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51: Geometry


Geometry is studied from many perspectives! This large area includes classical Euclidean geometry and synthetic (non-Euclidean) geometries; analytic geometry; incidence geometries (including projective planes); metric properties (lengths and angles); and combinatorial geometries such as those arising in finite group theory. Many results in this area are basic in either the sense of simple, or useful, or both!

There is separate page for constructibility questions (i.e. compass-and-straightedge constructions).

There is a separate page for a triangulation problem (in the geographers' sense, not the topologists'). Included on that page is information about determining location by distances from fixed points.


A bibliography (and some related web sites) on the history of geometry is available from David Joyce.

See the article on NonEuclidean geometry at St Andrews.

Applications and related fields

As appealing as questions on simple geometry are, they are often mathematically speaking rather trivial, so we have little material here. Some of the meatier issues in "geometry" are easily classified somewhere else. Thus you'll have to look elsewhere for topics on algebraic geometry and differential geometry; material on topology is with manifolds. It's a fine line between differential geometry and analysis on manifolds; topics like, "Can you hear the shape of a drum" (e.g. genus of a surface) are found in global analysis too, as are topics on fractals. Many questions which are at first blush geometric can also be considered group-theoretic. Likewise, some questions from vector geometry amount to so much linear algebra. Also relevant: a link to a separate page on spheres. The conic sections are also studied in algebraic geometry. Numerical questions about geometry show up in number theory (the "Geometry of Numbers"); look there for questions of the sort, "can we find arrangements of points making certain distances/areas/volumes integral (or rational)?" such as topics involving Pythagorean triples.

For geometric questions involving polyhedra and regular solids see (also) Polytopes and polyhedra. This also includes topics about polygons. Many questions about them arise in computational geometry (computer graphics -- 68U05), for which there is also separate page.

Solid geometry is placed here (actually in 51M05) because it mirrors elementary plane geometry, but spherical geometry is primarily on the page for general convex geometry.

Other topics in related areas: differential geometry, geometric topology (Manifolds), trigonometry (Special Functions).

The distinction between sections 51 and 52:Convex geometry and their subfields is perhaps slight. [Schematic of subareas and related areas]

This image slightly hand-edited for clarity.


A section heading "48 Geometry" was replaced in 1958 with section "50 Geometry"; it was in turn replaced in 1970.

Browse all (old) classifications for this area at the AMS.

Textbooks, reference works, and tutorials

It is something of a challenge to define the real scope of "geometry" and to cover it with any justice. Some notable efforts:

Just for fun: David Wells; The Penguin Dictionary of Curious and Interesting Geometry; Penguin Group. 1991.

There are eight Geometry Forum Newsgroups geometry.*

Software and tables

Note that most of most popular instructional software and a great deal of research materials are limited to real (or complex) geometries and so are more appropriately mentioned on the appropriate subpages.

This section of this index page lists existing software which can assist with the understanding of geometry. Discussions of the mathematics behind the development of such software are part of "computational geometry"; see 68U05: Computer Graphics.

Other web sites with this focus

Selected topics at this site

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Last modified 1999/05/12 by Dave Rusin. Mail: feedback@math-atlas.org