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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?