Return to Mathematics IndexHow 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)BackgroundThe 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.