20 May 2008

Origami Pentagon

A regular dodecahedron has faces which are regular pentagons. Looking around for a good method to construct pentagons by paper folding I came across one which seemed nice. Unfortunately, the method is not exact (as often purported to be) but only approximate. The approximation is good, but, if on top of the mathematical error one adds the physical one, the resulting pentagon is often visibly non-canonical.

The approximate method can be described as follows:

Start with a square. Let A, B be the middle points of two adjacent sides and C the vertex farthest away from these two points. Let M be the middle of the segment BC. Consider the shaded rectangle whose diagonal is MC and rotate it 90 degrees counterclockwise so C goes to C1. Pick the point K so that KC1:KM = 1:3. The "pentagon" has vertices A, B, C, K and the symmetric of K.

To see that it's not a regular pentagon, let the side of the original square have length 8u where u is the square root of 2 (for no good reason other than that I wanted to end up with integers). Then, using the Pythagorean theorem several times, we can see that AB equals sqrt(256)=16, but BK equals sqrt(250) which equals 15.81... (Pretty good approximation.) Notice that MK passes through the middle of BC and is perpendicular to it, so BK equals CK. By symmetry, 4 of the 5 sides of the pentagon have length 15.81... and only one has length 16.

Does anyone know an exact (and simple...) origami method for constructing regular pentagons?

1 comment:

  1. http://origami.oschene.com/cp/Decagon%20SCP.pdf

    I saw this on a calendar last year, I lost it for a while and found it again on the net. It works!



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It's about counting, but when things get too large.
Put otherwise, it's about addition of positive numbers, but when these numbers are far too many.

The principle of dynamic programming

max_{x,y} [f(x) + g(x,y)] = max_x [f(x) + max_y g(x,y)]

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