# Map

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

## Idea

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

### Graph embedding

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

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

### Graph-encoded map

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

Another topology-free definition of maps can be made in terms of permutations acting on a set.[6] A rotation system is a triple ${\displaystyle (X,\psi ,\phi )}$ such that:

• X  is a set. Elements of the set are called darts.
• ψ  is a permutation on X  such that ${\displaystyle \psi (\psi (x))=x}$ 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 ${\displaystyle y=\psi ^{n}(x)}$ 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 ${\displaystyle \psi (x)}$.
• A face is the orbit a dart traces out by alternating between φ  and ψ . Precisely it is the set of darts ${\displaystyle \left\{\varphi \circ (\psi \circ \varphi )^{i}(x),(\psi \circ \varphi )^{i}(x)\mid i\in \mathbb {Z} \right\}}$.

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 ${\displaystyle \varphi ^{-1}}$.

## Polyhedral maps

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.