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Topic: Construction of a Regular Polygon (Read 2317 times) |
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Ronno
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This is not really a puzzle. I have a geometric (compass and straight edge) construction that can closely approximate regular polygons of any number of sides. I want to know why this method works. All replies are welcome.
AB is the diameter of a circle with center O. CAB is an equilateral triangle. D divides AB in the ratio 2/(n-2) where n is the number of sides of the polygon. That is, AD/AB=2/n. CD extended cuts the circle at E. AE is the side of the polygon that is replicated along the circle.
The attached picture is for n=7. As you may see, the last point differs slightly from A. This construction works very well for all n that I have tested.
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Eigenray
wu::riddles Moderator
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Re: Construction of a Regular Polygon
« Reply #1 on: Oct 2nd, 2008, 4:41am » |
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If E were the correct point, and AB = n, then according to my calculations
AD = n sin( /n) [ cos(/n) +  3 sin(/n) ] / [ 3 + sin(2/n) ]
= 1.996 for n=7.
AD is exactly 2 for n=2,3,4,6; for n  5 we have
AD = 2.001, 2.000, 1.996, 1.992, 1.986, 1.980, 1.974, 1.968, ...
But AD actually converges to /sqrt3 = 1.814 as n   :
AD = /3 + 2/(3n) + O(1/n2).
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Ronno
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Re: Construction of a Regular Polygon
« Reply #2 on: Oct 2nd, 2008, 5:26am » |
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Thanks. It seems I was going in the reverse way. I tried comparing the angle AOE from my construction with 2Pi/n.
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