In mathematics, non-Archimedean geometry is any of a number of forms of

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

in which the

axiom of Archimedes In abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some algebraic structures, such as ordered or normed groups, and fields. The property, typic ...

is negated. An example of such a geometry is the

Dehn plane In geometry, Max Dehn introduced two examples of planes, a semi-Euclidean geometry and a non-Legendrian geometry, that have infinitely many lines parallel to a given one that pass through a given point, but where the sum of the angles of a triangle ...

. Non-Archimedean geometries may, as the example indicates, have properties significantly different from

Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...

. There are two senses in which the term may be used, referring to geometries over

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

s which violate one of the two senses of the

Archimedean property In abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some algebraic structures, such as ordered or normed groups, and fields. The property, typica ...

(i.e. with respect to order or magnitude).

Geometry over a non-Archimedean ordered field

The first sense of the term is the geometry over a

non-Archimedean ordered field In mathematics, a non-Archimedean ordered field is an ordered field that does not satisfy the Archimedean property. Examples are the Levi-Civita field, the hyperreal numbers, the surreal numbers, the Dehn field, and the field of rational function ...

, 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 In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry: ''If a line segmen ...

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 Value or values may refer to: Ethics and social * Value (ethics) wherein said concept may be construed as treating actions themselves as abstract objects, associating value to them ** Values (Western philosophy) expands the notion of value beyo ...

,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. or

ultrametric space In mathematics, an ultrametric space is a metric space in which the triangle inequality is strengthened to d(x,z)\leq\max\left\. Sometimes the associated metric is also called a non-Archimedean metric or super-metric. Although some of the theorems ...

. 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 In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extensio ...

. Intuitively, in such a space, distances fail to "add up" or "accumulate".

References

Fields of geometry