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Topology: Finding the Shape of the Universe

Topology: Finding the Shape of the Universe

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Topology: Finding the Shape of the Universe

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  1. Topology: Finding the Shape of the Universe A presentation by: Fionnghuala Eley O’Reilly Megan Gage Jeneé Dampier Kenmond Pang

  2. The Goal

  3. The Goal • Finding the shape of the universe

  4. The Goal • Finding the shape of the universe • Takes a substantial amount of math

  5. The Goal • Finding the shape of the universe • Takes a substantial amount of math • Uses a large amount of math

  6. The Goal • Finding the shape of the universe • Takes a substantial amount of math • Uses a large amount of math • Encountered problems

  7. Problems We Faced

  8. Problems We Faced • Is it even possible to figure out the shape of the universe if we live in it?

  9. Problems We Faced • Is it even possible to figure out the shape of the universe if we live in it? • How could we tell its shape from the inside?

  10. Topology

  11. Topology • Shape and preserving the figure

  12. Topology • Shape and preserving the figure • Geometry vs. Topology

  13. Topology • Shape and preserving the figure • Geometry vs. Topology • Why is it important in our problem?

  14. Classification

  15. Classification • Dimensional Classification

  16. Classification • Dimensional Classification • The Zeroth Dimension

  17. Classification • Dimensional Classification • The Zeroth Dimension • A point

  18. Classification • Dimensional Classification • The Zeroth Dimension • The First Dimension

  19. Classification • Dimensional Classification • The Zeroth Dimension • The First Dimension • Lines and circles

  20. Classification • Dimensional Classification • The Zeroth Dimension • The First Dimension • The Second Dimension

  21. Classification • Dimensional Classification • The Zeroth Dimension • The First Dimension • The Second Dimension • Squares and polygons

  22. Classification • Dimensional Classification • The Zeroth Dimension • The First Dimension • The Second Dimension • The Third Dimension

  23. Classification • Dimensional Classification • The Zeroth Dimension • The First Dimension • The Second Dimension • The Third Dimension • Cubes and polyhedra

  24. Classification • Dimensional Classification • The Zeroth Dimension • The First Dimension • The Second Dimension • The Third Dimension • The Fourth Dimension

  25. Classification • Dimensional Classification • Classifying Figures

  26. Classification • Dimensional Classification • Classifying Figures • “Dividing by 2”

  27. Classification • Dimensional Classification • Classifying Figures • “Dividing by 2” • Points divide lines

  28. Classification

  29. Classification • Dimensional Classification • Classifying Figures • “Dividing by 2” • Points divide lines • Lines divide polygons

  30. Classification

  31. Classification • Dimensional Classification • Classifying Figures • “Dividing by 2” • Points divide lines • Lines divide polygons • Polygons divide polyhedra

  32. Classification A cube is 3 dimensional because it can be split in two by a polygon.

  33. Classification • Dimensional Classification • Classifying Figures • “Dividing by 2” • Points divide lines • Lines divide polygons • Polygons divide polyhedra • Ultimately helps classify shapes

  34. Classification • Dimensional Classification • Classifying Figures • “Dividing by 2” • The Euler Characteristic

  35. Classification • Dimensional Classification • Classifying Figures • “Dividing by 2” • The Euler Characteristic • The Greek letter χ (chi)

  36. Classification • Dimensional Classification • Classifying Figures • “Dividing by 2” • The Euler Characteristic • The Greek letter χ (chi) • χ=f+p-e

  37. Classification • Dimensional Classification • Classifying Figures • “Dividing by 2” • The Euler Characteristic • The Greek letter χ (chi) • χ=f+p-e • Our conclusions:

  38. Classification • Dimensional Classification • Classifying Figures • “Dividing by 2” • The Euler Characteristic • The Greek letter χ (chi) • χ=f+p-e • Our conclusions: • Polyhedra, polygons, annuli, etc.

  39. Classification • Why is classification important? • Narrows which shapes are possibilities

  40. Classification • Why is classification important? • Narrows which shapes are possible

  41. Conclusion • Goal

  42. Conclusion • Goal • Classification

  43. Conclusion • Goal • Classification • The Euler Charactersitic

  44. Conclusion • Goal • Classification • The Euler Characteristic • What comes next?