From:Robin Chapman <rjc@noether.ex.ac.uk>Date:Fri, 17 Oct 1997 08:01:44 GMTNewsgroups:sci.math.researchSubject:Re: Volumes of pieces of dodecahedron.

David Epstein wrote: > > Take a regular dodecahedron and set it down on a horizontal surface. > Then there are four levels of vertices, one at the bottom face, > one the top face, and two levels in between. Now saw through the > dodecahedron horizontal at the two interior levels. The dodecahedron > is sawn into three pieces, of which the top and bottom piece are > congruent. How can one show that the three pieces each have the same > volume without doing a long and horrible integration? > > Is this a special case of some more general theorem? Let the dodecahedron have height h and face area A. One can decompose it into 12 pentagonal pyramids of base area A and height h/2. Then the volume of the dodecahedron is 12((h/2)A/3) = 2Ah. It suffices to show that the bottom piece has volume 2Ah/3. This piece is a frustum of a pentgonal pyramid. Its top face has area tau^2 A where tau =(1+sqrt(5))/2. (This comes from tau being the ratio of diagonal to side of a regular pentagon.) Let h_1 and h_2 be the heights of the bottom and central pieces. Then 2h_1 + h_2 = h. We need to determine h_1. But the ratio of (h_1 + h_2) to h_1 is the same as the ratio of the distance of the vertices A_1 and A_2 from the side A_3 A_4 in the regular pentagon A_1 A_2 A_3 A_4 A_5, and is seen to be tau. Then we find h_1 = tau^{-2}h and h_2 = tau^{-3}h. (Naturally everything is in golden ratio). We can express the frustum in question as the "difference" of two pentagonal pyramids. Let d be the height of the smaller of these, so that d + h_1 is the height of the larger. Then the ratio of d + h_1 to d is the same as the ratio of the sidelengths of the bases of the pyramids, i.e., it is tau. Thus d = tau^{-1}h and d + h_1 = h. Thus the large pyramid has volume (tau^2 A)h/3 and the small one has volume A tau^{-1}h/3. Thus the frustum has volume (tau^2-tau^{-1})Ah/3 = 2Ah/3. Not an integration in sight! I'm sure Euclid could have come up with such an argument. -- Robin Chapman "256 256 256. Department of Mathematics O hel, ol rite; 256; whot's University of Exeter, EX4 4QE, UK 12 tyms 256? Bugird if I no. rjc@maths.exeter.ac.uk 2 dificult 2 work out." http://www.maths.ex.ac.uk/~rjc/rjc.html Iain M. Banks - Feersum Endjinn