Polyhedra Plato, Archimedes, Euler Part 1

Polyhedra: Plato, Archimedes, Euler Department of Mathematics and Statistics 2020 Regular Polygons Definition A polygon is a planar region R bounded by a finite number of straight line segments that together form a loop. If all the line segments are congruent, and all the angles between the line segments are also congruent, we say R is a regular polygon. Polyhedra Definition A polyhedron is a spatial region S bounded by a finite number of polygons meeting along their edges, and so that the interior is a single connected piece. So these are not polyhedra: But these are: Regular Polyhedra Definition Let S be a polyhedron. If I all the faces of S are regular polygons, I all the faces of S are congruent to each other, and I all the vertices of S are congruent to each other, we say S is a regular polyhedron. Cube (Regular Hexahedron) Regular Octahedron Another Regular Polyhedron Regular Tetrahedron Two More Regular Polyhedra Regular Dodecahedron Regular Icosahedron And They’re Dual: Some Ancient History (of Regular Polyhedra) I Pythagoras of Samos, c.570–c.495 BCE: Knew about at least three, and possibly all five, of these regular polyhedra. I Theaetetus (Athens), 417–369 BCE: Proved that there are exactly five regular polyhedra. I Plato (Athens), c.426–c.347 BCE: Theorizes four of the solids correspond to the four elements, and the fifth (dodecahedron) to the universe/ether. I Euclid (Alexandria), 3xx–2xx BCE: Book XIII of The Elements discusses the five regular polyhedra, and gives a proof (presumably from Theaetetus) that they are the only five. The five regular polyhedra are often called the Platonic solids. There Can Be Only Five: Setup Let S be a regular polyhedron. Define integers n, k by: I the faces of S are regular n-gons I k faces meet at each vertex 1. n ≥ 3 and k ≥ 3. 2. S is completely determined by the numbers n and k. 3. S must be convex: For any two points in S, the whole line segment between the two points is contained in S. 4. Since S is convex, the total of angles meeting at a vertex of S is less than 360 degrees.   180(n − 2) ◦ 5. Each angle of a regular n-gon has measure . n [Triangle: 60◦ , Square: 90◦ , Pentagon: 108◦ , Hexagon: 120◦ .] There Can Be Only Five: Payoff Part 1 Faces of S are regular n-gons and k faces meet at each vertex n, k ≥ 3, but total angles at each vertex must be less than 360◦ . n = 3 (triangles) k = 3 triangles per vertex: Tetrahedron k = 4 triangles per vertex: Octahedron k = 5 triangles per vertex: Icosahedron k ≥ 6 triangles per vertex: NO! Angles ≥ 360◦ There Can Be Only Five: Payoff Part 2 Faces of S are regular n-gons and k faces meet at each vertex n, k ≥ 3, but total angles at each vertex must be less than 360◦ . n = 4 (squares) k = 3 squares per vertex: Cube k ≥ 4 squares per vertex: NO! Angles ≥ 360◦ There Can Be Only Five: Payoff Part 3 Faces of S are regular n-gons and k faces meet at each vertex n, k ≥ 3, but total angles at each vertex must be less than 360◦ . n = 5 (pentagons) k = 3 pentagons per vertex: Dodecahedron k ≥ 4 pentagons per vertex: NO! Angles > 360◦ n ≥ 6 (hexagons and more) k ≥ 3 n-gons per vertex: NO! Angles ≥ 360◦ There Can Be Only Five: Summary of the Proof Faces of S are regular n-gons and k faces meet at each vertex n, k ≥ 3, but total angles at each vertex must be less than 360◦ . k n 3 4 5 3 ≥6 — 4 — — — 5 ≥6 — — — — — — — The Five Regular Polyhedra Faces of S are regular n-gons and k faces meet at each vertex Possible choices for (n, k):