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��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

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