Petrie maps and Petrie configurations Jurij Kovič, University of Primorska, Slovenia EuroGIGA Final Conference Berlin, February 20, 2014

The aim of the talk is to explain how some maps (we will name them Petrie maps ) can be used to obtain examples of configurations (we will name them Petrie configurations ): Petrie maps Petrie configurations.

Motivation Goal: To develop a method for constructing examples of large configurations – with many symmetries. Question: How to do it? Idea: Use as a „starting point“ some class of combinatorial object with many symmetries (e.g. Platonic solids) and somehow „transform“ them into configurations.

1. Basic definitions Our topic connects two branches of mathematics, one old, one new: the theory of configurations has its: “classical age” “renaissance” “dark age” “golden age” 1990-now (B. Grünbaum, Configurations of points and lines) the theory of flag graphs of maps i s developing from around Let us briefly review some of their basic concepts.

incidence structure: a triple (B, P, I) consisting of B … the set of blocks B (or lines) P … the set of points P and I … an incidence relation, a subset of B x P

geometrical representation of incidence structure (B, P, I) : P and B are represented by points and lines in the Euclidean plane

In a lineal incidence structure each two lines intersect in at most one point.

A configuration (n k ) is: - a lineal incidence structure - consisting of n lines and n points; - each line is incident with k points, - each point is incident with k lines However, if only the condition of lineality is not fulfilled, we will call such an incidence structure a “ non-lineal (n k ) configuration”.

The Pappus configuration (one of the three (9 3 ) configurations) can be geometrically represented by points and lines in the Euclidean plane.

Some (n 3 ) configurations are not geometrically realizable by points and lines in the Euclidean plane, for example the Fano plane (7 3 ) :

map : a cellular embedding of a graph into a compact surface; it consists of the sets V of vertices, E of edges and F of faces polyhedral map : a map belonging to a polyhedron

flag Ф =(v,e,f) = a triple (vertex, edge, face) geometrically, flags can be represented as triangles (obtained by a barycentric subdivision of faces)

The three adjacent flags Ф 0, Ф 1, Ф 2 differ from Ф only in a vertex, an edge or a face.

flag graph F(M) of a map M; its vertices are flags, its edges (labeled 0,1,2 or colored blue, yellow and red) connect adjacent flags

Involutions s 0, s 1, s 2 carry each flag Ф of the flag graph F(M) into its three adjacent flags: Ф 0 = s 0 (Ф), Ф 1 = s 1 (Ф), Ф 2 = s 2 (Ф). They generate a group Mon(M) satisfying the relations: s 0 2 = s 1 2 = s 2 2 = (s 0 s 2 ) 2 = id.

The orbit of the flag Ф consists of all flags Ψ for which there is an edge-colors preserving automorphism of the flag graph F(M) sending Ф into Ψ. A k-orbit map has exactly k orbit of flags, e.g. a 1-orbit map has only 1 orbit of flags.

Example: the polyhedral net of a 10-orbit polyhedral map of the Archimedean solid ( ) called the snub cube.

2. Petrie walks and Petrie lines Definition 1. Choose any flag Ф in the flag graph F(M). Let (i, j, k) be any of the six permutations of the numbers (0,1,2). The corresponding Petrie walk is the sequence of flags (Φ, Φ i, (Φ i ) j, ((Φ i ) j ) k, (((Φ i ) j ) k ) i …).

An oriented Petry cycle is a Petrie walk in which the last flag is equal to the first one.

Definition 2. Choose any flag Ф in the flag graph F(M). Let (i, j, k) be any of the six permutations of the numbers (0,1,2). The cycle in F(M) whose vertices belong to the smallest oriented Petry cycle with length m ≡ 0 (mod 3) is the Petry cycle P i,j,k (Φ).

Proposition 1. Choose any flag Φ. There are at most three Petrie cycles incident with Φ. Proof. The following pairs of Petrie cycles are equal: P 0,1,2 (Φ) = P 2,1,0 (Φ), P 1,2,0 (Φ) = P 0,2,1 (Φ), P 2,0,1 (Φ) = P 1,0,2 (Φ).

Petrie walks and Petrie cycles may be represented as curves (meeting a flag Φ (v, e, f) in 1, 2 or 3 “arcs” near the points v, e, f)

Proposition 2. A flag Φ may be incident with 1, 2 or 3 Petrie cycles in 6 possible ways. There are 6 types of flags corresponding to the 6 ways how the Petrie cycles may meet a given flag.

Definition 3. A Petry cycle is called simple if it meets each flag of the flag graph F(M) at most once. Proposition 3. If all the Petry cycles are simple, each flag meets each Petrie cycle at most once (it is of the type I ).

Theorem 1. If M is a 1-orbit map then all the Petry cycles P in F(M) have the same length L(P). Proof. L(P 2,0,1 ( Φ )) = L(P 1,2,0 ( Φ)), since P 2,0,1 ( Φ ) and P 1,2,0 ( Φ) differ only in each third flag. L(P 1,2,0 ( Φ)) = L ( P 1,2,0 ( Φ 0 ) since Φ and Φ 0 are in the same orbit. L(P 0,1,2 ( Φ )) = L(P 1,2,0 ( Φ 0 )), since this is the same Petry cycle.

Definition 4. Given a Petrie cycle P i,j,k (Φ) = = (Φ, Φ i, (Φ i ) j, ((Φ i ) j ) k, (((Φ i ) j ) k ) i,… ) = = (Φ 1, Φ 2, Φ 3, Φ 4, Φ 5, … Φ 3m ) and a chosen flag Φ s of it the corresponding Petrie line is the set (with at most 3 elements): L i,j,k (Φ s ) = {Φ s, Φ s+ m(mod 3m), Φ s + 2m(mod 3m) }.

Some Petrie lines may be “degenerate” − incident with less than 3 points

Definition 4. A Petrie map is a map in which all the Petrie lines are non-degenerate. The corresponding incidence structure whose points correspond to flags of the flag graph F(M) and whose lines correspond to Petrie lines is called the Petrie configuration of M and is denoted P(M). Remark. It is possible to show that any such P(M) is really a non-lineal (n 3 )-configuration.

3. Existence of Petrie maps Proposition 5. If each Petry cycle in F(M) is simple, then M is a Petrie map. Proposition 6. If M is a 1-orbit map and at least one of its Petry cycles is simple, then all of its Petry cycles are simple, hence M is a Petrie map. Proof. If P 0,1,2 ( Φ ) is simple then P 1,2,0 ( Φ ) = P 1,2,0 ( Φ 0 ) = P 0,1,2 ( Φ ) is simple, and likewise P 2,0,1 ( Φ ) is simple, too. Proposition 7. All the polyhedral maps of the five Platonic solids are examples of Petrie maps. Proof. They are all 1-orbit maps; it suffices to show that they all have a simple Petrie cycle.

Octahedron and icosahedron have simple Petrie cycles:

The (n 3 ) configuration of dodecahedron is non-lineal:

Two Petrie lines of the same flag in a cube are the same

Petrie cycles of the tetrahedron

The incidence table of points and lines of the non-lineal configuration (24 3 ) of the tetrahedron

4. Structure of Petrie configurations From the above examples the structure of Petrie configurations could be guessed and then proven. Theorem 2. If all the Petrie cycles in F(M) has length 3m, then for any flag Φ : a) if m = 3(3n), then L 0,2,1 (3n) = L 2,0,1 (3n), b) if m = 3(3n +1), then Φ 1,0,2 (3n +1) = Φ 1,2,0 (3n + 1) c) if m = 3(3n + 2), then Φ 0,2,1 (3n+2) = Φ 2,0,1 (3n+2). Thus, every Petrie configuration is a non-lineal (n 3 ) configuration.

Proof. If m = 3(3n), then L 0,2,1 ( Φ ) = L 2,0,1 ( Φ ) = L 1,0,2 (Φ), since Φ 0,2,1 (3m) = Φ 2,0,1 (3m) for every m: if m = 3(3n +1), then Φ 1,0,2 (3n +1) = Φ 1,2,0 (3n + 1) if m = 3(3n + 2), then Φ 0,2,1 (3n+2) = Φ 2,0,1 (3n+2).

Also other (n 3 )-configurations, called s-Petrie configurations, can be obtained from maps: Their l ines L s ( Φ ) contain the s-th flags in the Petrie cycles P 0,1,2 ( Φ ), P 1,2,0 ( Φ ), P 2,0,1 ( Φ ).

At least for s = 2,3,4 the lines L s ( Φ ) are non- degenerate (contain 3 different elements). For other maps (e.g. on the torus) this is not always true.

3-Petrie lines in the cube satisfy the lineality condition, hence the corresponding 3-Petrie configuration is lineal.

5. Open questions, future work 1) Which maps of Archimedean solids have only simple Petrie circuits? 2) Which uniform solids are Petrie maps? 3) Study other examples of (n k )-configurations obtained by a similar method from k-regular graphs whose edges are labeled with 1,2,…,k, e.g. from Cayley graphs of some groups with k generators. Such configurations will very likely „inherit“ many symmetries from their „parents“.

Thank you!