What is Geometry and Topology?

Geometry is the study of shapes and spaces. Most people are aware of the standard objects of Euclidean geometry: lines, circles, polygons and of familiar notions such as angles, parallel lines, and congruent figures. In its modern form geometry has a much wider scope, reaching into higher dimensions and encompassing a broad range of current ideas. The subject of topology is concerned with those features of geometry which remain unchanged after twisting, stretching or other deformations of a geometrical space. It includes such problems as coloring maps, distinguishing knots and classifying surfaces and their higher dimensional analogs. The influence of topology is also important in other mathematical disciplines such as dynamical systems, algebraic geometry (the study of polynomial equations in many variables) and certain aspects of analysis and combinatorics.

More Information:

What is Topology? - An essay by Neil Strickland.

Steven Strogatz explains topology in his New York Times article.

What is Topology? - Robert Bruner's answer.

What is Algebraic Topology? - Joseph Neisendorfer's answer.

History of Topology in the History of Mathematics archive at Saint Andrews University in Scotland.

A Circular History of Knot Theory. - An essay by Bill Menasco.

What is Riemannian Geometry? - Answered by Christina Sormani.

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