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© Copyright 2002, Jim Loy
The puzzle: On the left is an object made up of five
irregular tetrahedra, glued together at some of their faces. All of the edges
are of length one, except for the pole AB. How long is AB? And what is CD/AB
(CD is the circumradius of the pentagon, the radius of the circumcircle)?
The solution: This is an application of
The Pythagorean Theorem. Not counting
points A and B, the other five points form the vertices of a regular pentagon
(see The Regular Pentagon). In that
article (addendum #1), we see that segment CD (diagram on the right) is equal
to the golden ratio, phi = (1+sqr(5))/2. That addendum shows how that fact is
deduced. Using the Pythagorean theorem, we can find x in the diagram. Phi^2 =
(1/2)^2+(x+y)^2 and x^2 = (1/2)^2+y^2 are two equations in two unknowns, and we
should be able to find x = sqrt(50+10 sqrt(5))/10.
In the original diagram, F would be at the center of the pole AB. Triangle CAF is a right triangle with sides 1, x, and AB/2. Using the Pythagorean theorem again, then the quadratic formula, we find that AB = sqrt(50-10 sqrt(5))/10 or about 0.5257311121.
Now we have reason to believe that this ugly answer cannot be simplified. Looking up the circumradius of a pentagon with unit sides in a math tables book, we find that it is sqrt(50+10 sqrt(5))/10 (same as we got above for x). Surely someone would have simplified that in the thousands of years that that has been know. And if one can't be simplified, then the other probably cannot. An expression of the form sqrt(a+sqrt(b)) can sometimes be simplified by trying this: sqrt(a+sqrt(b)) = sqrt(x) + sqrt(y). Squaring both sides, we get a+sqrt(b) = x+y + 2sqrt(xy). If there is a simple expression like this, then a = x+y and b = xy sqrt(2). So we can solve for a and b. Presumably, that doesn't help with our answer, above.
So, what is CD/AB in our first diagram? It is sqrt(50-10sqrt(5))/sqrt(50+10sqrt(5)). That can be simplified. 1/sqrt(a+b) = sqrt(a-b)/a^2-b^2, which should help. And we find that it is (1+sqrt(5))/2, also known as the Golden Ratio.
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