what surface has only one side

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what surface has only one side

Post  tom sawyer on Mon Apr 02, 2012 6:47 pm



The Möbius strip has several curious properties. A line drawn starting from the seam down the middle will meet back at the seam but at the "other side". If continued the line will meet the starting point and will be double the length of the original strip. This single continuous curve demonstrates that the Möbius strip has only one boundary.
Cutting a Möbius strip along the center line yields one long strip with two full twists in it, rather than two separate strips; the result is not a Möbius strip. This happens because the original strip only has one edge that is twice as long as the original strip. Cutting creates a second independent edge, half of which was on each side of the scissors. Cutting this new, longer, strip down the middle creates two strips wound around each other, each with two full twists.
If the strip is cut along about a third of the way in from the edge, it creates two strips: One is a thinner Möbius strip — it is the center third of the original strip, comprising 1/3 of the width and the same length as the original strip. The other is a longer but thin strip with two full twists in it — this is a neighborhood of the edge of the original strip, and it comprises 1/3 of the width and twice the length of the original strip.
Other analogous strips can be obtained by similarly joining strips with two or more half-twists in them instead of one. For example, a strip with three half-twists, when divided lengthwise, becomes a strip tied in a trefoil knot. (If this knot is unravelled, the strip is made with eight half-twists in addition to an overhand knot.) A strip with N half-twists, when bisected, becomes a strip with N+1 full twists. Giving it extra twists and reconnecting the ends produces figures called paradromic rings.
A strip with an odd-number of half-twists, such as the Möbius strip, will have only one surface and one boundary. A strip twisted an even number of times will have two surfaces and two boundaries.
If a strip with an odd number of half-twists is cut in half along its length, it will result in a longer strip, with the same number of loops as there are half-twists in the original. Alternatively, if a strip with an even number of half-twists is cut in half along its length, it will result in two conjoined strips, each with the same number of twists as the original. Rolling Eyes

Is it the inside or outside

tom sawyer
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Re: what surface has only one side

Post  Guest on Mon Apr 02, 2012 6:55 pm

The top.... it can never be a side or the bottom.

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