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Modern geometry


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Modern geometry

  1. 1. Lourise Archie Subang
  2. 2.  Hyperbolic geometry was created in the first half of the nineteenth century in the midst of attempts to understand Euclids axiomatic basis for geometry. It is one type of non-Euclidean geometry, that is, a geometry that discards one of Euclids axioms. Hyperbolic Geometry
  3. 3. Euclid’s famous Five 1. Each pair of points can be joined by one and only one straight line segment. 2. Any straight line segment can be indefinitely extended in either direction. 3. There is exactly one circle of any given radius with any given center. Hyperbolic Geometry
  4. 4. Euclid’s famous Five 4. All right angles are congruent to one another. 5. If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side on which the angles are less than two right angles. Hyperbolic Geometry
  5. 5. For two thousand yearsmathematicians attempted to deducethe fifth postulate from the foursimpler postulates. In each case onereduced the proof of the fifthpostulate to the conjunction of thefirst four postulates with anadditional natural postulate that, infact, proved to be equivalent to thefifth. Hyperbolic Geometry
  6. 6.  Proclus (ca. 400 a.d.) used as additional postulate the assumption that the points at constant distance from a given line on one side form a straight line. The Englishman John Wallis (1616- 1703) used the assumption that to every triangle there is a similar triangle of each given size. Hyperbolic Geometry
  7. 7.  The Italian Girolamo Saccheri (1667- 1733) considered quadrilaterals with two base angles equal to a right angle and with vertical sides having equal length and deduced consequences from the (non- Euclidean) possibility that the remaining two angles were not right angles. Hyperbolic Geometry
  8. 8.  Johann Heinrich Lambert (1728{1777) proceeded in a similar fashion and wrote an extensive work on the subject, posthumously published in 1786. G¨ottingen mathematician K¨astner (1719-1800) directed a thesis of student Kl¨ugel (1739-1812), which considered approximately thirty proof attempts for the parallel postulate.LINK Hyperbolic Geometry
  9. 9. The resulting postulate “Given a line and a point not on it, there is more than one line going through the given point that is parallel to the given line.” This is to hyperbolic as to fifth postulate to Euclidian Geometry. Hyperbolic Geometry
  10. 10. The consequences  Equidistant curves on either side of a straight line were in fact not straight but curved;  Similar triangles were congruent;  Angle sums in a triangle were less than 180, or sometimes all three angles have 0 measure. Hyperbolic Geometry
  11. 11. The Five Big Names  The amateurs were jurist Schweikart and his nephew Taurinus (1794-1874). By 1816 Schweikart had developed, in his spare time, an “astral geometry" that was independent of the fifth postulate.  The professionals were Carl Friedrich Gauss (1777-1855), Nikola Ivanovich Lobachevski (1793-1856), and Janos (or Johann) Bolyai (1802-1860).LINK Hyperbolic Geometry
  12. 12.  Gauss, the Bolyais, and Lobachevski developed non-Euclidean geometry axiomatically on a synthetic basis. Lobachevski developed a non- Euclidean trigonometry that paralleled the trigonometric formulas of Euclidean geometry. Leonhard Euler, Gaspard Monge, and Gauss in their studies of curved surfaces later laid the analytic study of hyperbolic non-Euclidean geometry. Hyperbolic Geometry
  13. 13. Why Hyperbolic?  Thenon-Euclidean geometry of Gauss, Lobachevski, and Bolyai is usually called hyperbolic geometry because of one of its very natural analytic models. LINK Hyperbolic Geometry
  14. 14. Models in H.G.  There are four models commonly used for hyperbolic geometry: a. the Klein model, b. the Poincaré disc model, c. the Poincaré half-plane model, d. and the Lorentz model, or hyperboloid model. LINK Hyperbolic Geometry
  15. 15.  The Klein model, also known as the projective disc model and Beltrami-Klein model, uses the interior of a circle for the hyperbolic plane, and chords of the circle as lines.  This model has the advantage of simplicity, but the disadvantage that angles in the hyperbolic plane are distorted.  The distance in this model is the cross-ratio, which was introduced by Arthur Cayley in projective geometry.LINK Hyperbolic Geometry
  16. 16.  The Poincaré disc model, also known as the conformal disc model, also employs the interior of a circle, but lines are represented by arcs of circles that are orthogonal to the boundary circle, plus diameters of the boundary circle.LINK Hyperbolic Geometry
  17. 17.  The Poincaré half-plane model takes one-half of the Euclidean plane, as determined by a Euclidean line B, to be the hyperbolic plane (B itself is not included). The Lorentz model or hyperboloid model employs a 2- dimensional hyperboloid of revolution (of two sheets, but using one) embedded in 3-dimensional Minkowski space.LINK Hyperbolic Geometry
  18. 18. Other Topics  Generalizing to Higher Dimensions  Stereographic Projection  Geodesics  Isometries and Distances  The Space at Infinity  Geometric Classifications of Isometries LINK Hyperbolic Geometry
  19. 19. Curious Facts aboutHyperbolic Space Fact1. In the three conformal models for hyperbolic space, hyperbolic spheres are also Euclidean spheres; however, Euclidean and hyperbolic sphere centers need not coincide. Hyperbolic Geometry
  20. 20. Curious Facts aboutHyperbolic Space Fact 2. In the hyperbolic plane, the two curves at distance r on either side of a straight line are not straight. Hyperbolic Geometry
  21. 21. Curious Facts aboutHyperbolic Space Fact 3. Triangles in hyperbolic space have angle sum less than ; in fact , the area of a triangle with angles , , and is − − − (the Gauss{Bonnet theorem). Given three angles , , and whose sum is less than , there is one and only one triangle up to congruence having those angles. Consequently, there are no nontrivial similarities of hyperbolic space. Hyperbolic Geometry
  22. 22. Curious Facts aboutHyperbolic Space Fact 4. If PQR is a triangle in hyperbolic space, and if x is a point of the side PQ, then there is a point y an element of PR in union of QR such that the hyperbolic distance d(x; y) is less than ln (1 +√2); that is, triangles in hyperbolic space are uniformly thin. Hyperbolic Geometry
  23. 23. Curious Facts about Hyperbolic Space Fact 5. For a circular disk in the hyperbolic plane, the ratio of area to circumference is less than 1 and approaches 1 as the radius approaches infinity. That is, almost the entire area of the disk lies very close to the circular edge of the disk. Both area and circumference are exponential functions of hyperbolic radius. Hyperbolic Geometry
  24. 24. Curious Facts about Hyperbolic Space Fact6. In the half-space model H of hyperbolic space, if S is a sphere centered at a point at infinity x 2 @H, then inversion in the sphere S induces a hyperbolic isometry of H that interchanges the inside and outside of S in H. Hyperbolic Geometry
  25. 25. Visualizing Hyperbolic Geometry 1997: Daina Taimina, Cornell
  26. 26. Visualizing Hyperbolic Geometry Hyperbolic Geometry
  27. 27. HyperbolicGeometry