Chapter 6 Alternative Concepts for Parallelism

# Chapter 6 Alternative Concepts for Parallelism - Chapter 6...

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Chapter 6 Alternative Concepts for Parallelism: Non- Euclidean Geometry 6.1 Historical Background of Non-Euclidean Geometry 6.2 An Improbable Logical Case 6.3 Hyperbolic Geometry: Angle Sum Theorem 6.3 ½ Hyperbolic Geometry Enrichment and Workbook Homework 7 1

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6.1 Historical Background of Non-Euclidean Geometry 6.2 An Improbable Logical Case About 575 B.C. Pythagoras wrote his book on Geometry. Some of the material was known in other cultures centuries before he wrote it down, of course. His was the first axiomatic approach to organizing the material. Interestingly, he did as much work as he could before introducing the Parallel Postulate. Many people have interpreted this progression in his work as indicating a level of discomfort with the Parallel Postulate. It’s not really possible to know what he was really thinking. About 400 years after the birth of Christ, Proclus, a Greek philosopher and head of Plato’s Academy, wrote a “proof” that derived the Parallel Postulate from the first 4 Postulates. thereby setting the tone for research in Geometry for the next 1400 years. Johann Gauss, the great German mathematician, actually realized that there was another choice of axiom but didn’t choose to publish his work for fear of getting into the same trouble as other scholars had with the Catholic Church. Around 1830, two young mathematicians published works on Hyperbolic Geometry – independently of one another. The world took no note of them. In 1868, the Italian Beltrami found the first model of Hyperbolic Geometry and in 1882, Henri Poincare developed the model we’ll study. 120 years later, Hyperbolic Geometry is finally making it into high school textbooks. My favorite is a text that St. Pius X used in the 90’s. If you ever get a chance to look at it – it’s just terrific. And it includes a section on Spherical Geometry as well: Geometry, second edition by Harold Jacobs. ISBN: 0-7167-1745-X (copyright 1987). Note that St. Pius isn’t a flagship diocese school – they actually have a very full section of “Algebra half”, the course for the kids not ready for Algebra I. In the 1400 years of work on the axioms of Absolute Geometry there was always a special group of people who endeavored to prove that the Parallel Postulate was actually a theorem. In fact, Saccheri and Lambert who both came so close to realizing that there was an alternate geometry out there waiting to be discovered never got all the way past believing that the axiom was a theorem. On page 427 is a list of statements that are equivalent to P1, the Euclidean Parallel Postulate. Here they are: The area of a triangle can be made arbitrarily large. The angle sum of all triangles is a constant. The angle sum of any triangle is 180. Rectangles exist. A circle can be passed through any 3 noncollinear points.
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## This note was uploaded on 02/22/2012 for the course MATH 5397 taught by Professor Staff during the Spring '08 term at University of Houston.

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Chapter 6 Alternative Concepts for Parallelism - Chapter 6...

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