The Nature and Interrelationship of Various Geometries

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The Nature and Inter-relationship of Various
Geometries
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Geometry

• The geometry taught today in school is a confused
  mixture of Euclid's and Descartes'. But the
  teacher is not to blame.

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Geometry

• Having evolved from antiquity from often-used
  methods for measurement of figures drawn mainly
  on plane surfaces, the methods and principles
  became distilled, for ready reference and use, as
  mathematical propositions, particularly in Greece
  in the peace and prosperity of the few centuries
  following the rule of Pericles. Then Euclid, in
  around 300 B.C., gathered, improved, and
  systematically wrote down all that was known in
  geometry to his day. The work - called The
  Elements - attempted to develop geometry from the
  firm foundation of axioms and succeeded in great
  measure to provide rigorous demonstrations -
  proof - of the mathematical results loosely
  proved by his predecessors.

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Euclidean Geometry

• While most of the proofs in Euclid's work were
  correct, blemishes were discovered in some proofs
  one of which being the very first proposition in
  the Elements. However, these blemishes were not
  due to erroneous deduction but tacit assumptions
  or "intuitively obvious" facts that were not
  justified by the axioms.

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Euclidean Geometry

• Considerations of these blemishes culminated in
  1899 with David Hilbert's proposal of a revised
  axiom system that would not only preserve the
  validity of the proofs in the Elements but which
  was in conformity with the modern notion of the
  axiomatic method as proceeding from a set of
  undefined terms, definitions, and the axiom
  statements on the undefined terms. The Euclidean
  Geometry of today is the geometry based on this
  revised axiom system or other equivalent systems
  since proposed.

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Euclids Axiomatics

• Undefined terms point, line, plane
• Common Notions
• Things equal to the same thing are equal.
• If equals are added to equals, the results are
  equal.
• If equals are subtracted from equals, the results
  are equal.
• Things that coincide with one another are equal
  to one another.
• The whole is greater than the part.

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Euclids Axiomatics

• Undefined terms point, line, plane
• Postulates
• Exactly one straight can be drawn from any point
  to any other point.
• A finite straight line can be extended
  continuously in a straight line.
• A circle can be formed with any center and
  distance (radius).
• All right angles are equal to one another.
• If a straight line falling on two straight lines
  makes the sum of the interior angles on the same
  side less than two right angles, then the two
  straight lines, if extended indefinitely, meet on
  that side on which the angle sum is less than two
  right angles.

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Euclidean Geometry

• Although Euclid's axioms needed to be revised to
  make all his proofs correct, Euclid's Geometry
  remained for two millennia as the example and
  model of the deductive method of mathematics. But
  the beauty of Euclid's geometry is that it does
  not make use of numbers to measure lengths or
  angles or areas or volumes. Instead, it deals
  with points, lines, triangles, circles, and the
  relationships among these.

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Analytic Geometry

• Even as the discussion on the Parallel Postulate
  of Euclid was raging on, Rene Descartes, in the
  year 1637, discovered that the ruler and compass
  constructions of Euclidean Geometry correspond to
  the solution of linear and quadratic equations in
  algebra on a Real Cartesian Plane constructed as
  follows. A point is an ordered pair of real
  numbers. The set of all such pairs is the
  Cartesian Plane. The set of points (a,0) is
  called the x-axis, and the set of points (0,b) is
  called the y-axis. The intersection of these axes
  is called the origin. A line in this plane is the
  subset defined by an equation of the form ax by
  c 0, with a and b not both zero.

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Analytic Geometry

• Descartes gave four (unmarked) ruler and compass
  constructions using line segments a and b to
  construct ab, ab, a/b and va, and showed that
  all construction problems of Euclidean Geometry
  can be done in his Cartesian Plane. Since this
  geometry in the Real Cartesian Plane satisfies
  all the axioms of Hilbert, it follows that the
  Cartesian Plane is Euclidean geometry using
  numbers.

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Analytic Geometry

• The method of solution used in finding points
  from given points in the Real Cartesian Plane is
  called Analytic Geometry. But Analytic Geometry
  is not a Geometry in the sense of a theory like
  Euclidean Geometry that is derived from an axiom
  set. Rather, Analytic Geometry is a method of
  doing geometry problems using algebra.

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Neutral Geometry

• Euclid's axioms without the fifth - which is the
  parallel axiom often referred to simply as the
  Fifth - or Hilbert's axioms without Euclid's
  Fifth is today called Neutral Geometry. In this
  geometry, the most important theorem is the
  Exterior Angle Theorem that the measure of the
  exterior angle of a triangle is greater than
  either of the non-adjacent interior angles.
  Compare this to the case in Euclidean Geometry
  which is Neutral Geometry Euclid's Fifth axiom.
  In Euclidean Geometry, the measure of the
  exterior angle is equal to the sum of the
  measures of the nonadjacent interior angles.

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Neutral Geometry

• The weak-looking Exterior Angle Theorem of
  Neutral Geometry implies the Alternate Interior
  Angle Theorem. That is, if two lines are
  intersected by a transversal such that a pair of
  alternate interior angles formed are congruent,
  then the lines are parallel. This theorem has the
  immediate corollary that two lines perpendicular
  to the same line are parallel. This corollary
  gives another corollary which can be said to be
  the culminating result of Neutral
  GeometryThere is at least one line parallel to
  a given line through a point not on that line.

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Neutral Geometry

• An important result of Neutral Geometry is also
  that the angle sum in a triangle is at most two
  right angles. The corresponding situation in
  Euclidean Geometry is that the angle sum in a
  triangle is two right angles.

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Hyperbolic Geometry

• In the two millennia since Euclid the greatest
  ongoing mathematical discussion was not on the
  blemishes in some of Euclid's proofs, but on the
  independence of the Fifth Axiom of Euclid which
  meant that given a line and a point not on it,
  exactly one line can be drawn through the point
  and parallel to the line. Gauss, Bolyai,and
  Lobachevsky separately and independently(?)
  discovered the new geometry - Hyperbolic Geometry
  - that assumed the negation of the Fifth Axiom.
  This established the independence of the Fifth
  axiom of Euclidean Geometry and laid to rest the
  longest discussion in the history of mathematics
  to that date.

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Elliptic Geometry

• A non-Euclidean Geometry is a geometry that has
  the negation of the Parallel Postulate as one of
  its axioms. An example is hyperbolic geometry. It
  turns out that hyperbolic geometry allows more
  than one parallel line through a point off a
  given line. Elliptic Geometry allows no parallel
  line to a given line.

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Elliptic Geometry

• The name "elliptic" is misleading, for it has no
  direct connection with the curve called ellipse.
  The name is used as an analogy of the following
  result. A central conic is called an ellipse or
  hyperbola according as it has no asymptote or two
  asymptotes. Analogously, a non-Euclidean plane is
  said to be elliptic or hyperbolic according as
  each of its lines contains no point at infinity
  or two points at infinity. A model of Elliptic
  Geometry is the surface of a sphere with great
  circles as lines and antipodal points identified.

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Elliptic Geometry

• Note that, if Elliptic Geometry allows no
  parallel lines, it can not have Neutral Geometry
  inside it as is the case with hyperbolic
  geometry. For in Neutral Geometry, we have the
  Alternate Interior Angle Theorem which implies
  that there is at least one parallel line through
  a point off a given line. If we can not have
  Neutral Geometry as part of Elliptic Geometry, we
  ask what axiom of Neutral Geometry is requiring
  the existence of parallel lines.

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Elliptic Geometry

• Since the Alternate Interior Angle Theorem of
  Neutral Geometry makes parallel lines necessary,
  we investigate its proof. Its proof depends on
  the Exterior Angle Theorem. So we ask "what does
  the proof of the Exterior Angle Theorem depend
  on?" It turns out it relies on three major
  components triangle congruence, angle addition,
  and plane separation. Since triangle congruence
  and angle addition are at the core of geometry,
  we do not seek to modify these. The remaining
  candidate for removal is plane separation. So we
  remove those axioms that imply plane separation.
  That would result in a geometry where we are able
  to pass from one side of a line to the other
  without crossing the line. And that is the
  geometry on the surface of the sphere with great
  circles as lines. Since the model identifies
  antipodal points on great circles, it is called a
  single elliptic geometry.

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Affine Geometry

• Affine Geometry takes for its axioms only
  Euclid's axioms I and II. This Geometry is
  motivated by the fact that its results hold not
  only in Euclidean Geometry but also in Hyperbolic
  Geometry and the Minkowskian Geometry of time and
  space. The propositions that hold in Affine
  Geometry are those that are preserved by parallel
  projection from one plane to another - for
  example the first 28 propositions of Euclid, then
  29 and 33-45, and some others in books
  III(1-19,25,28-30), IV(4-9), and
  VI(1,2,4,9,10,24-26)

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Projective Geometry

• This is the parent of all infinite geometries in
  that one can get all those geometries by
  appropriate restrictions and modifications of
  projective geometry. The projective geometry in
  the plane begins with the following four axioms
• 1. There exists at least one line.2. Every line
  contains at least three points. 3. Any two
  distinct points lie on a unique line.4. Any two
  lines meet in at least one point.5. There exist
  three noncollinear points.

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Projective Geometry

• The most elegant property of projective geometry
  is the principle of duality, which means that
  every definition remains relevant and every
  theorem remains valid when we consistently
  interchange the words point and line (and
  simultaneously interchange lie on and pass
  through, join and intersection, collinear and
  concurrent, etc.).

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Projective Geometry

• To establish the duality it suffices to verify
  that the axioms imply their own duals. As a
  result, given a theorem and its proof, we
  immediately can assert the dual theorem. It is a
  useful exercise to verify that the dual of an
  axiom is itself either an axiom or a theorem
  derivable from the remaining axioms.

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Finite Geometries

• These are systems comprising a finite number of
  points and satisfying some of the axioms of
  Euclidean Geometry. Examples are Fano Geometry, ?
  .

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Geometries Defined Over Fields

• The Real Cartesian Plane satisfies all the axioms
  of Euclidean Geometry. The word "Real" is
  important here. The points in the Cartesian Plane
  we discussed were ordered pairs of real numbers.
  The set of Real Numbers is an example of an
  algebraic structure called a field. So we could
  ask if we can use any field in place of the Real
  Numbers. It turns out, depending on the field,
  only some of the Hilbert axioms may hold in the
  associated Cartesian Plane. However, all of these
  various Cartesian Planes provide various
  geometries.

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Geometries Defined Over Fields

• So what are the fields whose associated Cartesian
  Planes give exactly Euclidean Geometries? It
  turns out that the smallest such field is
  obtained from the set of rational numbers by
  adjoining square roots as well as the addition,
  multiplication and square roots of the rationals.
  The resulting field is called a Hilbert's Field
  and the Cartesian Plane associcated with this
  field is called a Hilbert Plane.

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Relationship of the Various Geometries
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The Erlangen Program
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Felix Christian Klein(1849 1925)
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The problems of nineteenth century geometry

• Is there one 'geometry' or many? Since Euclid,
  geometry had meant the geometry of Euclidean
  space of two dimensions (plane geometry) or of
  three dimensions (solid geometry). In the first
  half of the nineteenth century there had been
  several developments complicating the picture.
  Mathematical applications required geometry of
  four or more dimensions the close scrutiny of
  the foundations of the traditional Euclidean
  geometry had revealed the independence of the
  Parallel Axiom from the others, and non-Euclidean
  geometry had been born and in projective
  geometry new 'points' (points at infinity, points
  with complex number co-ordinates) had been
  introduced.

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The Erlangen Program

• An influential research programme and manifesto
  was published in 1872 by Felix Klein, under the
  title Vergleichende Betrachtungen über neuere
  geometrische Forschungen. This Erlangen Program
  (Erlanger Programm) Klein was then at Erlangen,
  Germany proposed a new kind of solution to the
  problems of geometry of the time.

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The Solution

• In abstract terms the solution was to use
  symmetry as an underlying principle, and to state
  first that different geometries could co-exist,
  because they dealt with different types of
  propositions and invariances related to different
  types of symmetry and transformation.

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The Solution

• By abstracting the underlying groups of
  symmetries from the geometries, the relationships
  between them can be re-established at the group
  level.

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Invariants by Type of Geometry

• Geometry Transformation
  Invariants
• Group
• Projective
  cross-ratio
• 
  plane at infinity
• Affine
  relative distances
• 
  along direction
• 
  parallelism
• 
  
• Metric
  relative distances
• 
  angles
• 
  absolute conic
• Euclidean
  absolute distances

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Shapes which are equivalent to a cube for the
different geometric ambiguities
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Influence on later work

• The long-term effects of the Erlangen programme
  can be seen all over pure mathematics (see tacit
  use at congruence (geometry), for example) and
  the idea of transformations and of synthesis
  using groups of symmetry is, of course, now
  standard too in physics.

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Influence on later work

• When topology is routinely described in terms of
  properties invariant under homeomorphism, one can
  see the underlying idea in operation. The groups
  involved will be infinite-dimensional in almost
  all cases - and not Lie groups - but the
  philosophy is the same. Of course this mostly
  speaks to the pedagogical influence of Klein.
  Books such as those by H.S.M. Coxeter routinely
  used the Erlangen Program approach to help
  place geometries. In pedagogic terms, the
  program became transformation geometry, a mixed
  blessing in the sense that it builds on stronger
  intuitions than the style of Euclid, but is less
  easily converted into a logical system.

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Influence on later work

• In his book Structuralism (1970) Jean Piaget
  says, "In the eyes of contemporary structuralist
  mathematicians, like Bourbaki, the Erlangen
  Program amounts to only a partial victory for
  structuralism, since they want to subordinate all
  mathematics, not just geometry, to the idea of
  structure."

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Marius Sophus Lie(1842 1899)
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