Simanaitis Says

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GEOMETRIES OTHER THAN EUCLID’S PART 1

FOR TWO THOUSAND years, geometry codified by Euclid, a Greek mathematician who flourished c. 300 B.C., gave a model of reality for the likes of Johannes Kepler, Gottfried Wilhelm Leibnitz, and Sir Isaac Newton.

However, by the mid-19th-century, mathematicians discovered geometries other than Euclid’s. And, in the early 20th century, such non-Euclidian geometries found application in science’s latest descriptions of reality. It’s quite a tale, highlighted at SimanaitisSays in Part 1 today and Part 2 tomorrow, with tidbits gleaned from The Mathematical Experience, by Philip J. Davis and Reuben Hersh, other books around here, and my usual Internet sleuthing.

The Mathematical Experience, by Philip J. Davis and Reuben Hersh, Birkhäuser, 1995, reprinted 2002.

Geometry’s Dual Nature. Since the days of ancient Egypt, geometry has had a duality: It represents the real world and also serves as an intellectual discipline. Geometry’s applications allowed redefinition of land covered by annual floods of the Nile. And its deductions following from assumed principles made geometry a logical endeavor.

Euclid of Alexandra, flourished 300 B.C., Greek mathematician, author of The Elements.

Euclid’s Postulates. Euclid identified geometric principles with a collection of postulates, things assumed, and theorems, statements logically derived. His first five postulates assume things about points, lines, circles, right angles, and parallelism, all stated in a graphical way.

1. Any two points determines a straight line.
2, Any straight line may be extended indefinitely.
3, A circle may be drawn with any given point as center and any given radius.
4. All right angles are equal.
5. Given a line L and a point P not on L, then through P there exists one and only one line parallel to L.

Euclid’s Fifth Postulate. This particular formulation is known as the Playfair Axiom, after John Playfair, mathematician and eldest brother of Scottish engineer, secret agent, and sometimes con man William Playfair.

Eschewing Euclid’s Fifth. Mathematicians were long vexed by the Fifth Postulate being a less intuitive assumption than the others. Why one and only one line? Equivalently, how do we know that parallel lines never meet?

Attempts were made to prove the Fifth Postulate directly from Euclid’s other assumptions. No luck. In the 18th and early 19th century, mathematicians even tried the reductio ad absurdum approach, namely, deny the Fifth and see whether this leads to a contradiction.

Better than contradiction, this search led to new worlds of geometry, with wonders and applications only dreamt of. We’ll see them tomorrow in Part 2. ds

© Dennis Simanaitis, SimanaitisSays.com, 2018

2 comments on “GEOMETRIES OTHER THAN EUCLID’S PART 1

  1. Bob
    August 15, 2018

    Parallel lines DO meet! That’s the basis for every protest response presented by Schumacher, Senna and many other F1 pilotos!Cheers, Bob

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