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Hyperbolic Geometry

Hyperbolic Geometry. Introduction, Propositions (Theorems). Hyperbolic Geometry. Hyperbolic Geometry is represented by the first four Euclidean Postulates of Geometry plus a Hyperbolic Fifth Postulate. Hyperbolic Geometry.

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Hyperbolic Geometry

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  1. Hyperbolic Geometry Introduction, Propositions (Theorems)

  2. Hyperbolic Geometry • Hyperbolic Geometry is represented by the first four Euclidean Postulates of Geometry plus a Hyperbolic Fifth Postulate.

  3. Hyperbolic Geometry • Previous work was done to prove Hyperbolic Geometry is just as consistent as Euclidean Geometry. (In math, consistency is enough.)

  4. Hyperbolic Geometry • Non-Euclidean Geometry publishing history • Gauss: Developed fundamental theorems, but told his friends to keep it quiet. Never published. • Bolyai: Published the work, as an appendix, The Science Of Absolute Space (1832) and absolutely nothing else. • Lobachevsky: Lectured about it, then published On The Principles of Geometry (1829) and other items. • These people had contact with each other, and were thinking about similar things at the time. Do mathematicians really “own” what they discover?

  5. Hyperbolic Geometry • The first 28 Propositions don’t use 5th Postulate, and are valid in Hyperbolic Geometry. • Now we look at Hyperbolic’d propositions which don’t deal with parallel lines. Recall: Saccheri quadri. with 90o base angles, two equal sides. • Theorem H29 The summit angles of a Saccheri quadrilateral are equal.

  6. Hyperbolic Geometry • Theorem H30 The line joining midpoints of base and summit are perpendicular to both. • Theorem H31 The angle-sum of a triangle does not exceed two right angles.

  7. Hyperbolic Geometry • Theorem H32 The summit angles of a Saccheri quadrilateral are not obtuse. • Theorem H33 Given a quadrilateral with a base. If there are two un-equal arms, then the summit angles are also un-equal.Conversely: Greater summit angle is on the opposite of the greater arm.

  8. Hyperbolic Geometry • Hyperbolic Parallel Postulate (HP5) The summit angles of a Saccheri quadrilateral are acute. • Some consequences: • Parallel lines may not be equidistant. • More than one line can pass through a point and be parallel to a particular line. • Similar triangles are also congruent.

  9. Hyperbolic Geometry • Theorem H34 In a Saccheri quadrilateral: • The summit is longer than the base. • The segment joining their midpoints is shorter than either arms.

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