Geometry ofInfinite Graphs

Jim Belk

Bard College

A graph is a set vertices connected by edges.

Graphs

This graph is finite, since there are a finite numberof vertices.

This graph is infinite.

Graphs

So are these.

Graphs

square grid cubical grid

And these.

Graphs

infinite honeycomb infinite 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 Plane Square Grid

The most familiar kind of geometry is Euclidean geometry.

Geometry

Euclidean Plane Square Grid

The most familiar kind of geometry is Euclidean geometry.

Geometry

Euclidean Plane Square 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 occurswhen 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 occurswhen is a circle.

IsoperimetricInequality

Let be a region in the plane.

The Isoperimetric Problem

Isoperimetric Theorem

The maximum area occurswhen 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

Infinite Tree

Infinite Tree

Infinite Tree

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

My Favorite Graphs

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

Three Dimensions

There are only three two-dimensional geometries:

• Spherical geometry

• Euclidean geometry

• Hyperbolic geometry

In three dimensions, there are eight geometries.

In three dimensions, there are eight geometries.

These were discovered

by Bill Thurston in the

1970’s

They are known as the

Thurston geometries.

1982 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 2006.

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

PSL(2) Geometry

Heisenberg Geometry