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Maps are a concrete definition of a polyhedron based on its topological surface.

Idea[edit | edit source]

The icosahedron as a tiling of the sphere.

We can think about polyhedra as tilings of a surface. Euclidean tilings are tilings of the plane, and convex polytopes are tilings of a sphere-like surface. We'd like to extend this idea to tile more surfaces than these. Maps are a topological way of extending this concept of tiling a space so that we can tile all sorts of surfaces.

A map is a division of a surface into vertices, edges and faces. Vertices should be single points (0-balls), edges should look like line segments (1-balls) and faces should look like discs (2-balls). We want these divisions to be nicely behaved somehow, so making every point on a surface a vertex isn't a map, decomposing the surface into only edges isn't a map et cetera. We want maps to look somewhat like polyhedra.

Definitions[edit | edit source]

Graph embedding[edit | edit source]

A map is a graph embedding of a connected multi-graph (allows multiple edges and self-loops)[note 1] onto a compact connected 2-manifold, such that every connected component of the complement of the embedding is homeomorphic to an open disc.[1][2][3] These connected components are the faces of the polyhedron.[2]

Topological[edit | edit source]

Topologically, a map can be defined as a 2-cell decomposition of a compact connected 2-manifold.[2]

Graph-encoded map[edit | edit source]

The graph-encoded map of a tetrahedron on the left and on the right the same map with its e  edges removed to show a graph made only of cycles of size 4.

A map can also be defined without reference to topology at all as a graph-encoded map. A graph-encoded map or gem is a finite properly edge 3-colored graph, with colors v , e  and f  such that the subgraph generated by the edges v  and f  form cycles of size 4.[4]

Graph-encoded maps have a bijective correspondence with finite maps defined in terms of graph embeddings, thus these definitions are equivalent.[5]

Rotation systems[edit | edit source]

The rotation system of a tetrahedron. Red double-tipped arrows indicate the φ  permutation, blue bi-directional arrows indicate the ψ  permutation.

Another topology-free definition of maps can be made in terms of permutations acting on a set.[6] A rotation system is a triple such that:

  • X  is a set. Elements of the set are called darts.
  • ψ  is a permutation on X  such that for every dart x  and ψ  has no fixed points.
  • φ  is a permutation on X .

From here we can define vertices, edges and faces:

  • A vertex is an orbit of a dart under φ . That is for some dart x  it is the set of all darts such that for some integer n .
  • An edge is an orbit of a dart under ψ . Since ψ  is an involution each edge has two darts: x  and .
  • A face is the orbit a dart traces out by alternating between φ  and ψ . Precisely it is the set of darts .

Two elements are incident on another if their intersection is non-empty.

Rotation systems correspond exactly to orientable maps. That is every orientable map under the other definitions is expressible as a rotation system and vice versa.[7] Rotation systems can be extended to make a definition of map that encompasses non-orientable maps by making some edges reverse orientation. Edges that reverse orientation are called barred edges[7] and can be indicated in diagrams by drawing a bar through the edge.[7] Intuitively when determining the faces of a map crossing barred edges causes the direction of the φ  permutation to reverse replacing it with .

Polyhedral maps[edit | edit source]

A polyhedral map is a map whose skeleton has no self-loops or multiple edges, such that the intersection of any two distinct elements is either:[8]

  • empty
  • a single vertex
  • a single edge (including its two vertices)

Polyhedral maps are dyadic and don't permit degeneracies such as dihedra and hosohedra. Polyhedral maps are thus a subset of abstract polyhedra.

External links[edit | edit source]

Notes[edit | edit source]

  1. Some authors require that the graph be simple (i.e. without multiple edges or self-loops) with every vertex having degree at least 3, other authors refer to these maps as singular maps. This wiki follows the latter convention, using the term map to refer to a broader class of structures.

References[edit | edit source]

Bibliography[edit | edit source]

  • Bonnington; Little (1995). The Foundations of Topological Graph Theory (1 ed.). Springer. doi:10.1007/978-1-4612-2540-9. ISBN 978-1-4612-7573-2.
  • Brehm, Ulrich; Schulte, Egon (1997), Polyhedral Maps
  • Lins, Sóstenes (1982). "Graph-encoded maps". Journal of Combinatorial Theory. Series B. 32 (2): 171–181. doi:10.1016/0095-8956(82)90033-8. MR 0657686.
  • Nedela, Roman (2007), Maps, Hypermaps, and Related Topics (PDF)
  • Wilson, Steve (2012). "Maniplexes: Part 1: Maps, Polytopes, Symmetry and Operators". Symmetry. doi:10.3390/sym4020265.