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### Platonic Solids

```
Date: 01/01/97 at 15:53:30
From: A. Kale Rodabough
Subject: "Regular" 3-D shapes

A friend gave me a gift this holiday season.  It is two cubes with
numbers on them which are used to count down the days from the first
of December to Christmas (i.e., 25 to 00).  I thought it would be neat
to have three cubes with numbers on them to count down all the days of
the year until Christmas.  I realized this was impossible.  But, if a
seven-sided "cube" exists, it IS possible.

The numbers on the first "cube" are: 1,2,3,4,6-(doubles as a 9),8,
and 0. The numbers on the second "cube" are:  1,2,3,5,6(9),7, and 0.
The numbers on the third "cube" are: 1,2,3,4,5,7, and 8. The
difficulty comes in getting the numbers 333, 222, and 111. Also,
every two number combination (i.e., 11,22,33,44,55,66(99),77,88,00) is
difficult to get. I also need to make sure the digits for each number
are available for every day of the year (i.e., placement of digits is
important).  My only problem is how to "cut out" a regular seven-sided
"cube".

I got to thinking about the regular polyhedra that I know
(tetrahedron, cube, dodecahedron, and the 20-sided die in the game
"Scattergories").  How does someone make a "regular" 5-hedron, or
7-hedron, or for that matter N-hedron?  I can see that each polyhedron
has a "center" where n-rays extend out from.  The angles between each
consecutive ray are equal.  If you "cut off" each ray so that each
line-segment has equal length, and "place" infinite planes on each
"end-point", the lines where the planes intersect define the
corners/edges of the n-hedron.  However, I can only visualize the
previously mentioned polyhedra.  Can you send me a picture, or
description, of a regular 7-hedron?  Any other info you could send me
would be of great interest!

Kale Rodabough
```

```
Date: 01/01/97 at 18:42:13
From: Doctor Pete
Subject: Re:

Hi,

I am very sorry to tell you that there are only 5 "regular" solids.
By "regular," I mean having the following properties:

1) They are is convex; i.e., they have no dimples.
2) The faces are all regular n-gons, each of the same type.
3) Each vertex has the same arrangement of faces around it.

With this in mind, the names of these solids (called Platonic solids)
are:

faces  edges  vertices   face type
---------------------------------------------------------
tetrahedron             4      6     4       triangle (3)
hexahedron(cube)        6     12     8       square   (4)
octahedron              8     12     6       triangle (3)
dodecahedron           12     30    20       pentagon (5)
icosahedron            20     30    12       triangle (3)

If we eliminate requirement (3), then we obtain many more, in
particular the convex deltahedra.  The tetrahedron, octahedron, and
icosahedron, having triangular faces, are a subset of the convex
deltahedra.  These have 4, 6, 8, 10, 12, 14, 16, and 20 sides.  In any
case, a 7-hedron cannot be "regular."

http://www.ugcs.caltech.edu/~peterw/studies/platonic/

-Doctor Pete,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Geometry
High School Polyhedra

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