```How Many Regular Polyhedrons Are There In This or Any Universe?

Lawrence E Freeman             Kenwood Academy
5015 Blackstone Avenue
Chicago, IL 60615
1-312-536-8850 (school main)

Background

The idea of "regularity" is a very old one in geometry. It dates back to the
ancient Greek mathematicians/philosophers.  (See, for example, Plato's theory of
"ideals"). Regular polygons are thus convex polygons whose vertex angles are all
equal (or congruent) and whose sides are likewise all congruent.

The first geometry textbook, Euclid's Elements, assumed convexity without
mention of that concept.  We shall take convexity in its intuitive sense. Convex
polygons have a very neat property:  Take any vertex and draw all possible
diagonals within the polygon. This process subdivides  the polygon of n sides into
n-2 non-overlapping triangles.  Since a triangle's three vertex angles have a sum
of 180 degrees, an n-sided convex polygon's n vertices must have an angle sum of
180(n-2) degrees.  Now, if our n-sided polygon is also regular, each of its n
congruent vertex angles must have a measure of one-nth of this angle sum.

Expressing this fact for several regular polygons we obtain the following
data:
Number of            Degree measure of
Sides               any vertex angle

3                      60 degrees
4                      90
5                     108
6                     120
Note that as the number of sides increases, the degree measure of a vertex angle
does likewise.  (Thought experiment for the thoughtful reader:  The exterior
angles of a convex polygon become smaller as the number of sides increases, but
what do you suppose happens to the sum of all the exterior angles of a convex
polygon, regular or not?  Can you prove or disprove your hunch? Try it!)

Regular Polyhedrons

A polyhedron is a "solid" three-dimensional  figure analogous to the two-
dimensional polygon discussed above.  Polyhedrons have vertices, edges, and faces
which to Euclid had dimensions of zero, one, and two respectively. If a polyhedron
has faces which are regular and congruent polygons -- all of them -- and if at
each vertex exactly the same number of  faces meet, then we have a"regular"
polyhedron.  The question is, "exactly how many such 'critters' are there?"
Obviously the number is infinite if size is considered, so we shall eliminate that
consideration and ask merely how many "truly" different regular polyhedrons can
exist.

More nomenclature is needed (Sorry).  In two-dimensional plane geometry,
angles are just angles, but in three dimensions life gets more complicated: When
two planes intersect, they intersect in a line.  Pick any point on such a line of
intersection (edge) and in each plane construct a line perpendicular to the line
of intersection.  The angle between these two perpendiculars is the "DIHEDRAL
ANGLE" of the two intersecting planes.  In regular polyhedrons these dihedral
angles are all equal (congruent). Computation of their measure can get quite
2
complicated, but isn't essential for this project.

At each vertex (point or corner) of our polyhedron there is a solid or
TRIHEDRAL angle.  How such angles are measured -- if indeed that concept even
applies -- isn't known to this writer.  What is clear is that a trihedral angle
must be the meeting point of three or more planes --faces -- of the polyhedron.
And recall that all faces are congruent polygons having congruent vertex angles
according to the above-mentioned table and formula.

Let us examine how trihedral angles can be made: Assemble three or more
polygons so that they meet along common sides with one common vertex. If the sum
of the vertex angles is less than 360 degrees, then there is a gap between the
outer, unmatched edges. Closing up that gap by joining the two unmatched edges
yields a trihedral angle.  Working with regular polygons makes our job of analysis
possible....  We will start with the simplest regular polygon, the equilateral
triangle, manufacture all possible trihedral angles from just  that unit, and then
move up as far as needed to  the point where the angle sum equals or exceeds 360
degrees.

This last statement is the key to the proof (solution or answer to the
initial question).  If the angle sum equals 360, then there will be no gap to be
closed. In such a situation, the trihedral angle degenerates to a plane instead of
a "bulge", and a polyhedron can't exist.  Should more regular polygons be added to
the assembly, they will OVERLAP, and such a "creature" would have a negative gap.
It CERTAINLY can't be folded to  produce a trihedral angle.

Three, four, or five equilateral triangles can thus fit around a trihedral
angle.  These are the vertices of the regular tetrahedron, octahedron, or
icosahedron respectively.  We may be certain that no other regular polyhedra can
exist having equilateral triangles for faces.

Next, move up to the regular quadrilateral -- the square.  At least three
squares must comprise this trihedral angle; and that is all for the square because
if a fourth square is added to the assembly, the  angle sum is exactly 360
degrees.  The only regular polyhedron having squares as faces is the best-known,
the cube. The count is now four regular polyhedra; onward....

The regular pentagon has five sides, and each vertex angle has measure of 108
degrees.  Three regular pentagons attached as before yield 324 degrees and a gap
of only 36 degrees.  When this small gap is closed we obtain a vertex (trihedral)
angle of the last possible regular polyhedron, the celebrated regular dodecahedron
of twelve faces.

Our task is now over because one must next try to construct trihedral angles
out of regular hexagons, regular heptagons, and regular polygons of greater
numbers of sides (each with a correspondingly greater vertex angle measure). Three
regular hexagons have a vertex angle sum of exactly 360 degrees, and they won't
fold into a trihedral angle because there is no gap.  The angle sum of three
vertex angles of a regular heptahedron is greater than 360 degrees. three vertex
angles of a regular heptahedron is greater than 360 degrees, so nomertex angles
of a regular heptahedron is greater than 360 degrees. more regular polygons need
be examined.

To sum up, there are only five possible regular polyhedra.  Period.

This ended the matter for the ancients.  Until about two hundred years ago
that is.  Then it was noticed that no one had ever explicitly called for CONVEX
polyhedra.  Johannes Kepler and later Poinsot found it possible to add to the
roster of regular polyhedra by creation of "dimpled" and/or stellated regular
3
polyhedra. The additional regular polyhedra won't be discussed in this report, but
information about them may be found in the Bibliography.

Here is a summary of the five convex regular polyhedra:
Number       Number        Number
Reg. Polyhedron            of Faces    of Vertices    of Edges

Tetrahedron                  4 T's*         4             6
Cube                         6 S's*         8            12
Octahedron                   8 T's          6            12

Dodecahedron                12 P's*        20            30
Icosahedron                 20 T's         12            30
*: T = Equilateral triangle; S = Square; P = Regular Pentagon
Notice two things about this table:  First, in every case the numbers of
faces, vertices and edges satisfy the Euler formula, F + V = E + 2.(Look up a
proof of this sometime). Secondly, from the spacing of the table note that the
numbers of faces and vertices interchange between Cube and Octahedron and between
Dodecahedron and Icosahedron. Such relationships lead to study of the topic of
"duality," basic to advanced Euclidean geometry and also virtually the foundation
of "projective geometry" (an advanced extension of geometry).

If wire models having transparent faces are made of each of these polyhedra,
of appropriate "size," then  each of the pair of duals will so fit inside of each
other that a vertex of one will lie at the center of a face of the other of the
dual pair.

BIBLIOGRAPHY

Cundy, H. M. & Rollett, A. P. "Mathematical Models." London: Oxford University
Press, 1961 (second edition).

Holden, Alan. "Shapes, Spaces and Symmetry."  New York: Columbia
University Press, 1971.

Olson, Alton T. "Mathematics Through  Paper Folding.:  Reston VA: National
Council of Teachers  of  Mathematics, 1975 (Revision of earlier work by
Donovan Johnson).

Rademacher, Hans, and Toeplitz, Otto. "The Enjoyment of Mathematics."
Princeton NJ: Princeton University Press, 1970.

Wenninger, Magnus,  "Polyhedron Models".  London: Cambridge University
Press, 1971.

(end of bibliography section. what follows was once typed onto
the disk, i thought, but doesn't seem to be here, now that i want it......
##&\$)^^^^12345678)  SO I WASTED TIME RETYPING THE GOD DAMNED STUFF!

OBJECTIVES
To demonstrate a simple to prove but surprising fact of elementary geometry.
To illustrate one method of mathematical (logical) proof -- Cauchy's
celebrated  method of cases in which all possibilities are studied one-by-one.
To create a climate in which student, teacher or both can extend their
knowledge by framing additional conjectures (plausible hypotheses) worthy of
investigation.

MATERIALS
Tag board, cutting instrument (scissors, "Xacto" knife or single-edge razor
4
blade), and paste or (preferred) "Scotch" brand  "Magic Transparent Tape."
Models of completed polyhedra, each relevant trihedral angle, and student
sets of the latter to be folded, handled, and examined.

STRATEGIES
For  the basic theorem ("How Many...?"), employ the Cauchy method of cases to
illustrate the virtue of patience.  Hands-on work completes the proof almost
faultlessly, very convincingly, and (surprise!) theoretically correct.
Class brainstorming ought to then produce a host of additional conjectures
in the realm  of efficient coloring of faces, dualism, wire models, best
approximation of a sphere, measurement  of dihedral angles, computation of the
measures of such angles, the "Euler formula" and its spookiness, etc.
Finally, even highly competitive, high achievers soon see the virtue of
cooperation in the manufacture of their own models of trihedral angles, entire
polyhedra, stellations, and coloring schemes.  The smarter the student, the
quicker the realization that mass production saves time and labor.

121121112111121111121111112111111121111111121111111112111111111121111111111
12.......
(The above irrational thought ends this exercise in futility.  I tried
hard to shift the last three sections to the beginning of the paper but
inexperience, lack of time to undo what looked like fatal errors  made that
impossible.......).
..
...
....     Larry Freeman 10:37 AM, July 30, 1986.

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