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January 18th, 2014, 07:21 AM   #1
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Spirangles (polygon spirals) and graph theory question

Hi, So this is a question that I thought of when I was about 14 and haven't had answered since (2 years of waiting!).

I was drawing the Fibonnaci squares (like this http://mathforum.org/dr.math/faq/fibonacci.squares1.gif; I don't know what it's called) and coloured the first square and spiraled outwards to another square that wasn't touching the first square. I realized that this applied to any 'square spiral' (cf. http://imgur.com/a/KxIEG). I continued experimenting and found a trend with different shapes.

I put the sequence that appears in Wolfram Alpha and got this: http://imgur.com/a/Tcsme.

I have asked multiple teachers but none of them knew and the only clue I got was that it might have something to do with graph theory.

Can anyone explain what the sequence is and why the shapes' sequences are paired (if n is the number of sides differences in sequences are: lens and triangle: n+1,n+1,n+1... ; square and pentagon: n+1,2,n+1,2... ; hexagon and heptagon: n+1,2,2,n+1,2,2... ; octogan and nonagon: n+1,2,2,2,n+1,2,2,2... )?

I looked at the very basics of graph theory and tried it: http://imgur.com/a/PzPGG. I haven't tried with other shapes yet though.

(I asked this question on the math subreddit on reddit and only got this response: "For the n-gons where n is an even number, all of this makes perfect sense. You jump n+1 to get to the next "shell," and then jump forward 2 exactly n/2 times to fill in every other piece in that shell, then jump to the next shell with another n+1 jump. The odd numbers are curious, though. I might come back to this. EDIT: You may as well as concentric circles to this. It's a weird case, but a case nonetheless.")

Thanks!
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January 18th, 2014, 11:02 AM   #2
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Re: Spirangles (polygon spirals) and graph theory question

Update: I looked further into graph coloring and the four-color theorem. I realised I was adding the vertices wrong in my first edit. I should have done it this way: http://upload.wikimedia.org/wikipedi..._Graph.svg.png. I then realised that apart from the lens and concentric circles, all the graphs only contained triangular circuits. This means that the four-color theorem applies to the polygonal spirangles. It explains why the triangle has differences in the sequence of: +4,+4,+4... It haven't linked this to the other spirangles yet though.
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