In mathematics, non-Archimedean geometry is any of a number of forms of geometry in which the axiom of Archimedes is negated. An example of such a geometry is the Dehn plane. Non-Archimedean geometries may, as the example indicates, have properties significantly different from Euclidean geometry.
Geometry over a non-Archimedean ordered field
The first sense of the term is the geometry over a non-Archimedean ordered field, or a subset thereof. The aforementioned Dehn plane takes the self-product of the finite portion of a certain non-Archimedean ordered field based on the field of rational functions. In this geometry, there are significant differences from Euclidean geometry; in particular, there are infinitely many parallels to a straight line through a point—so the parallel postulate fails—but the sum of the angles of a triangle is still a straight angle.
Intuitively, in such a space, the points on a line cannot be described by the real numbers or a subset thereof, and there exist segments of "infinite" or "infinitesimal" length.
Geometry over a non-Archimedean valued field
The second sense of the term is the metric geometry over a non-Archimedean valued field, or ultrametric space. In such a space, even more contradictions to Euclidean geometry result. For example, all triangles are isosceles, and overlapping balls nest. An example of such a space is the p-adic numbers.
Intuitively, in such a space, distances fail to "add up" or "accumulate".
- Robin Hartshorne, Geometry: Euclid and beyond (2000), p. 158.
- Hilbert, David (1902), The foundations of geometry (PDF), The Open Court Publishing Co., La Salle, Ill., MR 0116216
- Conrad, B. "Several approaches to non-archimedean geometry. In p-adic Geometry (Lectures from the 2007 Arizona Winter School). AMS University Lecture Series." Amer. Math. Soc., Providence, RI 41 (2008): 78.