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Topic: ways a curving line can cross itself
Replies: 2   Last Post: Nov 8, 2010 6:18 PM

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digory

Posts: 2
Registered: 10/31/10
ways a curving line can cross itself
Posted: Oct 31, 2010 11:47 PM
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if you draw a line on a piece of paper
(not a perfectly straight one but a line that can curve as it likes)
and if that line bends around and crosses itself once
there are two possible configurations
either it will hide both the start and end of the line inside the loop
or it will leave them outside the loop
if it crosses itself twice there are twelve possible configurations
these twelve form three families
if A is a time that the line passes through the first crossing
and B is a time it passes through the second crossing
these three families are
AABB (which has four members)
ABAB (which has two)
ABBA (which has six)
each letter appears twice because that is what a crossing is, a time the line comes to the same place twice.
there is no such family as BAAB for example since it is just the wrong way to write ABBA
using this notation it would appear that their are 15 possible families of three-crossing figures but two are equivalent to each other (if the start and end of the line are the same) and two have no members because they are impossible to actually draw.
-these are very interesting shapes to play with i encourage you to take a stroll in this little world and draw them yourself (i'll post a picture if i can get my scanner working)
-has this been studied? if so do you know what this set of figures is called? or is it equivalent to some other system that has been studied?
-the closest thing i could find was knot theory but that is quite different, as far as i can tell it has only a cosmetic similarity.



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