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January 18th, 2014, 07:21 AM  #1 
Newbie Joined: Jan 2014 Posts: 2 Thanks: 0  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 ngons 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! 
January 18th, 2014, 11:02 AM  #2 
Newbie Joined: Jan 2014 Posts: 2 Thanks: 0  Re: Spirangles (polygon spirals) and graph theory question
Update: I looked further into graph coloring and the fourcolor 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 fourcolor 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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