Euclidean geometry

Euclidean geometry, also called "flat" or "parabolic" geometry, is named after the Greek mathematician Euclid. Euclid's text Elements is an early systematic treatment of this kind of geometry, based on axioms (or postulates). This is the kind of geometry familiar to most people, since it is the kind usually taught in high school.

This system is an axiomatic system, which hoped to prove all the "true statements" as theorems in geometry from a set of finite number of axioms.

The five postulates/axioms of the Euclidean system are:

  • Any two points can be joined by a straight line.
  • Any straight line segment can be extended indefinitely in a straight line.
  • Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
  • All right angles are congruent.
  • If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate.

Euclidean geometry is distinguished from other geometries by the parallel postulate, which is more easily phrased as follows
Through a point not on a given straight line, one and only one line can be drawn that never meets the given line.

The parallel postulate was a subject of deep contention among modern mathematicians in the middle ages. Many mathematicians, argued that the fifth postulate was redundant, and could be deduced independently from the first four postulates. They tried to prove this fact by "not assuming" the postulate to hold true, and try and arrive at this postulate as a theorem by means of inference rules alone. O

Other mathematicians proposed that the statement was indeed a postulate, and tried to prove their suggestions, by "negating the postulate" (call the negation NOT P) as an axiom, hoping to arrive at a contradiction. If a contradiction is achieved with Y, a known theorem, (previously deduced from the other postulates) ie: if it leads to a situation - Y AND NOT Y, or if a contradiction is achieved directly with the assumed NOT P ie : if we arrive at P AND NOT P, this would mean that the assumption of negation of P was wrong (Proof by contradiction), and hence the parallel postulate needs to be "assumed to be true".

However, both factions of mathematicians were stumped in their efforts to achieve a definite answer to the question of "whether the parallel postulate is an axiom of geometry". The disbelievers, could not successfully prove that it is not. Neither could the believers arrive at a contradiction, by negating it. Curiously, however, by negating the fifth postulate in various ways, they extended geometry to represent non-planar universes. (See Non-Euclidean geometry for further explication)

In particular, this postulate separates Euclidean geometry from hyperbolic geometry, where many parallel lines could be drawn through the point, and from elliptic and projective geometry, where no parallel lines exist. (Euclidean geometry does, however, share the parallel postulate with some other geometries, such as certain finite geometries and affine geometry.)

Since Euclid's time, other mathematicians have laid out more thorough axiomatic systems for Euclidean geometry, such as David Hilbert and George Birkhoff.

Modern Concept of Euclidean Geometry

Today Euclidean geometry is usually constructed rather than axiomatized, by means of analytic geometry. A rectangular coordinate system maps each point in Euclidean space with a unique list of n real numbers (x1,...,xn), so we can define it to be the set of all such lists (Rn). We also define a metric (distance function) d by

which you might recognise as an application of the Pythagorean Theorem (see also Euclidean distance). This turns Rn into a metric space. Maps that preserve the distance between all pairs of points are called isometries, and include reflections, rotations, translationss, and compositions thereof. In matrix notation any of these have the form

where A is an orthogonal matrix and b is a column vector. Isometries are taken as the congruencess of Euclidean geometry - that is, we only consider properties preserved by them. That way we do not have to worry about the precise origin or axes, but still consider distances, angles, and so forth.

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