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Exploring Non-Euclidean Mathematics: Riemann vs. Euclid

Discover the contrast between Riemann's non-Euclidean geometry and Euclid's classical axioms in this engaging presentation. Learn about the mathematical systems, including Godel's Incompleteness Theorem, that challenge traditional views. Dive into the world of curved spaces and endless lines to unlock the mysteries of geometry. Explore the fascinating history of mathematics and its evolution through contradictory theories. Understand why despite contradictions, Euclidean mathematics remains a foundational tool in our world. Unravel the secrets of mathematical thinking and the enduring power of Euclidian principles.

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Exploring Non-Euclidean Mathematics: Riemann vs. Euclid

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  1. Mathematics Non-euclidian Maths Powerpoint Templates

  2. Once upon a time Euclid had • 5 axioms • It shall be possible to draw a straight line joining any 2 points. • A finite straight line may be extended without limit in either direction. • It shall be possible to draw a circle with a given centre and through a given point. • All right angles are equal to one another. • There is just one straight line through a given point which is parrallel to a given line.

  3. But then along came Riemann and his Contraries • Two points may determine more than one line (axiom 1) • All lines are finite in length but endless • (axiom 2) • There are no parallel lines • (axiom 5) a. All perpendiculars to a straight line meet at one point. b. Two straight lines enclose an area. c. The sum of the angles of any triangle is greater than 180 degrees.

  4. Georg Friedrich Bernard Riemann (1822-66)

  5. But then along came Riemann and his Contraries

  6. Albert agrees…space is curved

  7. Riemann • Two points may determine more than one line (axiom 1)

  8. Riemann • All lines are finite in length but enless (axiom 2)

  9. Riemann • There are no parallel lines • (axiom 5)

  10. M.C. Escher

  11. M.C. Escher

  12. Riemann vs Euclid Neither have even been proven wrong, and yet they contradict each other??? Systems work within their own axiom sets Godel (1906-78), Austrian

  13. Kurt Godel (1906-78) – Incompleteness Theorm It is impossible to prove that any mathematical system is free from contradiction He didn’t prove that Maths has contraditions, just that it’s impossible to prove it doesn’t

  14. Kurt Godel – Incompleteness Theorm However, Maths hasn’t had any contraditions since it’s formalisation 2500 yrs ago – so most mathematicians mostly ignore Godel’s theorm

  15. So if Riemann, Godel & Einstien have contradicted Euclidian maths, why do we still use it? Because it works! Why does it work?

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