Robbie Robinson, Professor of Mathematics, George Washington University Georgetown Math Club March 1, 2016 Leonardo da Vinci, illustrations for Luca Pacioli’s 1509 book The Divine Proportion. From: http://www.georgehart.com/virtual-polyhedra/leonardo.html Melencolia I Albrecht Dürer, 1514 Anatoly Fomenko 1945- Nicholas Neufchatel, Johannes Neudorfer and son 1561 M. C. Escher Stars Salvador Dali, The Last Supper, 1955 Tony Robbin, 2000-10 http://tonyrobbin.home.att.net/work.htm The 2-dimensional case… Ehrhard Bohne, 2000 http://www.math-inf.uni-greifswald.de/mathematik+kunst/mathematiker_bohne.html\ Pas de Deux, Nat Friedman, Fractal Stone Print http://www.albany.edu/~artmath/cetl/am97/ Theorem 1. Let A1, A2, …, An be the interior angles of a polygon. Let o Sn=A1+A2+…+An. Then Sn=180 (n-2). A2 A3 A4 A1 Example. For n=4 we have o o o o. A1+A2+A3+A4 = 180 (n-2)=180 (4-2)=180 (2)=360 Lemma. Let B1, B2, …, Bn be the exterior o angles of a polygon. Then B1+B2+…+Bn= 360 B2 B3 B1 B4 Note: this does not depend on the number of sides! n Sn Rn Sn = 180(n-2) 3 180 60 � � − 2 � = " = 180 4 360 90 " � � 5 540 108 Rn is the angle of a regular n-gon 6 720 120 �" → 180 as � → ∞ 7 900 128.5… 8 1080 135 9 1260 140 10 1440 144 Schlflaly symbol {n,k}={n/k}={9,3} � − 2� � = 180 " � Vertex e.g. Edge Dodecahedron Face Vertices Dihedral angle (plural) Vertex: interior angle/angle deficit wire frame or stick figure solid (perspective) Schlegel web diagram diagram Convex polyhedra projected to sphere=tiling of sphere. non-convex polyhedra The five regular polyhedra Regular polyhedra: All f aces are the same regular polygon. All vertices the same. Name V E F p q tetrahedron 4 6 4 3 3 hexahedron 8 12 6 4 3 “cube” octahedron 6 12 8 3 4 dodecahedron 20 30 12 5 3 Icosahedron 12 30 20 3 5 § Well known throughout ancient world, as far back as Neolithic era . § Proclus Lycaeus (412–485 AD) credits Pythagoras (570-495 BC) with classifying them. § Plato (428-348 BC) wrote about them in the dialogue Timaeus. Plato ascribed them to the elements Earth, Air, Fire and Water. § Aristotle (384–322 BC) replaces ”universe” with ”aether.” Aristotle also says cube and tetrahedron fill space. Tetrahedron is incorrect. § Euclid (c 325-265 BC) completely characterized them in his Elements. § Fra Luca Bartolomeo de Pacioli writes The Divine Proportion illustrated by Leonardo da Vinci (1452-1519) § Kepler (1571-1630) proposes Platonic solids as geocentric model of solar system (Mercury, Venus, Mars, Saturn and Jupiter). Scotland c 1500 BC Etruscan (500 BC) and Roman Johannes Kepler (1571-1630), Illustration from Harmonice Mundi (1619) Illustrates Plato’s (incorrect) theory that the solids represent four elements Earth, Air, Fire & Water… plus Universe http://www.georgehart.com/virtual-polyhedra/kepler.html Leonardo da Vinci, illustrations for Luca Pacioli’s 1509 book The Divine Proportion. From: http://www.georgehart.com/virtual-polyhedra/leonardo.html Luca Pacioli, Jacopo de Barbari, 1495 http://www.math.nus.edu.sg/aslaksen/teaching/math-art-arch.shtml Golden Rectangle 12 3 1 : 4 Johannes Kepler, Mysterium Cosmographicum 1596 http://www.georgehart.com/virtual-polyhedra/kepler.html § a= interior angle of vertex. § d= angle deficit of vertex. a=3*A5 =324o d=36o 53 Vertex • p = 5 •q = 3 Observation 1. To be a vertex of a polyhedron, the degree q > 2 (otherwise flat). Observation 2. To be a vertex of a polyhedron, a<360o o 1. dodecahedron First. p = 5, A5=108 q = 3 q = 4 q = 2 a= 3 * 108o a= 4 * 108o = 324o = 432o >360o o Next. p = 3, A3=60 3. octahedron 2. tetrahedron q = 4 q = 2 q = 3 a= 4 * 600 a= 3 * 60o = 240± = 1800 4. icosahedron q = 5 q = 6 a= 5 * 60o a= 6 * 600 = 300o = 360o o Next. p = 4, A3=90 5. cube q = 3 q = 4 q =2 a= 3 * 90o a= 4 * 90o = 270o = 360o Case. p > 5. If p = 6. If p > 6. q = 2 q = 2 q = 3 q = 3 a= 360o a= 360o No more regular polyhedra! Conclusion: The only regular polyhedra (sphere tilings) are the five Platonic solids. Ve r te x f i g u re an d tangential polygon. http://mathworld.wolfram.com/DualPolyhedron.html M. C. Escher Stars (1948) n = 3 n = 4 n = 6 There are only three… The proof is the same. {5,4} {5,5} {5,6} ... There are infinitely many of these. Here are the first three using hyperbolic pentagons, identified by their Schlafli symbol. Theoretical interlude… Name V E F V-E+F tetrahedron 4 6 4 2 cube 8 12 6 2 octahedron 6 12 8 2 dodecahedron 20 30 12 2 icosahedron 12 30 20 2 truncated tetrahedron 12 18 8 2 rhombic dodecahedron 22 40 20 2 soccer ball (truncated icosahedron) 60 90 32 2 pentagonal prism 10 15 7 2 • We see that V-E+F = 2 in many cases. How do we know this always holds? • This fact, called Euler’s formula, was discovered by Leonard Euler, c 1750. Euler’s Theorem. For any polyhedron (regular or otherwise) the vertices, edges and faces satisfy, V – E + F = 2. Start with Schlagel diagram. It’s a plane W = 1 graph. W = # faces filled. W = # faces filled. E1= # edges removed (Step 1). Now the graph is a tree. V=E2+1 W = 1+E1 F=W Thus: V-E+F=2 Now consider the dodecahedron again… •Each vertex is type 53 o • For Pentagon, A5= 108 • Vertex interior total: a=3 * 108o = 324o • Vertex deficit: d= 360o – 324o = 36o Totals: I = V * a=20 * 324o = 6480o D = V * d= 20 * 36o = 720o Name V E F I D tetrahedron 4 6 4 720o 720o hexahedron 8 12 6 2160o 720o “cube” octahedron 6 12 8 1440o 720o dodecahedron 20 30 12 6480o 720o Icosahedron 12 30 20 36000 720o Descartes Theorem. For any polyhedron (regular or otherwise) D=åd= 7200 vertices Proved by René Descartes c 1640 Corollary. For any polyhedron (regular or otherwise) I=åa= 3600(V-2) vertices Semi-regular solids §Definition: §Two or more regular polygons as faces. §Only one vertex type. § Identified by their “vertex symbol” §The 13 Archimedean solids §Two infinite families §Prisms §Anti-prisms §Archimedes(287–212 BC).No account by Archimedes survives. §Pappus of Alexandria (290-350 AD) describes the 13 in his book Synagoge (“Collection”) and attributes them to Archimedes. § Johannes Kepler (1571-1630) describes and draws Harmonices Mundi. M. C. Escher (1959) Circle Limit Regular, but faces can be star polygons and can intersect. Rediscovered by Louis Poinsot in 1809. Christoph Jamnitzer (1563-1618) German Christmas stamps (1973) Floor of San Marco, Venice (c 1085) H. S. M. Coxeter (1907-2003) and John Flinders Petrie (1907-1972) discovered in 1929: Regular Skew Apeirohedra {4,6|4} {6,4|4} {6,6|3} mucube muoctahedron mutetrahedron Polytopes Six convex: first four discovered by Ludwig Schlafli (1814-1895) Simplex orthoplex tesseract octaplex dodecaplex tetraplex 5-cell 8-cell 16-cell 24-cell 120-cell 600-cell Edmund Hess (1843–1903) published the complete list in 1883 including 4 more non-convex cases. Salvador Dalí (1954) Crucifixion (Corpus Hypercubus) Picasso (1910) Portrait of Daniel-Henr y Kahnweiler § H. S.M. Coxeter, Polytopes, 1973, Dover § Peter Cromwell, Polyhedra, 1997/2001Cambridge University Press. § Daud Sutton, Platonic & Archimedean Solids, 1998, Walker and Company. § Wenning er, Magnus, Polyhedron Models, 1971, Cambridge University Press. Fr. Wenninger is a Benedictine monk and mathematician at St. John’s Abby, Collegeville, MN.