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Euclidean geometry

2007 Schools Wikipedia Selection. Related subjects: Mathematics

   Euclid
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   Euclid

   Euclidean geometry is a mathematical well-known system attributed to
   the Greek mathematician Euclid of Alexandria. Euclid's text Elements
   was the first systematic discussion of geometry. It has been one of the
   most influential books in history, as much for its method as for its
   mathematical content. The method consists of assuming a small set of
   intuitively appealing axioms, and then proving many other propositions
   ( theorems) from those axioms. Although many of Euclid's results had
   been stated by earlier Greek mathematicians, Euclid was the first to
   show how these propositions could be fitted together into a
   comprehensive deductive and logical system.

   The Elements begin with plane geometry, still often taught in secondary
   school as the first axiomatic system and the first examples of formal
   proof. The Elements goes on to the solid geometry of three dimensions,
   and Euclidean geometry was subsequently extended to any finite number
   of dimensions. Much of the Elements states results of what is now
   called number theory, proved using geometrical methods.

   For over two thousand years, the adjective "Euclidean" was unnecessary
   because no other sort of geometry had been conceived. Euclid's axioms
   seemed so intuitively obvious that any theorem proved from them was
   deemed true in an absolute sense. Many other consistent formal
   geometries are now known, the first ones being discovered in the early
   19th century. It also is no longer taken for granted that Euclidean
   geometry describes physical space. An implication of Einstein's theory
   of general relativity is that Euclidean geometry is only a good
   approximation to the properties of physical space if the gravitational
   field is not too strong.

Axiomatic approach

   Following a precedent set in the Elements, Euclidean geometry has been
   exposited as an axiomatic system, in which all theorems ("true
   statements") are derived from a finite number of axioms. Near the
   beginning of the first book of the Elements, Euclid gives five
   postulates (axioms):
    1. Any two points can be joined by a straight line.
    2. Any straight line segment can be extended indefinitely in a
       straight line.
    3. Given any straight line segment, a circle can be drawn having the
       segment as radius and one endpoint as centre.
    4. All right angles are congruent.
    5. Parallel postulate. If two lines 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.

   These axioms invoke the following concepts: point, straight line
   segment and line, side of a line, circle with radius and centre, right
   angle, congruence, inner and right angles, sum. The following verbs
   appear: join, extend, draw, intersect. The circle described in
   postulate 3 is tacitly unique. Postulates 3 and 5 hold only for plane
   geometry; in three dimensions, postulate 3 defines a sphere.
   A proof from Euclid's elements that, given a line segment, an
   equilateral triangle exists that includes the segment as one of its
   sides. The proof is by construction: an equilateral triangle ΑΒΓ is
   made by drawing circles Δ and Ε centered on the points Α and Β, and
   taking one intersection of the circles as the third vertex of the
   triangle.
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   A proof from Euclid's elements that, given a line segment, an
   equilateral triangle exists that includes the segment as one of its
   sides. The proof is by construction: an equilateral triangle ΑΒΓ is
   made by drawing circles Δ and Ε centered on the points Α and Β, and
   taking one intersection of the circles as the third vertex of the
   triangle.

   Postulate 5 leads to the same geometry as the following statement,
   known as Playfair's axiom, which also holds only in the plane:

          Through a point not on a given straight line, one and only one
          line can be drawn that never meets the given line.

   Postulates 1, 2, 3, and 5 assert the existence and uniqueness of
   certain geometric figures, and these assertions are of a constructive
   nature: that is, we are not only told that certain things exist, but
   are also given methods for creating them with no more than a compass
   and an unmarked straightedge. In this sense, Euclidean geometry is more
   concrete than many modern axiomatic systems such as set theory, which
   often assert the existence of objects without saying how to construct
   them, or even assert the existence of objects that probably cannot be
   constructed within the theory.

   The Elements also include the following five "common notions":
    1. Things that equal the same thing also equal one another.
    2. If equals are added to equals, then the wholes are equal.
    3. If equals are subtracted from equals, then the remainders are
       equal.
    4. Things that coincide with one another equal one another.
    5. The whole is greater than the part.

   Euclid also invoked other properties pertaining to magnitudes. 1 is the
   only part of the underlying logic that Euclid explicitly articulated. 2
   and 3 are "arithmetical" principles; note that the meanings of "add"
   and "subtract" in this purely geometric context are taken as given. 1
   through 4 operationally define equality, which can also be taken as
   part of the underlying logic or as an equivalence relation requiring,
   like "coincide," careful prior definition. 5 is a principle of
   mereology. "Whole," "part," and "remainder" beg for precise
   definitions.

   In the 19th century, it was realized that Euclid's ten axioms and
   common notions do not suffice to prove all of theorems stated in the
   Elements. For example, Euclid assumed implicitly that any line contains
   at least two points, but this assumption cannot be proved from the
   other axioms, and therefore needs to be an axiom itself. The very first
   geometric proof in the Elements, shown in the figure on the right, is
   that any line segment is part of a triangle; Euclid constructs this in
   the usual way, by drawing circles around both endpoints and taking
   their intersection as the third vertex. His axioms, however, do not
   guarantee that the circles actually intersect, because they are
   consistent with discrete, rather than continuous, space. Starting with
   Moritz Pasch in 1882, many improved axiom systems for geometry have
   been proposed, the best known being those of Hilbert, George Birkhoff,
   and Tarski.

   To be fair to Euclid, the first formal logic capable of supporting his
   geometry was that of Frege's 1879 Begriffsschrift, little read until
   the 1950s. We now see that Euclidean geometry should be embedded in
   first-order logic with identity, a formal system first set out in
   Hilbert and Wilhelm Ackermann's 1928 Principles of Theoretical Logic.
   Formal mereology began only in 1916, with the work of Lesniewski and A.
   N. Whitehead. Tarski and his students did major work on the foundations
   of elementary geometry as recently as between 1959 and his 1983 death.

The parallel postulate

   To the ancients, the parallel postulate seemed less obvious than the
   others; verifying it physically would require us to inspect two lines
   to check that they never intersected, even at some very distant point,
   and this inspection could potentially take an infinite amount of time.
   Euclid himself seems to have considered it as being qualitatively
   different from the others, as evidenced by the organization of the
   Elements: the first 28 propositions he presents are those that can be
   proved without it.

   Many geometers tried in vain to prove the fifth postulate from the
   first four. By 1763 at least 28 different proofs had been published,
   but all were found to be incorrect. In fact the parallel postulate
   cannot be proved from the other four: this was shown in the 19th
   century by the construction of alternative ( non-Euclidean) systems of
   geometry where the other axioms are still true but the parallel
   postulate is replaced by a conflicting axiom. One distinguishing aspect
   of these systems is that the three angles of a triangle do not add to
   180°: in hyperbolic geometry the sum of the three angles is always less
   than 180° and can approach zero, while in elliptic geometry it is
   greater than 180°. If the parallel postulate is dropped from the list
   of axioms without replacement, the result is the more general geometry
   called absolute geometry.

Treatment using analytic geometry

   The development of analytic geometry provided an alternative method for
   formalizing geometry. In this approach, a point is represented by its
   Cartesian (x,y) coordinates, a line is represented by its equation, and
   so on. In the 20th century, this fit into David Hilbert's program of
   reducing all of mathematics to arithmetic, and then proving the
   consistency of arithmetic using finitistic reasoning. In Euclid's
   original approach, the Pythagorean theorem follows from Euclid's
   axioms. In the Cartesian approach, the axioms are the axioms of
   algebra, and the equation expressing the Pythagorean theorem is then a
   definition of one of the terms in Euclid's axioms, which are now
   considered to be theorems. The equation

          |PQ|=\sqrt{(p-r)^2+(q-s)^2}

   defining the distance between two points P = (p,q) and Q = (r,s) is
   then known as the Euclidean metric, and other metrics define
   non-Euclidean geometries.
   A disproof of Euclidean geometry as a description of physical space. In
   a 1919 test of the general theory of relativity, stars (marked with
   lines) were photographed during a solar eclipse. The rays of starlight
   were bent by the Sun's gravity on their way to the earth. This is
   interpreted as evidence in favor of Einstein's prediction that gravity
   would cause deviations from Euclidean geometry.
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   A disproof of Euclidean geometry as a description of physical space. In
   a 1919 test of the general theory of relativity, stars (marked with
   lines) were photographed during a solar eclipse. The rays of starlight
   were bent by the Sun's gravity on their way to the earth. This is
   interpreted as evidence in favour of Einstein's prediction that gravity
   would cause deviations from Euclidean geometry.

As a description of physical reality

   Euclid believed that his axioms were self-evident statements about
   physical reality. However, Einstein's theory of general relativity
   shows that the true geometry of spacetime is non-Euclidean. For
   example, if a triangle is constructed out of three rays of light, then
   in general the interior angles do not add up to 180 degrees due to
   gravity. A relatively weak gravitational field, such as the Earth's or
   the sun's, is represented by a metric that is approximately, but not
   exactly, Euclidean. Until the 20th century, there was no technology
   capable of detecting the deviations from Euclidean geometry, but
   Einstein predicted that such deviations would exist. They were later
   verified by observations such as the observation of the slight bending
   of starlight by the Sun during a solar eclipse in 1919, and
   non-Euclidean geometry is now, for example, an integral part of the
   software that runs the GPS system. It is possible to object to the
   non-Euclidean interpretation of general relativity on the grounds that
   light rays might be improper physical models of Euclid's lines, or that
   relativity could be rephrased so as to avoid the geometrical
   interpretations. However, one of the consequences of Einstein's theory
   is that there is no possible physical test that can do any better than
   a beam of light as a model of geometry. Thus, the only logical
   possibilities are to accept non-Euclidean geometry as physically real,
   or to reject the entire notion of physical tests of the axioms of
   geometry, which can then be imagined as a formal system without any
   intrinsic real-world meaning.

Logical status

   Euclidean geometry is a first-order theory. That is, it allows
   statements that begin as "for all triangles ...," but it is incapable
   of forming statements such as "for all sets of triangles ..."
   Statements of the latter type are deemed to be outside the scope of the
   theory.

   We owe much of our present understanding of the properties of the
   logical and metamathematical properties of Euclidean geometry to the
   work of Alfred Tarski and his students, beginning in the 1920s. Tarski
   used his axiomatic formulation of Euclidean geometry to prove it
   consistent, and also complete in a certain sense: every proposition of
   Euclidean geometry can be shown to be either true or false. Gödel's
   theorem showed the futility of Hilbert's program of proving the
   consistency of all of mathematics using finitistic reasoning. Tarski's
   findings do not violate Gödel's theorem, because Euclidean geometry
   cannot describe a sufficient amount of arithmetic for the theorem to
   apply. Although Hilbert thought Euclidean geometry could be put on a
   firmer foundation by rewriting it in terms of arithmetic, in fact
   Euclidean geometry is complete and consistent in a way that Godel's
   theorem tells us arithmetic can never be.

   Although complete in the formal sense used in modern logic, there are
   things that Euclidean geometry cannot accomplish. For example, the
   problem of trisecting an angle with a compass and straightedge is one
   that naturally occurs within the theory, since the axioms refer to
   constructive operations that can be carried out with those tools.
   However, centuries of efforts failed to find a solution to this
   problem, until Pierre Wantzel published a proof in 1837 that such a
   construction was impossible.

   Absolute geometry, formed by removing the parallel postulate, is also a
   consistent theory, as is non-Euclidean geometry, formed by alterations
   of the parallel postulate. Non-Euclidean geometries are consistent
   because there are Euclidean models of non-Euclidean geometry. For
   example, geometry on the surface of a sphere is a model of an
   elliptical geometry, carried out within a self-contained subset of a
   three-dimensional Euclidean space.

Classical theorems

     * Ceva's theorem
     * Heron's formula
     * Nine-point circle
     * Pythagorean theorem
     * Tartaglia's formula

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