geometry

geometry

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Geometry

, or literally, the measurement of land, is among the oldest
and largest areas of mathematics. It is as old as civilization itself
— even when texts and traditions have been lost, such monuments as
Stonehenge and the

pyramids

of Egypt and South America stand as mute
to the geometrical knowledge of the
ancients. Over the centuries, geometry has grown from its humble
origins in land measurement to a study of the properties of space in
the widest sense of the . In

addition

to the
familiar three-dimensional space in which we move and breathe, modern
geometers routinely consider spaces of more than three

dimensions

,
even infinite-dimensional and fractional dimensional spaces, curved
spaces, discrete spaces, non-commutative spaces,

infinitesimal

spaces,
and many other of spaces.

For this reason, it is quite difficult to provide a precise definition
of geometry. In this survey of geometry, we shall indicate several
approaches to the subject. We start with the synthetic (or axiomatic)
approach to

Euclidean geometry

not only because that is historically
the oldest, but because it is the approach one is most likely to
encounter first. After this, we move on to other approaches in
roughly an of increasing mathematical
sophistication.

In this survey, our goal is to give the reader an overview of the
different of geometry, the

concepts

and
techniques used, and the sort of results which are proven. In order
to make this

accessible

to a wide audience, we have assumed the
minimum of knowledge on the part of the reader necessary to understand
and appreciate the topics presented in a meaningful way. Since our
goal is to present the substance and flavor of the subjects discussed
as opposed to giving a comprehensive and detailed account, we
sometimes omit technical details in the interest of clarity. To
compensate for this shortcoming, we have included
to entries in which the interested reader may
find more detailed and

rigorous

treatments of the topics discussed
here 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/6977

Coordinate 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 of
calculus. In fact, its early history is indistiguishable from that of
calculus — it is a matter of personal taste whether one chooses to
regard Fermat’s method of drawing

tangents

and finding extrema as a
contribution to calculus or differential geometry; the pioneering work
of Barrow and Newton on calculus was presented in a geometrical

language

; Halley’s 1696 paper in which he announces his discovery that

∫

d

⁢

x

x

=

log

⁡

x

+

C

is entitled

quadrature

of the

hyperbola

.

It is only later on, when calculus became more algebraic in outlook
that one can begin to make a meaningful

separation

between the
subjects 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/6977

Coordinate 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

References

1

D. Hilbert

:

Grundlagen der Geometrie

. Neunte Auflage, revidiert und ergänzt von Paul Bernays.  B. G. Teubner Verlagsgesellschaft, Stuttgart (1962).

Title

geometry

Canonical name

Geometry

Date of creation

2013-03-22 13:19:01

Last modified on

2013-03-22 13:19:01

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

46

Author

rspuzio (6075)

Entry type

Topic

Classification

msc 51-00

Classification

msc 51-01

Synonym

Egyptian geometry

Related topic

FiniteProjectivePlane4

Related topic

ProjectivePlane2

Related topic

PointFreeGeometry

Related topic

ComparisonOfCommonGeometries

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DeBruijnErdHosTheorem

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Defines

Greek geometry

Defines

Euclidean geometry

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