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I'd never seen a golden triangle as the basis of a logarithmic spiral before...just the rectangle like I wrote about a couple of weeks ago. A golden triangle is an isosceles triangle where the length of the longer sides is approximately 1.6180339 times the length of the shorter side...or exactly (1+sqr(5)/2) times as long as shown here:
The vertex angle is 36 degrees and the two angles at the base are 72 degrees each. If you put ten of these triangles together you get a decagon:
And if you put five together, base angle to base angle, you get a pentagram:
Let's get back to the logarithmic spiral which is what got this whole thing started in the first place.
From Wikipedia:
The golden triangle is used to form a logarithmic spiral. By bisecting the base angles, a new point is created that in turn, makes another golden triangle. The bisection process can be continued infinitely, creating an infinite number of golden triangles. A logarithmic spiral can be drawn through the vertices. This spiral is also known as an equiangular spiral, a term coined by René Descartes. "If a straight line is drawn from the pole to any point on the curve, it cuts the curve at precisely the same angle," hence equiangular.And it looks something like this:
Math can be a beautiful thing!
Interesting days
Tomorrow - Mitten Tree Day, Miners' Day and Pawnbrokers Day
Next Monday - Gingerbread House Day and Poinsettia Day
January 5 - Whipped Cream Day and Bird Day
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