if m is greater than 

 unity it is knotted. 



ADDENDUM. 



Note. Page 64. One method of arranging 16 counters in 15 lines, as 

 stated in the text, is as follows. Draw a regular re-entrant pentagon 

 vertices A u A it A 3 , A it A 6 , and centre 0. The sides intersect in five 



* Vorstudien ear Xopplogie, Die Studien, Gottingen, 1847, part x. 

 B- E. 6 



