Non-Archimedean geometry

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In mathematics, non-Archimedean geometry [1] 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.

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There are two senses in which the term may be used, referring to geometries over fields which violate one of the two senses of the Archimedean property (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, 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. [2]

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, [3] 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".

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<span class="mw-page-title-main">Space (mathematics)</span> Mathematical set with some added structure

In mathematics, a space is a set with some added structure. While modern mathematics uses many types of spaces, such as Euclidean spaces, linear spaces, topological spaces, Hilbert spaces, or probability spaces, it does not define the notion of "space" itself.

Foundations of geometry is the study of geometries as axiomatic systems. There are several sets of axioms which give rise to Euclidean geometry or to non-Euclidean geometries. These are fundamental to the study and of historical importance, but there are a great many modern geometries that are not Euclidean which can be studied from this viewpoint. The term axiomatic geometry can be applied to any geometry that is developed from an axiom system, but is often used to mean Euclidean geometry studied from this point of view. The completeness and independence of general axiomatic systems are important mathematical considerations, but there are also issues to do with the teaching of geometry which come into play.

In absolute geometry, the Saccheri–Legendre theorem states that the sum of the angles in a triangle is at most 180°. Absolute geometry is the geometry obtained from assuming all the axioms that lead to Euclidean geometry with the exception of the axiom that is equivalent to the parallel postulate of Euclid.

In algebra, a Pythagorean field is a field in which every sum of two squares is a square: equivalently it has Pythagoras number equal to 1. A Pythagorean extension of a field is an extension obtained by adjoining an element for some in . So a Pythagorean field is one closed under taking Pythagorean extensions. For any field there is a minimal Pythagorean field containing it, unique up to isomorphism, called its Pythagorean closure. The Hilbert field is the minimal ordered Pythagorean field.

In mathematics, a non-Archimedean ordered field is an ordered field that does not satisfy the Archimedean property. Such fields will contain infinitesimal and infinitely large elements, suitably defined.

References

  1. Robin Hartshorne, Geometry: Euclid and beyond (2000), p. 158.
  2. Hilbert, David (1902), The foundations of geometry (PDF), The Open Court Publishing Co., La Salle, Ill., MR   0116216
  3. 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.