Veech surface

In mathematics a translation surface is a surface obtained from identifying the sides of a polygon in the Euclidean plane by translations. An equivalent definition is a Riemann surface together with an holomorphic 1-form.

These surfaces arise in dynamical systems where they can be used to model billiards, and in Teichmüller theory. A particularly interesting subclass is that of Veech surfaces (named after William A. Veech) which are the most symmetric ones.

A translation surface is the space obtained by identifying pairwise by translations the sides of a collection of plane polygons.

Here is a more formal definition. Let ${\displaystyle P_{1},\ldots ,P_{m}}$ be a collection of (not necessarily convex) polygons in the Euclidean plane and suppose that for every side ${\displaystyle s_{i}}$ of any ${\displaystyle P_{k}}$ there is a side ${\displaystyle s_{j}}$ of some ${\displaystyle p_{l}}$ with ${\displaystyle j\not =i}$ and ${\displaystyle s_{j}=s_{i}+{\vec {v}}_{i}}$ for some nonzero vector ${\displaystyle {\vec {v}}_{i}}$ (and so that ${\displaystyle {\vec {v}}_{j}=-{\vec {v}}_{i}}$. Consider the space obtained by identifying all ${\displaystyle s_{i}}$ with their corresponding ${\displaystyle s_{j}}$ through the map ${\displaystyle x\mapsto x+{\vec {v}}_{i}}$.

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