37TH ANNUAL AMATYC CONFERENCE A Modern Existence Proof Through Graphs Austin, Texas November 10, 2011 The Platonic Solids A Modern Existence Proof Through Graphs Dr. Jean Nicolas Pestieau Assistant Professor of Mathematics SUNY – Suffolk County Community College

The platonic solids (shown above) are the five convex polyhedrons that can be constructed with congruent regular polygonal faces.

Historical Background These solids are named after the great Greek philosopher Plato, who discusses them in his dialogue Timaeus (c. 360 B.C.) and associates each of the five classical elements – earth, fire, air, water and aether - with one of these solids. It was Plato’s contemporary Theaetelus who first described these solids mathematically and proved their existence. Theaetelus’s work later became the basis for the final book of Euclid’s Elements (c. 300 B.C.), where Euclid proves the existence and uniqueness of these solids (by construction).

Preliminary Definitions A set is a collection of distinct objects, none of which is the set itself. We assume these objects are well defined and call them the elements of the set. A graph G = (V, E) consists of two sets: a non-empty finite set V, called its vertex set, and a (possibly empty) set E, called its edge set. The elements of E, called edges, always pair two elements of V, called vertices. We denote by e and v the respective cardinalities of sets E and V.

Preliminary Definitions (cont’d) A graph G is said to be isomorphic to another graph H if there exists a one-to-one correspondence between their vertex sets that preserves adjacency. For example, the cyclic graph C3 is isomorphic to a triangle and the complete graph K4 is isomorphic to any plane drawing of a tetrahedron. A graph is said to be planar if it is isomorphic to a graph that can be drawn in a plane (or on the surface of a sphere) without edge-crossings. If no such isomorphism exists, then the graph is said to be nonplanar. For example, K4 is planar but K5 is nonplanar.

Preliminary Definitions (cont’d) When a planar graph is drawn in the plane without edge-crossings, it cuts the plane into regions called the faces of the graph. We denote the number of such faces by f. The degree of a vertex is the number of edges that are incident to it. A graph is said to be regular of degree d if all its vertices have the same degree. For example, any cyclic graph is regular of degree 2 and any complete graph on n vertices is regular of degree n-1.

The Complete Graphs K1 K2 K3 K4 … K5 K6 K7

Preliminary Definitions (cont’d) A walk in a graph joining vertices v1 and vn is a sequence of (not necessarily distinct) vertices v1, v2, v3, …, vn in which v1 is joined by an edge to v2, v2 is joined by an edge to v3, …, and vn-1 is joined by an edge to vn. A graph is said to be connected if every pair of vertices in the graph is joined by a walk. Otherwise the graph is said to be disconnected.

Preliminary Definitions (cont’d) A graph G is said to be platonic if it satisfies the following five conditions: 1) G is planar. 2) G is connected. 3) G is regular. 4) G has the property that every edge borders on two different faces. 5) G has the property that all of its faces are bounded by the same number of edges, which we denote by n.

Let us now investigate which graphs are platonic. K1 is (trivially) platonic. All the cyclic graphs Cn are also platonic. Q: Are there any others? A: Remarkably, there exists only five other platonic graphs. This last point constitutes our main result.

Claim: There exists only five non-trivial, non-cyclic platonic graphs. Moreover, these graphs (which have at least three faces) are isomorphic to the plane drawings of the five platonic solids constructed by Euclid in the 13th book of The Elements. We now turn to Richard J. Trudeau’s proof of this result (Introduction to Graph Theory, 1976). We should note that Trudeau proves the existence of these graphs, but not their uniqueness.

Preliminary Results Euler’s Polyhedral Formula. For any planar, connected graph with v vertices, e edges and f faces, v + f – e = 2. Proof. The proof is by induction on f . [See, for example, Trudeau] 

Preliminary Results (cont’d) Lemma 1. If G is a regular graph of degree d, then e = dv / 2. Proof. The result follows from the fact that the sum of the degrees of the vertices of any graph is 2e. Since G is regular of degree d, the sum of the degrees of its vertices is dv. This implies that 2e = dv, or e = dv / 2.

Preliminary Results (cont’d) Lemma 2. If G is a platonic graph, d is the degree of each vertex, n is the number of edges bounding each face, and f is the number of faces, then f = dv / n. Proof. Since G is platonic, each one of its edges borders on two different faces. Now, since every face of G is bounded by n edges, this implies that nf = 2e or, equivalently, that f = dv / n.

Main Result Theorem. There are only five platonic graphs having at least three faces. Existence Proof. Let G be a platonic graph with f > 2, where f is the number of faces of G. Let d be the degree of each vertex of G and n be the number of edges bounding each face of G.

Main Result (cont’d) If d = 0, then G is the graph consisting of only one vertex (i.e. K1). This (trivial) graph has only one face. Hence d > 0. If d = 1, then G does not exist. Hence d > 1. If d = 2, then G must be a cyclic graph. But any cyclic graph has exactly two faces. Hence d > 2.

Main Result (cont’d) Since G is connected, planar, and has each edge bordering on exactly two faces, we clearly have n > 2. We conclude that d, n are naturals greater than 2. Now, using Euler’s Polyhedral Formula in conjunction with Lemmas 1 & 2, we have the following identity: v + (dv / n) – (dv / 2) = 2, or 2nv + 2dv – ndv = 4n, or v(2n + 2d – nd) = 4n.

Main Result (cont’d) Since v and 4n are both positive, this implies the following strict inequality: 2n + 2d – nd > 0, or nd – 2n – 2d < 0, or nd – 2n – 2d +4 < 4, or (n – 2)(d – 2) < 4. We are now done with the proof! Note that the inequality above only admits five solution pairs (n,d) such that n and d are both naturals greater than 2.

The table below summarizes the topological features of the graphs that correspond to each of these five solution pairs. Note that we can now name the graph in each row in accordance with one of the Platonic solids (i.e. its historical counterpart). d n v e f Name 3 4 6 Tetrahedron 8 12 Hexahedron, or Cube 5 20 30 Dodecahedron Octahedron Icosahedron

The following are the plane drawings for each of the five Platonic solids. Note that these are isomorphic to each one of the graphs described in the previous table. Cube Tetrahedron Octahedron Dodecahedron Icosahedron

Thank You!