Paper cutting (ConceptTopic, 4)

From Hi.gher. Space

Revision as of 23:11, 19 November 2010

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 4-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 > 0, 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.

• Cutting a 1-twist in thirds and then cutting the new 1-twist in half produces two 4-twists, linked in multiple places.

See also

• Manifold