Paper cutting (ConceptTopic, 4)
From Hi.gher. Space
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- | This page documents the results cutting a twisted loop of paper | + | This page documents the results cutting a twisted loop of paper. |
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+ | == Cutting in half == | ||
http://img59.exs.cx/img59/2479/cut.png | http://img59.exs.cx/img59/2479/cut.png | ||
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== See also == | == See also == | ||
*[[Manifold]] | *[[Manifold]] | ||
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+ | [[Category:Topology]] |
Revision as of 20:14, 31 August 2008
This page documents the results cutting a twisted loop of paper.
Cutting in half
http://img59.exs.cx/img59/2479/cut.png
- A 0-twist (hose) goes to two separate 0-twists:
- A 1-twist (Möbius strip) goes to a long 2-twist:
- A 2-twist goes to two linked together 2-twists:
- A 3-twist goes to a long 8-twist containing a trefoil knot:
- A 4-twist goes to two linked together 4-twists:
- A 5-twist goes to a long 12-twist containing a knot with five crossings:
- A 6-twist goes to two linked together 6-twists:
General rule
We can deduce that, when n > 1, cutting an n-twist will produce a single strip with n crossings and 2(n+1) twists if n is odd, or two linked strips each with n twists if n is even.
Cutting in thirds
It is easy to see that cutting a loop into thirds rather than in half would be the same as above for an even number of twists. Therefore, the following concerns only loops with odd numbers of twists.
- A 1-twist goes to a short 1-twist linked to a long 4-twist:
- A 3-twist goes to a short 3-twist linked to a long 8-twist containing a trefoil knot:
(There are more crossings between the two loops than shown above)
Repeated cuttings
- Cutting a 3-twist in half produces an 8-twist with a trefoil knot. Cutting this in half produces two interlinked 8-twists, each with their own trefoil knot, but also crossing in multiple other places.