Geometry of Infinite Graphs Jim Belk Bard College

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

This graph is infinite. Graphs

So are these. Graphs square gridcubical grid

And these. Graphs infinite honeycombinfinite tree

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

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

The most familiar kind of geometry is Euclidean geometry. Geometry    Euclidean PlaneSquare Grid

The most familiar kind of geometry is Euclidean geometry. Geometry    Euclidean PlaneSquare Grid

The most familiar kind of geometry is Euclidean geometry. Geometry    Euclidean PlaneSquare Grid

Example: The Isoperimetric Problem

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

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

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

Isoperimetric Inequality Let  be a region in the plane. The Isoperimetric Problem  Isoperimetric Theorem The maximum area occurs when  is a circle. 

The Isoperimetric Problem Circle Double Bubble

The Isoperimetric Problem Quadratic In the plane, area is a quadratic function of perimeter.

On the Grid

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

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

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

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

Isoperimetric Theorem Theorem For the infinite grid:

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

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

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

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

More Examples

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

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

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

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

Infinite Tree

Isoperimetric Inequality:

More Geometry Distance in a graph  length of shortest path

More Geometry A shortest path is called a geodesic.

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.

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

My Favorite Graphs

Very similar to the hyperbolic plane!

The Hyperbolic Plane The hyperbolic plane is the setting for non-Euclidean geometry.    (half-plane model)

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

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

The Hyperbolic Plane  not shortest distance

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

The Hyperbolic Plane Hyperbolic “lines” are semicircles. 

The Hyperbolic Plane The hyperbolic plane is non-Euclidean. 

The Hyperbolic Plane The hyperbolic plane is non-Euclidean.    

My Favorite Graphs This graph is like a grid for the hyperbolic plane.

My Favorite Graphs This graph is like a grid for the hyperbolic plane.

My Favorite Graphs This graph is like a grid for the hyperbolic plane.

My Favorite Graphs This graph is like a grid for the hyperbolic plane.

My Favorite Graphs Isoperimetric Inequality:

Three Dimensions

There are only three two-dimensional geometries: Spherical geometry Euclidean geometry Hyperbolic geometry In three dimensions, there are eight geometries.

These were discovered by Bill Thurston in the 1970’s They are known as the Thurston geometries Fields Medalist William Thurston Three Dimensions

In three dimensions, there are eight geometries. These were discovered by Bill Thurston in the 1970’s They are known as the Thurston geometries. Three Dimensions Thurston Geometrization Conjecture: Any 3-manifold can be broken into pieces, each of which has one of the eight geometries.

In three dimensions, there are eight geometries. This was proven by Grigori Perelman in Three Dimensions Thurston Geometrization Conjecture: Any 3-manifold can be broken into pieces, each of which has one of the eight geometries.

In three dimensions, there are eight geometries. Many of the Thurston geometries can be modeled effectively with graphs. Three Dimensions

Euclidean Three-Space

Hyperbolic Three-Space

Solv Geometry

PSL(2) Geometry

Heisenberg Geometry