The Mathematical Influence of the Ancient Greeks

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The Mathematical Influence of the Ancient Greeks
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Obvious fingerprints of the Ancient Greeks

• The use of Greek symbols
• Pythagoras theorem
• Euclidean space
• Squaring the Circle

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Squaring the circle

• Antiphon the Sophist introduced a method of
  exhaustion
• in this way the area of the circle would be used
  up, and we should sometime have a polygon
  inscribed in the circle, the sides of which
  would, owing to their smallness, coincide with
  the circumference of the circle
• Antiphon therefore deserves an honourable place
  in the history of geometry - T L Heath

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The Idea of a Limit

• Euclid made Antiphons idea mathematically
  correct, by saying that if the process is
  continued far enough, the segments left over will
  be less than any given amount.

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Archimedean Spirals
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Archimedes Spiral

• If a straight line drawn in a plane revolves
  uniformly any number of times about a fixed
  extremity until it returns to its original
  position, and if, at the same time as the line
  revolves, a point moves uniformly along the
  straight line beginning at the fixed extremity,
  the point will describe a spiral in the plane.

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Using Archimedes Spiral to Square the Circle
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But how do you square a right angled triangle?
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But was there a way to square the circle using a
ruler and compass construction?

• This is essentially the same as asking whether pi
  a transcendental number
• A transcendental number is one that is not
  algebraic
• An algebraic number is one that is a root of a
  polynomial

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LindemannWeierstrass theorem (1882)

• If a1,...,an are distinct algebraic numbers,
  then the exponentials are linearly independent
  over the algebraic numbers.

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How does this help?
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Non-Euclidean spaces

• The main difference between Euclidean and
  non-Euclidean geometries is the nature of
  parallel lines.

• Although it is implossible to square the circle
  in Euclidean space (using ruler and compass),
  this does not mean it is impossible to do so in
  all non-Euclidean spaces.

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Gauss-Bolyai-Lobachevsky Space

• A non-Euclidean space with constant negative
  Gaussian curvature, that is, a hyperbolic
  geometry
• Gauss-Bolyai-Lobachevsky space is particularly
  noted for the fact that it is possible to square
  the circle in this geometry

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Thats all for now folks!