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