CLASSICAL GEOMETRY : More than 180 definition and theorem by : DANNY CALEGARI

A CRASH COURSE IN GROUP THEORY : A group is an algebraic object which formalizes the mathematical notion which expresses the intuitive idea of symmetry. We start with an abstract definition.


MODEL GEOMETRIES IN DIMENSION TWO : Euclid, who taught at Alexandria in Egypt and lived from about 325 BC to 265 BC, is thought to have written 13 famous mathematical books called the Elements. In these are found the earliest (?) historical example of the axiomatic method. Euclid proposed 5 postulates or axioms of geometry, from which all true statements about the Euclidean plane were supposed to inevitably follow. These axioms were as follows:

(1) A straight line segment can be drawn joining any two points.

(2) Any straight line segment is contained in a unique straight line.

(3) Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.

(4) All right angles are congruent.

(5) One and only one line can be drawn through a point parallel to a given line.


The topology of surfaces.


Geometry is a beast that can be approached from many angles. Four of the most important concepts that arise from our different primitive intuitions of geometry are symmetry, measurement, analysis, and continuity. We briefly discuss these four faces of geometry, and mention some fundamental concepts in each. Don’t worry if these concepts seem very technical or abstract — think of this section as an abstraction of the concrete notions found in the main body of the text.


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