**Formal Definition**

Formally, a rotation system is defined as a pair (σ,θ) where σ and θ are permutations acting on the same ground set *B*, θ is a fixed-point-free involution, and the group <σ,θ> generated by σ and θ acts transitively on *B*.

To derive a rotation system from a 2-cell embedding of a connected multigraph *G* on an oriented surface, let *B* consist of the *darts* (or *flags*, or *half-edges*) of *G*; that is, for each edge of *G* we form two elements of *B*, one for each endpoint of the edge. Even when an edge has the same vertex as both of its endpoints, we create two darts for that edge. We let θ(*b*) be the other dart formed from the same edge as *b*; this is clearly an involution with no fixed points. We let σ(*b*) be the dart in the clockwise position from *b* in the cyclic order of edges incident to the same vertex, where "clockwise" is defined by the orientation of the surface.

If a multigraph is embedded on an orientable but not oriented surface, it generally corresponds to two rotation systems, one for each of the two orientations of the surface. These two rotation systems have the same involution θ, but the permutation σ for one rotation system is the inverse of the corresponding permutation for the other rotation system.

Read more about this topic: Rotation System

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### Famous quotes containing the words definition and/or formal:

“The *definition* of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.”

—Samuel Taylor Coleridge (1772–1834)

“It is in the nature of allegory, as opposed to symbolism, to beg the question of absolute reality. The allegorist avails himself of a *formal* correspondence between “ideas” and “things,” both of which he assumes as given; he need not inquire whether either sphere is “real” or whether, in the final analysis, reality consists in their interaction.”

—Charles, Jr. Feidelson, U.S. educator, critic. Symbolism and American Literature, ch. 1, University of Chicago Press (1953)