Euclidean geometry

The Euclidean geometry is the first we are familiar, descriptive geometry of the plane or three-dimensional space. However, the term has very different aspects and allows generalizations. Named is this mathematical branch of geometry by the Greek mathematician Euclid of Alexandria.

• 2.1 Hilbert's approach
• 2.2 Geometry and Reality in Hilbert
• 2.3 Other axiom systems

• 3.1 Non-Euclidean geometries and the reality

The geometry of Euclid

In the narrowest sense, Euclidean geometry is the geometry that Euclid set out in the elements.

About two thousand years geometry has been taught by this axiomatic structure. The phrase " more geometrico " (Latin: " to the nature of the ( Euclidean ) geometry " ) is still used as an indication of a strictly deductive reasoning.

Euclid is this as follows:

Definitions

The book begins with some definitions, such as:

• A point is that which has no part.
• A line is a breadthless length.
• A straight line is a line that always lies with respect to the points on her equal.

Similarly plane angles are defined, inter alia.

In addition to these more or less ideological definitions of basic terms, there are also definitions that are to be understood in the modern sense as a word introductions, because they are needed in the following text abbreviated, such as for parallels: "Parallel are straight lines that lie in the same plane while if they are extended in both directions to infinity, take no side by side. "

Overall, give the members 35 definitions.

Postulates

According to the more descriptive definitions of the five rather defining postulates follow. It asks for here,

• That could be drawn from any point after each point the distance
• That one can coherently extend a limited straight line straight,
• That one could draw the circle with any center and distance,
• That all right angles are equal to each other,
• That if a straight line with two straight lines effecting while cutting that domestic emerging on the same side angles would together less than two right angles, then the two straight lines would meet if extended to infinity on the side, on the subject, the angle together less than two right are (in short: that to a straight line through a given point, which lies outside of this line, at most one to parallel straight line should exist, see parallel postulate ).

Euclid's axioms

At the five listed geometric postulates to connect multiple logical axioms, for example:

• What is equal to the same, is also equal to each other.
• If for like same is added, the whole being equal.
• If it is taken from equals the same, the residues are the same.

Problems and theorems

Building on Euclid treats problems now ...

And theorems ...

To solve a problem or proving a theorem only the definitions, postulates and axioms and previously proved theorems and constructions of previously solved problems in principle be used.

Geometry and reality in Euclid

As a Platonist Euclid was convinced that reflect the formulated by him postulates and axioms reality. According to Plato's theory they belong to an ontological level higher rank than the drawn in the sand figures, which are their pictures. The relationship between an imperfectly drawn circle, the perfect idea of the circle illustrates the difference between the sensible world and the intelligible (only mentally detectable ) world, which is illustrated in Plato's allegory of the cave.

Differences from a purely axiomatic theory

From today's perspective, do not satisfy the elements of the claim to an axiomatic theory:

• Purpose of the definitions ( as they relate to basic concepts ), it is in Euclid to relate to familiar geometric world of experience and motivate the postulates. The validity of such propositions is judged very differently. Strict Axiomatiker they deem unnecessary.
• The five postulates represent most of what today would be regarded as an axiom. As a basis for the conclusions drawn from them, they are not comprehensive enough and too inaccurate. - It should be noted that at least the first three " postulates " the possibility of certain constructions postulate (and not the applying of specific facts ). Euclid's axioms can also be referred to as a constructive axiomatics.
• Termed axioms statements concern the less geometry, but rather the logical foundations. In terms of a justification of the logic however, they are incomplete.

It follows that the conclusions of necessity use a variety of unspoken assumptions.

The modern axiomatic theory

In another sense, Euclidean geometry is a at the end of the 19th century created strictly axiomatic theory. The above problems were apparent when Russell, Hilbert and other mathematicians worked to a rigorous foundation of mathematics.

They were freed by David Hilbert in his work Foundations of Geometry ( Teubner 1899, numerous reprints ). Precursors were Hermann Grassmann, Moritz Pasch, Giuseppe Peano and others. Even after several other Hilbert axiom systems have been established for the Euclidean geometry.

Hilbert's approach

David Hilbert used " three different systems of things ", namely points, lines and planes, of which he only says: "We think ( it ) to us ." These things should " be considered " " in three basic relations " with each other, namely, " lie ", " between " and " congruent ". To link these " things" and " relations ", he then represents 20 axioms in five groups:

• Eight axioms of logic (incidence)
• Four axioms of arrangement (order)
• Five axioms of congruence ( congruence )
• The axiom of parallels ( parallel axiom )
• Two axioms of continuity ( Archimedean axiom and axiom of completeness )

Geometry and reality in Hilbert

As a representative of the formalism Hilbert declared it irrelevant what these points, lines and planes to do with reality. The meaning of the basic terms is determined to satisfy the axioms. So he begins the section on the axioms of logic with the sentence: " The axioms of this group make between things introduced above: points, lines and planes a shortcut here and are as follows: ... " The definitions of the basic concepts done implicitly.

On the other hand, Hilbert explained in the introduction to his work: " The present study is a new attempt to draw up a complete and simple as possible system of axioms for geometry ... ". With this reference to the geometry he makes it clear that it is not him by any formalism, but rather a clarification of what Euclid meant by " geometry " and what we all know as the properties of the surrounding us space. - This specification is Hilbert completely successful, and she turns out to be much more complicated than Euclid guessed.

Other axiom systems

Later established axiom systems are basically equivalent to the Hilbert. They take into account the advancement of mathematics.

A possible axiomatization is given by the axioms of absolute geometry together with the following axiom which is equivalent to the parallel postulate, assuming the rest of the axioms of absolute geometry:

See also: Meschkowskis system of axioms of Euclidean geometry

Euclidean and non-Euclidean geometry

Furthermore, the term Euclidean geometry serves as a counter-concept to the non-Euclidean geometries

The impetus was thereby addressing the parallel postulate. After it had been centuries earlier unsuccessfully tried to recycle this fifth postulate of Euclid on a simpler, the Hungarian János Bolyai and the Russian Nikolai Ivanovich Lobachevsky concluded in 1830 that a denial of this fifth postulate would lead to logical contradictions if this actually simpler statements could be traced. So the two mathematicians denied this postulate and defined their own (replacement) postulates that led unexpectedly to a logic completely flawless geometric system - the non-Euclidean geometries: " Not the evidence, however, was so disturbing, but his rational byproduct that already soon to overshadow him, and almost everything in mathematics: Mathematics, the cornerstone of scientific certainty, had become uncertain at once. It now had to deal with two conflicting visions unimpeachable scientific truth ", which led to a deep crisis in the sciences ( Pirsig, 1973).

The exact formulation of the " hyperbolic " axiom that in the geometry of Lobachevsky, hyperbolic geometry, takes the place of the parallel axiom is: " ... on a straight line does not pass through a point located at least two lines in this one plane and they do not cut ... "

Non-Euclidean geometries and the reality

Whether non-Euclidean geometries (there are several ) can describe the real space is answered differently. They are usually seen as a purely abstract mathematical theories that deserve only by the similarity of the concepts and systems of axioms called " geometry ".

These theories, however, have now been proven in theoretical physics as very relevant for the description of the reality of our universe.

The analytic geometry of the plane and space

In a coordinate system, a point can be represented as a pair ( in the planar geometry ), or as a triple of real numbers. A straight line or plane then is a set of such pairs of numbers (or - triples ) whose coordinates satisfy a linear equation. The built thereon analytic geometry of the real number plane or the real number space turns out to be completely equivalent to the axiomatic defined.

You can view the analytical geometry as a model for the axiomatic theory. Then it provides a proof of the consistency of the axiom system ( it being necessary to presuppose a consistent justification of the real numbers as a given ).

You can view the analytical approach but also as an independent ( and more comfortable ) reasons for the geometry; from this perspective is the axiomatic access only of historical interest. Bourbaki, for example (as well as Jean Dieudonné ) dispenses completely with the use of original geometric concepts and keeps with the treatment of topological vector spaces the issue settled.

Euclidean geometry as the science of measuring

Euclidian geometry and the geometry, in the segments and angles measures are assigned.

In the axiomatic structure of Euclidean geometry figures apparently come not before. However, it is determined how a congruent attaches to a track in the same direction, so this doubled - and consequently multiplied with an arbitrary natural number. There is also a structure to share a given distance in n equal parts. If now nor any route awarded as a unit distance, so it is therefore possible to construct routes whose metric is an arbitrary rational number. This is the essence of this ancient Greek arithmetic.

In other constructions, routes that have no rational number as a measure result. To have ( about the diagonal of the square on the unit path or its sections in the division for the golden section. ) This proved a testament to the incredibly high level of Greek mathematics even at the time of the Pythagoreans. Thus, the introduction of irrational numbers is required. 2,000 years later Hilbert's completeness axiom ensures that all real numbers can occur as measures of distances.

The definition of measures of angle is similar. The definition of a " radian " is omitted, since the full angle (or right angle) an objective measure exists. On the other hand, the division of the angle into equal parts is more problematic; nowhere to every rational angle measure can construct an angle. Even the trisection of an angle fails in general.

The thus introduced metric is equivalent to the Euclidean norm induced by the Euclidean metric of " analytical " or. For the given by their coordinate points and is therefore.

Measures of angles can be defined in analytic geometry over the scalar product of vectors.

Generalization to higher dimensions

As analytic geometry Euclidean geometry can be readily for any (even infinite) number of generalized dimensions.

The lines and planes then enter higher-dimensional linear point sets, which are called hyperplanes. ( In a narrower sense, a hyperplane of an -dimensional space is a possible " large ", ie -dimensional subspace. )

The number of dimensions is not restricted and it need not be finite. For each cardinal number you can define a Euclidean space of dimension.

Rooms with more than three dimensions are basically inaccessible to our imagination. They were not designed with the aim to represent human experience of space. Similar to the non-Euclidean geometries but were also found here references to theoretical physics: The space-time of special relativity can be represented as a four-dimensional space. In modern cosmology, there are explanations with considerably more dimensions.

Related areas

When you remove the 3rd and 4th Euclidean postulate (ie, on the terms " circuit " and " Right angle " ) or we restrict ourselves, for a more precise definition, on Hilbert's axioms of logic and the parallels, we obtain an affine geometry. It was developed by Leonhard Euler first time. The terms " distance" and " square " does not occur here, but probably track conditions and parallelism.

If one replaces the parallel axiom by fixing that two located in a plane straight line should always have a point of intersection, the result of the affine projective geometry.

If command and continuity axioms omitted, affine and projective geometries also a finite number of points can be made.

In the synthetic geometry of the notion of a Euclidean plane is generalized to exactly the layers whose affine coordinates lie in a Euclidean body are Euclidean planes.