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Inscribed Pentagon Illation (Posted on 20141101) 

At the outset, consider a circle with radius 1.
Using paper folding alone, inscribe a regular pentagon in this circle.
Solution

Comment 1 of 1

I’m
assuming that all we have is a paper circle. If we
had a circle marked on a much larger sheet it would be a
little easier.
Make two folds to form two diameters at right angles.
Consider these to be x and y axes with origin O.
Fold and double fold to form the creases x=1/2 & x=1/4.
Similarly form the creases y=1/2, y=1/4 & y=3/4.
Form a crease along the line joining P(0,1/4) to Q(1/4,3/4).
Now fold so that PQ maps to PQ’ where Q’ is on the y axis.
Form a crease through Q’ parallel to the x axis and denote
its points of intersection with the circle by A and B.
OQ’ = OP + PQ = ¼(1 + sqrt(5)) = cos 36^{o}, so /AOB = 72^{o}
and A, B are vertices of the required pentagon ABCDE.
Fold along OB so that A maps to C, and along OA so that B
maps to E. D is the point (0, 1)
Illation begets elation.

Posted by Harry
on 20141104 18:25:58 


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