geometry

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Geometry, or literally, the measurement of land, is among the oldestand largest areas of mathematics. It is as old as civilization itself— even when texts and traditions have been lost, such monuments asStonehenge and the pyramids of Egypt and South America stand as mute to the geometrical knowledge of theancients. Over the centuries, geometry has grown from its humbleorigins in land measurement to a study of the properties of space inthe widest sense of the . In addition to thefamiliar three-dimensional space in which we move and breathe, moderngeometers routinely consider spaces of more than three dimensions,even infinite-dimensional and fractional dimensional spaces, curvedspaces, discrete spaces, non-commutative spaces, infinitesimal spaces,and many other of spaces.

For this reason, it is quite difficult to provide a precise definitionof geometry. In this survey of geometry, we shall indicate severalapproaches to the subject. We start with the synthetic (or axiomatic)approach to Euclidean geometry not only because that is historicallythe oldest, but because it is the approach one is most likely toencounter first. After this, we move on to other approaches inroughly an of increasing mathematicalsophistication.

In this survey, our goal is to give the reader an overview of thedifferent of geometry, the concepts andtechniques used, and the sort of results which are proven. In orderto make this accessible to a wide audience, we have assumed theminimum of knowledge on the part of the reader necessary to understandand appreciate the topics presented in a meaningful way. Since ourgoal is to present the substance and flavor of the subjects discussedas opposed to giving a comprehensive and detailed account, wesometimes omit technical details in the interest of clarity. Tocompensate for this shortcoming, we have included to entries in which the interested reader mayfind more detailed and rigorous treatments of the topics discussedhere as well as related topics which had to be omitted to keep the of this entry within reasonable.

0.1 Axiomatic method (http://planetmath.org/AxiomaticGeometry)

0.2 Analytic and Descriptive Geometry

1. Euclidean geometry of plane

2. Euclidean geometry of space

3. http://planetmath.org/node/6977Coordinate systems

4. Topics on vectors

5. Index of entries on compass and straightedge constructions

0.3 Geometry as the study of invariants under certain transformations (http://planetmath.org/GeometryAsTheStudyOfInvariantsUnderCertainTransformations)

0.4 Differential geometry

Differential geometry studies geometrical objects using techniques ofcalculus. In fact, its early history is indistiguishable from that ofcalculus — it is a matter of personal taste whether one chooses toregard Fermat’s method of drawing tangents and finding extrema as acontribution to calculus or differential geometry; the pioneering workof Barrow and Newton on calculus was presented in a geometricallanguage; Halley’s 1696 paper in which he announces his discovery that${\displaystyle \int \frac{dx}{x}}=\log x+C$ is entitled quadrature of the hyperbola.

It is only later on, when calculus became more algebraic in outlookthat one can begin to make a meaningful separation between thesubjects of calculus and differential geometry.

Below are some main topic entries on PlanetMath on differential geometry:

1. Euclidean geometry of plane

2. Euclidean geometry of space

3. http://planetmath.org/node/6977Coordinate systems

4. Topics on vectors

5. Classical differential geometry

6. Bibliography for differential geometry

7. Fundamental concepts in differential geometry

8. Concepts in symplectic geometry

0.5 Algebraic geometry