1. More than Meets the Eye:
Geometry and Our Perception of Reality
Richard G. Ligo
The University of Iowa

11. Overview Introduction The shape of the Earth Determining Earth’s size
Making maps of the Earth
Curvature applied to reality
The shape of the Universe

12. Hints of the Earth’s shape
Lunar eclipses:

13. Hints of the Earth’s shape
Horizon of the ocean:

14. Hints of the Earth’s shape
Constellation visibility:

15. Hints of the Earth’s shape
Eratosthenes and the gnomon

16. Eratosthenes and the gnomon

17. Mapping the Earth
Theorema Egregium (Gauss) The Gaussian curvature of a surface is invariant under isometries.
Intuitively, the theorem says that a surface may be “bent” without stretching or squishing it and have the same Gaussian curvature.

18. Mapping the Earth
Definition: A surface is called developable if it has zero Gaussian curvature.

19. Maps: central stereographic projection

20. Maps: azimuthal equidistant projection

21. Maps: central cylindrical projection

22. Maps: equirectangular projection

23. Maps: Lambert cylindrical projection

24. Maps: Mercator projection

25. Derivation of the Mercator projection?

26. Derivation of the Mercator projection
Globe
Projection

27. Derivation of the Mercator projection

28. Derivation of the Mercator projection

29. Maps: Natural Earth projection

30. The curvature of a surface
K = 0
K < 0
K > 0
K ? 0

31. The curvature of a surface
K = 0 C = 2πr

32. The curvature of a surface
K > 0 C < 2πr

33. The curvature of a surface
K < 0 C > 2πr

34. The curvature of space

35. The shape of the universe

36. The shape of the universe

37. The shape of the universe

38. The shape of the universe

39. The shape of the universe

40. The shape of the universe?

41. References
Stewart, Ian (2001). Flatterland. Cambridge, MA: Perseus Publishing.
Oprea, John (2007). Differential Geometry and Its Applications. Washington, DC: Mathematical Association of America.
Osserman, Robert (1995). Poetry of the Universe. New York, NY: Anchor Books.