Phil 105: Science and Philosophy - Fall 2004 Lectures
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Topic Two. The Shape of Space
Previous Lecture
Lecture 9 – Non-Euclidean Geometry (9/27/04) pp35-42.
1. Geometries
Three kinds of 2D geometry: (left)
flat, plane or Euclidean; (centre) positive curvature, Riemannian,
spherical geometry; and (right) negative curvature, Lobachevskian,
'saddle' geometry. The figure shows ants walking along the
straight(est) lines – geodesics – of each surface.
- Euclidean geometry:
- Pythagorean theorem
- Single parallel postulate
- Riemannian geometry:
- Diagonal of a right angled triangle < √(sum of sides2)
- No parallels
- Lobachevskian geometry:
- Diagonal of a right angled triangle > √(sum of sides2)
- Many parallels through a point
- Measuring space: suppose light travels along geodesics and if you
pull string tight it follows a geodesic. What different things would
you expect to see in each space ...
- ... because of the differences in the 'Pythagorean theorem'?
- ... because of the differences in the 'parallel postulate'?
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