Editing Discrete mathematics (section)

==== Discrete geometry ====
[[Discrete geometry]] and combinatorial geometry are about combinatorial properties of ''discrete collections'' of geometrical objects. A long-standing topic in discrete geometry is [[tessellation|tiling of the plane]].

In [[algebraic geometry]], the concept of a curve can be extended to discrete geometries by taking the [[Spectrum of a ring|spectra]] of [[polynomial ring]]s over [[finite field]]s to be models of the [[affine space]]s over that field, and letting [[Algebraic variety|subvarieties]] or spectra of other rings provide the curves that lie in that space. Although the space in which the curves appear has a finite number of points, the curves are not so much sets of points as analogues of curves in continuous settings. For example, every point of the form <math>V(x-c) \subset \operatorname{Spec} K[x] = \mathbb{A}^1</math> for <math>K</math> a field can be studied either as <math>\operatorname{Spec} K[x]/(x-c) \cong \operatorname{Spec} K</math>, a point, or as the spectrum <math>\operatorname{Spec} K[x]_{(x-c)}</math> of the [[Localization of a ring|local ring at (x-c)]], a point together with a neighborhood around it. Algebraic varieties also have a well-defined notion of [[tangent space]] called the [[Zariski tangent space]], making many features of calculus applicable even in finite settings.