I have a strip of paper with parallel sides, a pencil and a straight edge. How can I construct a regular pentagram (star) with these? And for that matter, a pentagon? Well ... the pentagon is the proper way to go.

I have what I think is a start on the right track, but I got stuck and thought I would just share it:

Fact 1: Basically, we can use the strip of paper to give us a unit measure (just fold it in half and the fold will give you a segment of length one).

Fact 2: Once we have the fold in the paper, this allows us to construct right angles, since the fold will be at a right angle to the two parallel sides of the strip.

Fact 3: Once we have a unit length, we can also constuct lenghths 1/2, 1/4, etc. of a unit by folding the paper repeatedly in the other direction (in principle as many times as we want, in practice only about 7 times, but either way it is okay, since the construction below only requires us to fold it twice).

First, construct a right triangle with base 1 and height 1/2. The hypotenuse will then be √5/4. Now, mark off 1/4 unit on the hypotenuse, giving us a segment of length (√5 -1)/4. Now, Draw a line perpendicular to the segment of length (√5 - 1)/4 passing through one of the endpoints. Now use the strip of paper to construct a right triangle with the line of length (√5 - 1)/4 as base and unit length as hypotenuse. This triangle will have angles of 72 and 18 degrees.

Since a pentagon has five interior angles of 108 (180-72) degrees, I cannot help but feel like I am on the right track here, but I could not get it any farther. The problem is that I cannot find a way to modify the construction above so that I can construct 72 (or 108) degree angles at an arbitrary point.

Any ideas?

Posted by RoyCook on 2003-09-24 13:49:20