Geometry of Infinite Graphs

1. Geometry ofInfinite Graphs Jim Belk Bard College

2. Graphs A graph is a set vertices connected by edges. This graph is finite, since there are a finite numberof vertices.

3. Graphs This graph is infinite.

4. Graphs So are these. square grid cubical grid

5. Graphs And these. infinite honeycomb infinite tree

6. Geometry of Graphs Central Argument: It is possible to do geometry just with graphs! infinite honeycomb

7. Euclidean Geometry The most familiar kind of geometry is Euclidean geometry.    Euclidean Plane

8. Geometry The most familiar kind of geometry is Euclidean geometry.    Euclidean Plane Square Grid

9. Geometry The most familiar kind of geometry is Euclidean geometry.    Euclidean Plane Square Grid

10. Geometry The most familiar kind of geometry is Euclidean geometry.    Euclidean Plane Square Grid

11. Example: The Isoperimetric Problem

12. The Isoperimetric Problem Let  be a region in the plane. Given: perimeter    Question: What is the maximum possible area of ?

13. The Isoperimetric Problem Let  be a region in the plane. Given: perimeter    Question: What is the maximum possible area of ? Isoperimetric Theorem The maximum area occurswhen  is a circle. 

14. Isoperimetric Theorem The maximum area occurswhen  is a circle.  The Isoperimetric Problem Let  be a region in the plane.

15. Isoperimetric Theorem The maximum area occurswhen  is a circle.  The Isoperimetric Problem Let  be a region in the plane. IsoperimetricInequality 

16. The Isoperimetric Problem Circle Double Bubble

17. The Isoperimetric Problem In the plane, area is aquadratic function of perimeter. Quadratic

18. On the Grid

19. Some Definitions A region in the gridis any finite set of vertices. The area is just the number of vertices.

20. Some Definitions A region in the gridis any finite set of vertices. The area is just the number of vertices. The perimeter is the number of boundary edges.

21. Some Definitions A region in the gridis any finite set of vertices. The area is just the number of vertices. The perimeter is the number of boundary edges.

22. Some Definitions A region in the gridis any finite set of vertices. The area is just the number of vertices. The perimeter is the number of boundary edges.

23. Theorem For the infinite grid: Isoperimetric Theorem

24. Theorem For the infinite grid: Isoperimetric Theorem   Square

25. Theorem For the infinite grid: Isoperimetric Theorem   Quadratic Square

26. Theorem For the infinite grid: Isoperimetric Theorem Theorem For the plane: Quadratic

27. Theorem For the infinite grid: Isoperimetric Theorem Quadratic Idea: Plane area is comparable to grid area, and plane perimeter is comparable to grid perimeter.

28. More Examples

29. Three Dimensions In the cubical grid: # of vertices  volume # boundary edges surface area

30. Three Dimensions In the cubical grid: # of vertices  volume # boundary edges surface area

31. Three Dimensions In the cubical grid: # of vertices  volume # boundary edges surface area

32. Three Dimensions In the cubical grid: # of vertices  volume # boundary edges surface area    

33. Infinite Tree

34. Infinite Tree

35. Infinite Tree

36. Infinite Tree

37. Infinite Tree Isoperimetric Inequality:

38. More Geometry Distance in a graph  length of shortest path

39. More Geometry A shortest path is called a geodesic.

40. More Geometry With distance, you can make: • straight lines (geodesics) • polygons • balls (center point, radius ) The geometry looks very strange on small scales,but is interesting on large scales.

41. Things to Do • Volumes of Balls • Random Walks • Heat Diffusion • Flow of Water • Jumping Rabbits

42. My Favorite Graphs

43. My Favorite Graphs

44. My Favorite Graphs Very similar to the hyperbolic plane!

45. The Hyperbolic Plane The hyperbolic plane is the setting fornon-Euclidean geometry.   (half-planemodel) 

46. The Hyperbolic Plane Distances are much longer near the -axis.   (half-planemodel) 

47. The Hyperbolic Plane Distances are much longer near the -axis.  Euclidean Length Hyperbolic Length 

48. The Hyperbolic Plane not shortest distance 

49. The Hyperbolic Plane Hyperbolic “lines” are semicircles. shortest distance 

50. The Hyperbolic Plane Hyperbolic “lines” are semicircles. 