Deriving Uniform Polyhedra with Wythoff’s Construction Don Romano UCD Discrete Math Seminar

1. Deriving Uniform Polyhedra with Wythoff’s Construction Don Romano UCD Discrete Math Seminar 30 August 2010

2. Outline of this talk • Fundamentals of uniform polyhedra • Definitions and properties • Convex solids • Regular polyhedra • Nonconvex solids • Early research • Systematic approach for deriving uniform polyhedra • Spherical tessellations • Wythoff’s Construction • The complete enumeration

3. Definitions and a few properties Def. A polyhedron is a finite set of polygons such that every side of each belongs to just one other, with the restriction that no subset has the same property Def. A uniform polyhedron is made up of regular polygons and its vertices are transitive • Vertex transitivity means there is an isometry (rotation, reflection) that takes any vertex to any other • All vertices are congruent and lie on a sphere There are 75 uniform polyhedra • 18 convex, 57 nonconvex and an infinite set of prismatoids

4. Exhibit at London Museum of Science

5. Budzelaar Collection

6. Pawlikowski Collection

7. Convex Uniform Polyhedra • 5 Platonic Solids • Faces are regular polygons of only one kind • Symmetry groups form basis for all uniform polyhedra • Tetrahedral, Octahedral, Icosahedral • Known since antiquity — Euclid’s Elements • 13 Archimedean Solids • Faces can be of more than one kind (2 or 3) • Can be derived from Platonics by simple operations of truncation, rectification, and cantellation • Two enantiomorphic pairs (snubs) • First enumeration by J. Kepler (ca. 1600) • 2 infinite sets of convex prisms and antiprisms • Dihedral symmetry

8. Platonic and Archimedean Solids

9. Regular Polyhedra Def. A regular polyhedron is made up of only one kind of regular polygon and vertices are congruent • The 5 Platonic solids are regular • 4 Kepler-Poinsot solids are regular and nonconvex • 2 have star faces, 2 have star vertices • Derived by stellating or faceting Platonics • “Wayside shrines at which one should worship on the way to higher things” • Peter McMullen

10. Nonconvex Uniform Polyhedra • Can be derived by faceting Archimedean solids • Star polygons can be inscribed in faces • Removing one kind of polygon face and inserting others • Isomeghethic: same edge set • Many uniform polyhedra discovered between 1878 -1881 • Edmund Hess (2) • Johann Pitsch (18) • Albert Badoureau (37) • Max Brückner • Vielecke und Vielflache (1900) • Isogonal-isohedra a.k.a. ‘noble’ polyhedra

11. Brückner’s Noble Polyhedra and many more . . .

12. Spherical Tessellations • Spherical triangles are bounded by segments of great circles • The sum of the angles of a spherical triangle are greater than 180° and less than 540° • Area of spherical triangle A = r² E, where E is the spherical excess, that is, the sum of the angles minus 180° • Only four ways to cover the sphere (once) with congruent spherical triangles

13. Möbius Triangles Let the angles of a spherical triangle be π/p, π/q, π/r where p, q, r are integers The area of the spherical triangle [(1/p + 1/q + 1/r) -1] π must be positive Hence, 1/p + 1/q + 1/r > 1. Only possibilities for p, q, r are 2, 3, 4, 5 with the restriction that 4 and 5 cannot occur together These lead to the four fundamental spherical triangles which are known as Möbius Triangles: (2,3 3), (2,3,4), (2,3,5), (2,2,r) Repeated reflections in sides of triangles will tile a sphere exactly once

14. The Four Fundamental Spherical Tilings (2,3 3) (2,3,4) (2,3,5) (2,2,r)

15. Tetrahedral Symmetry |g| = 24

16. Octahedral Symmetry |g| = 48

17. Icosahedral Symmetry |g| = 120

18. Dihedral Symmetry |g| = 4n

19. Schwarz Triangles • Karl Schwarz (1873) • Proposed and solved problem of finding all spherical triangles which lead, by repeated reflections in their sides, to a set of congruent triangles covering the sphere a finite number of times • Extension of Möbius triangles where p, q, r are rational, but not necessarily integral • Still have 1/p + 1/q + 1/r > 1 (positive area) Admissible values for p, q, r are 2, 3, 3/2, 4, 4/3, 5, 5/2, 5/3, 5/4 with restriction that numerators 4 and 5 cannot occur together • Some triplets are reducible • Some do not cover the sphere a finite number of times • 44 distinct Schwarz triangles and 2 others of infinite variety • The density, d, of a Schwarz triangle is the number of times sphere is covered

20. Schwarz triangles are composed of fundamental Möbius triangles π/5 π/5 π/5 π/5 π/3 π/2 π/5 π/2 Schwarz triangle (5/2 2 3) density = 7 Schwarz triangle (5/2 2 5) density = 3

21. Schwarz Triangles — examples

22. Complete list of Schwarz triangles sorted by density Symmetry Groups 5 Tetrahedral 7 Octahedral 32 Icosahedral 2 Dihedral

23. Wythoff’s Construction Kaleidoscopic construction by tiling the sphere with a Schwarz triangles along with a specific point in the triangle Willem Wythoff (1907) applied this method to 4-dimensional problems • A point is chosen in Schwarz triangle • Repeated reflections of triangles produce multiple instances of that point around sphere • If suitable points are chosen, they will generate the vertices of a uniform polyhedron

24. Wythoff — point placements Points can be chosen in four ways, each with its own Wythoff Symbol p | q r Point is at a vertex P of triangle PQR p q | r Point is on side of PQ such that it bisects the angle at R p q r | Point is at the incenter of triangle PQR (intersection of angle bisectors) | p q r Point is the Fermat point and alternating triangles are used

28. The Fermat point Spherical triangles alternately black and white

30. Wythoff Symbol • p|qr • Quasi-regular (16 polyhedrons) • Vertex configuration {q, r, q, r, . . . q, r} • Regular if r = 2 or q = r • pq|r • Semi-regular (33 polyhedrons) • Vertex configuration {p, 2r, q, 2r} • pqr| • Even-faced (14 polyhedrons) • Vertex configuration {2p, 2q, 2r} • |pqr • Snub (11 polyhedrons) • Vertex configuration {3, p, 3, q, 3, r}

31. Non-Wythoffian Polyhedron • Great Dirhombicosidodecahedron • Discovered by J.C.P. Miller • ‘Miller’s Monster’ • Found by combining both enantiomorphs of |3 5/3 5/2 • Only uniform polyhedron with 8 faces surrounding each vertex • Largest number of faces (124) and edges (240) • Euler characteristic Χ = - 56 • Has 60 diametral squares that can be considered snub faces • Existence indicated no general reason for restriction to triangles as snub faces Vertex figure

32. Enumeration and Proof of Completeness H.S.M. Coxeter, M. S. Longuet-Higgins, J.C.P Miller • “Uniform Polyhedra”, Philosophical Transactions of the Royal Society of London, 1954 • Complete enumeration of the 75 • Conjectured that list was complete S.P. Sopov • “Proof of the Completeness of the Enumeration of Uniform Polyhedra”, Ukrain. Geom. Sbornik, 1970 J. Skilling • “The Complete Set of Uniform Polyhedra”, Philosophical Transactions of the Royal Society of London, 1975 • Computer search examined all possible polygon configurations for the basic symmetry groups • “Skilling’s Figure” was found by relaxing definition of uniform polyhedron to allow more than two faces at an edge Donald Coxeter John Skilling

33. Skillings Figure

34. Facial Intersections (Think inside the box!) Polyhedral density = 38 Individual surface segments = 3,000

35. Facial Intersections “Geometry is a skill of the eyes and hands as well as the mind.” - J. Pederson

36. Caution: Facial Intersections may be hazardous to your mental health!

37. A novice tackles Miller’s Monster (ca. 1973)

38. Uniform polytopesexist in higher dimensions!