Polyhedron Project

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.