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[Texts]## 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.

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.

This image slightly hand-edited for clarity.

- 51A: Linear incidence geometry
- 51B: Nonlinear incidence geometry
- 51C05: Ring geometry (Hjelmslev, Barbilian, etc.)
- 51D: Geometric closure systems
- 51E: Finite geometry and special incidence structures
- 51F: Metric geometry (distances, lengths, and angles; triangulation)
- 51G05: Ordered geometries (ordered incidence structures, etc.)
- 51H: Topological geometry
- 51J: Incidence groups
- 51K: Distance geometry
- 51L: Geometric order structures, see also 53C75
- 51M: Real and complex geometry, especially 51M04: Elementary Euclidean geometry (2-dimensional), 51M05: Generalized Euclidean geometry (3- and N-dimensional), and 51M15: Geometric Constructions.
- 51N: Analytic and descriptive geometry
- 51P05: Geometry and physics (should also be assigned at least one other classification number from Sections 70--86)

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.

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

- Coxeter, H. S. M., "Introduction to geometry", John Wiley & Sons, Inc., New York, 1989. 469 pp. ISBN 0-471-50458-0 MR90a:51001 (First imprint of second edition 1969: MR49#11369. First edition, 1961: MR23#A1251).
- Chern, Shiing Shen: "What is geometry?", Amer. Math. Monthly 97 (1990), no. 8, 679--686. MR91h:01003
- Atiyah, Michael: "What is geometry?", Math. Gaz. 66 (1982), no. 437, 179--184. MR84g:51001
- Eves, Howard: "A survey of geometry", Allyn and Bacon, Inc., Boston, Mass., 1972. 442 pp. MR48#1015
- Tarski, Alfred: "What is elementary geometry? The axiomatic method. With special reference to geometry and physics." Proceedings of an International Symposium held at the Univ. of Calif., Berkeley, Dec. 26, 1957-Jan. 4, 1958, pp. 16--29; Studies in Logic and the Foundations of Mathematics; North-Holland Publishing Co., Amsterdam 1959 MR21#4919
- Aleksandrow, A. D.: "What is geometry?" (Polish) Wiadom. Mat. (2) 1, (1955). 4--46. MR16,1044d
- Fejes Tóth, László: "What is 'discrete geometry'?" (Hungarian) Magyar Tud. Akad. Mat. Fiz. Oszt. Közl. 13 1963 229--238. MR28#4421

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

There are eight Geometry Forum Newsgroups geometry.*

- The Geometer's Sketchpad.
- Cabri-geometry is used for teaching secondary school geometry, but, equally important, is its use for university level instruction and as a tool by mathematicians in their research work.
- Cinderella, featured in the paper: "Automatic theorem proving of Geometric Theorems", H. Crapo and J. Richter-Gerbert.
- Downloadable software from the Geometry Center
- Pointer to software for n-dimensional geometric modelling
- Geometry packages for Mathematica, versions 2.2 and 3.0.

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.

- Dave Eppstein's Geometry junkyard.
- A useful collection of Geometry Formulas and Facts is taken from the CRC Standard Mathematical Tables and Formulas, and available at the The Geometry Center.
- Here are the AMS and Goettingen resource pages for area 51.

- Hilbert's axioms for geometries
- Geometries with a "betweenness" relation.
- Desargues' theorem, Pappus' theorem, and coordinatization
- Which points on a box are furthest apart (geodesic distance) -- it's not opposite corners!
- Citation for decomposition H=A u B u C of hyperbolic plane with A, B, C all congruent and A congruent to B u C
- Summary of multidimensional scaling (dimension reduction, singular-value decomposition, rather like principal component analysis) to pick out key data attributes -- or locate cities on a map.
- Example of computing the right affine change-of-variables
- ASCII-art version of the Fano plane (7 points on 7 lines)
- Automated proof of Pons Asinorum
- Using Fibonacci numbers to make a geometric puzzler
- Points in the plane which are integral distances apart

Last modified 2000/01/24 by Dave Rusin. Mail: