Polyhedra, Complexes, Nets, and Symmetry Egon Schulte Northeastern University, Boston Ateneo Workshop, May 2017 Polyhedra • Ancient history (Greeks), closely tied to symmetry. • With the passage of time, many changes in point of view about polyhedral structures and their symmetry. Many dif- ferent definitions!! Who knows what a polyhedron is? urrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr rru u r rrrrrrrrrrrrr r rrrrrrrrrrrr r rrrrrrrrrrrrr r rrrrrrrrrrrrr rrrrrru r rurrrrrrr u rrrrrrrrrrrrr r r rrrrrrrrrrrr r r r rrrrrrrrrrrrr r r r rrrrrrrrrrrrr r r r rurrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr r ru r u r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r rrrrrrrrrrrur r r rrrrrrrru u r rrrrrrrrrrrrr r r rrrrrrrrrrrr r r rrrrrrrrrrrr r r rrrrrrrrrrrrr r r rrrrrrrrrrrrr r r rrrrrrrrrrrr r rurrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr r yrrrrryurrrrrrrrrrrr r rrurrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr rrrrrrrrrrrr rrr r rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr r rrrrrrrrrrrrr r r rrrrrrrrrrrr r r rrrrrrrrrrrr r r rrrrrrrrrrrrr r rrrrrrrrrrrrrr r r rrrrrrrrrrrr r u rurrrr r r rurrrrrrrrr r r r r r r r r r r r r r r r r r r r r r r r r r r r r r u r ru r r rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr rrrrur r r r rrrrrrrrrrrr r r r rrrrrrrrrrrrr r r r rrrrrrrrrrrr r r rrrrrrrrrrrrrr u rrrrrrrrrru r urr rrrrrrrrrrrrr r rrrrrrrrrrrrr r rrrrrrrrrrrrr r u urrrrrrrrrrr rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr ur Platonic solids (solids, convexity) f3; 5g Definition: Set K is called convex if every line segment joining two points of K lies entirely in K. Plane: convex polygons Space: convex polyhedra | convex hull of finitely many points in 3-space. (Convex hull: smallest convex set that contains the given points.) More convex polyhedra: Archimedean solids, prisms and antiprisms • All have regular convex polygons as faces, and all vertices are surrounded alike. Kepler-Poinsot (star) polyhedra What's new here? Faces and vertex-figures can be regular star-polygons (for example, pentagrams). Are you still sure you know what a polyhedron is? Petrie-Coxeter polyhedra (sponges) What's new here? Infinite polyhedra (apeirohedra)! Faces still are convex polygons! Vertex-figures allowed to be skew (non-planar) polygons! Periodic! (Vertex-figure at vertex x: joins the vertices adjacent to x in the order in which they occur around x) Skeletal Approach • Initiated by Branko Gr¨unbaum(1970's), probably inspired by crystal chemistry. • Rid the theory of the psychologically motivated block that membranes must be spanning the faces! Allow skew faces (and skew vertex-figures)! Restore the symmetry in the definition of \polyhedron"! Graph-theoretical approach! • Basic question: what are the regular polyhedra in ordinary space? Answer: Gr¨unbaum-DressPolyhedra. • Later: group theory forces skew faces and vertex-figures! General reflection groups. • No membranes spanned into faces! Skeleton only! = • Polygon faces (simple polygons) Who said that faces have to be finite? zigzag polygon (planar) helical polygon Polyhedron (heading towards the definition) 3 A polyhedron P in E is a (finite or infinite) family of simple polygons, called faces, such that • each edge of a face is an edge of just one other face, • all faces incident with a vertex form one circuit (each vertex-figure is a simple closed polygon), • P is connected, • each compact set (each large cube, say) meets only finitely many faces (discreteness). All traditional polyhedra are polyhedra in this generalized sense. What is a polygon (for our purposes)? Finite (simple) polygon (p-gon, with p ≥ 3) • finite sequence of p distinct points v1; v2; : : : ; vp, called the vertices; • line segments [v1; v2]; [v2; v3];:::; [vp−1; vp] and [vp; v1], called the edges. A finite polygon is topologically a 1-sphere! Infinite (simple) polygon (apeirogon) • infinite sequence of distinct points : : : ; v−2; v−1; v0; v1; v2;::: , called the vertices; • the line segments [vi; vi+1] for each integer i; • each large cube meets only finitely many of these line segments. An infinite polygon is topologically a real line! Some skew (non-planar) hexagons! s s s s s s s s Four skew hexagons in the edge graph of the cube (red, blue, green, black). Some zigzag apeirogons! Four zigzag apeirogons in the edge graph of the square tes- sellation. Some definitions F any geometric figure in 3-space (possibly planar or linear) • A symmetry of F is an isometry of the ambient space that leaves F invariant. • The set of symmetries of F is a group called the symmetry group of F and denoted G(F). Isometries in the plane: reflection in a line, rotation about a point, translation, glide reflection Isometries in 3-space: reflection in a plane, rotation about a line, rotatory reflection, screw motion (rotation followed by a translation along the axis). Regular polygons • Polygon P is regular if its symmetry group G(P ) acts transitively on the set of flags (incident vertex-edge pairs). Then G(P ) is vertex-transitive (isogonal) and edge-transitive (isotoxal). But a priori flag-transitivity is stronger! edge-transitive octagon, not regular two vertex-transitive apeirogons, not regular What kind of regular polygons exist? 3 Regular polygons in E are either • finite planar (convex or star-) polygons, • finite non-planar (skew) polygons, • infinite planar zigzags, • infinite helical polygons. How about the symmetry groups? Generators? Later!! Polyhedron (formal definition) 3 A polyhedron P in E is a (finite or infinite) family of (finite or infinite) simple polygons, called faces, such that • each edge of a face is an edge of just one other face, • all faces incident with a vertex form one circuit (each vertex-figure∗ is a simple closed polygon), • P is connected (the edge graph is connected), • each compact set (each large cube, say) meets only finitely many faces (discreteness)y. ∗Vertex-figure at vertex x of P : polygon whose vertices are the vertices of P adjacent to x, and whose edges join two vertices u and v if and only if [x; u] and [x; v] are edges of a common face of P at x. yConsequence: Vertex-figures are finite! Highly symmetric polyhedra P • Faces are finite (planar or skew) or infinite (helical or zig- zags) polygons! Vertex-figures are finite (planar or skew) polygons! • Polyhedron P is regular if its symmetry group G(P ) is transitive on the set of flags. Flag: incident vertex-edge-face triple Explicitly: any incident vertex-edge-face triple can be mapped to any incident vertex-edge-face triple by a symmetry of P . @ @ @ w @ w @ w @ w @ @ @ @ @ @ • Flags seen as triangles @ @ @ @ @ @ @ @ @ w @ w @ w @ w @ @ @ @ @ @ @ u @ @ @ @ @ @ @ @ w @ uw @ u w @ w @ @ @ @ @ @ @ @ @ @ @ @ w w w w Exercise 1: If a polyhedron P is regular then G(P ) is also transitive, separately, on the { vertices (isogonal), edges (isotoxal), faces (isohedral), { vertex-edge pairs, vertex-face-pairs, and edge-face pairs! But a priori flag-transitivity is stronger! (Tilings by rhombi can be tran- sitive on vertices, edges, faces, but not regular!) Consequence: congruent faces, congruent edges, congruent vertex- figures!! Regular polyhedra @ @ @ w @ w @ w @ w @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ w @ w @ w @ w @ @ @ @ @ @ @ u @ @ @ @ @ @ @ @ w @ uw @ u w @ w @ @ @ @ @ @ @ @ @ @ @ @ w w w w Must be able to map each flag to each of its adjacent flags! Two flags are called adjacent if they differ in exactly one element. Exercise 2: If P is any polyhedron, then P is strongly flag- connected, meaning that for any two flags Φ and Ψ of P there is a sequence of flags Φ = Φ0; Φ1;:::; Φn = Ψ, all containing the elements common to both Φ and Ψ (if any), such that Φj and Φj+1 are adjacent for all j = 0; : : : ; n − 1. Exercise 3: If P is any polyhedron and Φ and Ψ are flags of P , then there exists at most one symmetry of P mapping Φ to Ψ. In particular, if P is regular, there exists exactly one symmetry of P mapping Φ to Ψ. Hierarchy of Symmetry • Top of hierarchy: regular polyhedra!! They have maximum possible symmetry, by Exercise 3(b). • P is chiral if G(P ) has two orbits on the flags such that adjacent flags are in distinct orbits. No longer able to map a flag to its adjacent flags! @ @ @ w @ w @ w @ w @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ w @ w @ w @ w @ @ @ @ @ @ @ u @ @ @ @ @ @ @ @ w @ uw @ u w @ w @ @ @ @ @ @ @ @ @ @ @ @ w w w w • P is Archimedean (or uniform) if G(P ) is vertex-transitive and P has regular polygons as faces. Analogues of the Archimedean solids!! Vertex-figures con- gruent! Faces all regular but possibly of several kinds. • Other interesting classes! Isohedral, isotoxal, isohedral polyhedra, for example! Some examples of regular polyhedra s s s s s s The Petrie duals of the cube. A regulars polyhedron with 8 vertices, 12 edges, 4 skew hexagonal faces. Type f6; 3g. Petrie duals (Petrials) of regular polyhedra P Petrie polygon of P : path along the edges of P such that any two, but no three, consecutive edges lie in a common face of P . Petrie dual P π of P : same vertices and same edges as P , but the faces are the Petrie polygons of P . (Almost always again a polyhedron.) • (P π)π = P , that is, the Petrie dual of the Petrie dual is the original polyhedron. • G(P π) = G(P ), but generators change as follows: (R0;R1;R2) ! (R0R2;R1;R2) The Petrie dual of the square tessellation. An infinite regular polyhedron with zig-zag faces. Type f1; 4g. Symmetry group of a regular polyhedron • Generated by reflections R0;R1;R2 in points, lines, or planes. u R2 u u R1 R0 3 Note: Every isometry of E which is an involution (its order is 2) is a reflection in a point, line, or plane! (A reflection in a line is a halfturn about the line!) Exercise 4: Let P be a regular polyhedron, and let Φ be a fixed flag (base flag) of P .