Date: 10/29/2003 at 00:25:39 From: Alan Subject: project ideas on a polyhedron Dear Dr Math, I read about the tetrahedron and it fascinated me. I would like to do a science project having someting to do with this. What can I try?
Date: 10/29/2003 at 06:43:53 From: Doctor Korsak Subject: Re: project ideas on a polyhedron Hello Alan, For your science project you can build a tetrahedron from cardboard or perhaps clear plastic sheets cut into four triangles that can be glued together. I gather that if you read about it, you know the arrangement of the four sides and six edges forming a so-called closed 2-D surface in 3-D space. Now, if you want something more exciting to add to that, make out of cardboard the only other known polyhedron (as far as I know) having the same property as the tetrahedron: every possible edge made by joining any two vertices of the polyhedron is already in the polyhedron. It is a 7 vertex polyhedron, as described in the following book: "Excursions Into Mathematics" by Anatole Beck, Michael N. Bleicher, Donald W. Crowe, Worth Publishers, 1969, pp.31-39. This polyhedron, therefore, could have 7 ants sitting at its vertices such that each ant can see every other ant. It is topologically equivalent to a doughnut, or a coffee cup. In case you have trouble finding the book, here is a description using coordinate geometry: Vertex x y z ----------------------- 1 3 3 0 2 3 -3 1 3 1 -2 3 4 -1 2 3 5 -3 3 1 6 -3 -3 0 7 0 0 15 The 14 faces of the polyhedron are the triangles comprised by the following triples of vertices: 126 235 356 346 467 237 267 156 245 124 134 137 457 157 Using 3-D coordinate geometry, you can compute the lengths of the triangle sides and cut them out of some cardboard, leaving tabs along edges for gluing them together. Better yet, you could carve a very intriguing figure out of a solid block of plexiglass, or perhaps wood. The above mentioned book displays a layout of partially joined triangles in Illustration 33 and actually shows the finished polyhedron in Illustration 32. Of course, if you have access to Mathematica or some other graphics software, you could do all kinds of fancy things like rotating this polyhedron in 3-D to view it! Please contact Dr. Math if you need further help. - Doctor Korsak, The Math Forum http://mathforum.org/dr.math/
Date: 10/30/2003 at 01:37:04 From: Alan Subject: Thank you Thank you! I will consult you again when I encounter other problems.
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