124 



Popular Science Monthly 



this you would probably jump to the 

 conclusion that if this larger ring should 

 again be bisected lengthwise a single ring 

 with circumference again doubled would 

 be formed. Attempt to prove your 

 prediction and you will discover that 

 two rings with the same circumference 

 as the bisected ring will be formed and 

 these two rings will be doubly linked 

 together. Why were the results so 

 dissimilar? This is the explanation. 

 The original paper was twisted one half 

 a turn before the ends were pasted 

 together. After the ring was bisected 

 and the larger ring had been formed, a 

 careful examination of it would have 

 shown that the effect was the same as 

 though it had been formed from a strip 

 one end of which had been given two 

 complete revolutions before being pasted 

 to the other end. This accounts for the 

 difiference in the results. 



At this point you will begin to wonder 

 what the results would be if the ring 

 should be divided into three parts 

 instead of two, or what difference it 

 would make if the paper should be given 

 two or three or four half turns. Also 

 you will probably be convinced that it 

 would be useless to predict results and 

 that only by original investigation would 

 you be able to arrive at correct con- 

 clusions. Here are a few of the cases 

 which you might investigate. 



Prepare another ring by twisting the 

 paper one half turn, and separate it 

 lengthwise into three parts of approxi- 

 mately equal width. The result will be 

 two rings linked together, one having 

 the same circumference as the original, the 

 other having twice that circumference. 



The next step would probably be to 

 form a ring by twisting the strip two 

 half turns or one complete revolution. 

 This ring, you will find, has two surfaces 

 and two edges. Bisecting it will give 

 two rings linked together. Bisecting 

 each of the two will give a total of four 

 rings arranged in two pairs, the pairs 

 being linked together and each individual 

 also being linked to its mate. 



A ring formed with three half turns 

 will, upon being bisected, form a single 

 ring with the circumference doubled, 

 but in the ring there will be tied a 

 simple knot. Bisecting again will give 

 two rings, with the same circumference 

 as at first, knotted as the parent ring 

 and doubly linked together. 



In a similar manner the experiments 

 may be continued almost indefinitely 

 until the rings become so complicated 

 that they cannot be handled success- 

 fully. If a strip of paper is to be 

 twisted several times in forming a ring, 

 a long piece should be used, and if the 

 ring is to be cut into several parts the 

 width must be increased accordingly. 



When a ring is to be re-bisected it 

 is advisable to mark each half with a 

 pencil in order to help in studying out 

 the relations. Where several rings are 

 linked or knotted together it is some- 

 times necessary to repeat the experiment 

 several times, first tearing off certain 

 rings next other rings before the com- 

 plicated results can be understood. 



In general it will be found that with 

 one, three, five or any odd number of 

 half turns the rings will have one surface 

 and one edge. When bisected they will 

 formoneringwith circumference doubled, 

 and, beginning with the third half turn, 

 the ring will be tied in a knot the com- 

 plexity of which increases with the num- 

 ber of half turns. If these rings are again 

 bisected a pair will be formed having 

 the same shape and circumference as 

 the original and, in addition, each will 

 be linked to the other. 



With two, four or any even number of 

 half turns the original rings will have 

 two surfaces and two edges. Upon 

 being bisected two rings will be formed 

 and they will be linked together in an 

 increasingly complicated manner as the 

 number of half turns increases. A 

 second bisection will yield two pairs of 

 rings, each pair having the same cir- 

 cumference and arrangement as the 

 parent ring and, in addition, each ring 

 being linked to its mate. 



The trisection of the rings also yields 

 interesting results and opens up new 

 possibilities. To begin with, the manner 

 of cutting into three parts rings formed 

 with an odd number of half turns is 

 different from that used with rings of 

 an even number. The former, it will be 

 remembered, have but one edge. Hence, 

 to trisect such a ring, begin by cutting 

 off a strip one third the width of the 

 original. Continuing the cut, you will 

 make two complete revolutions, finally 

 coming back to the starting point, and 

 the paper will apparently form a con- 

 tinuous ring. Upon unfolding it you 

 will find two rings linked together, one 



