Three pentagramshaped stars (the stars formed from the diagonals of a regular pentagon) are stacked up so that the bottom two ends of the tips touch the middle ends of the tips of the star below.
(See diagram.)
The distance from the top of the stack to the floor (where the bottom star's "feet" rest) is 4 feet.
What is the distance between the bottom two ends of the tips of the stack that touch the floor?
Let the distance between the bottom two ends = 1. Then the middle two ends = phi, and the top two ends = phi². Incidentally, the diagonal of the top pentagram = phi³. I used the value phi = 1.618033988. The distance, D, between any two ends (actually the side, S, of the associated pentagon) and the horizontal diagonal directly above would be sin72ºS. So, bottom D1=sin72º and middle D2=sin72ºphi. The height, H, of the top pentagram is the sum of incircle radius (0.68819096phi²) and circumcircle radius(0.850650807phi²) of the associated pentagon and = 4.028892501. Then D1+D2+H=6.518638329. Since the stack is 4 feet, 4/D1+D2+H = the actual distance between the bottom two ends which = 0.613625085 feet.

Posted by CeeAnne
on 20041002 17:48:25 