Surprises in high dimensions

Martin Lotz

Galois Group, April 22, 2015

Life in 2D

Ladd Ehlinger Jr. (dir). Flatland, 2007.

Life in 2D

Edwin A. Abbott. Flatland: A Romance of Many Dimensions, 1884.

The novella describes a two-dimensional world inhabited by geometric figures.

Flatlanders would see everything like this:

How would we go about describing the third dimensions to a flatlander?

What is dimension?

A point on a plane is determined by two numbers

A point on a plane is determined by three numbers

While we can’t “imagine” four or more perpendicular axes, we can speak of the space of real n-tuples (n>3) in geometric terms.

A point on a line can be specified using one number

The dimension of an object

The dimension can be defined as the number of parameters needed to describe an object. There’s no reason this should be restricted to 3!

11

What is dimension?

What is dimension?

11

22

00223

3

Requires a more

sophisticated notion of

dimension!

Why should we care?

Higher dimensional “objects” appear whenever we are dealing with systems that require more than three parameters to describe!

•Higher order differential equations reduce to first-order equations in higher dimensions;•The location of the hand of a robotic arm depends on various angles and lengths, and can be considered as a high-dimensional problem;•The price of stocks depends on many factors: it is a function in high-dimensional space;•Galois groups can appear as symmetry groups of higher dimensional geometric objects;•Countless other examples come to mind!

Visualising the fourth dimension

There are various strategies to visualize four or more dimensions.

Study projections of a higher-dimensional object:

•This is how we represent 3D objects on a screen!

Visualise the structure that defines a higher-dimensional object:

•combinatorial structure•symmetries

Interpret the fourth dimension as time.

Polyhedra

The Platonic solids

Polyhedra

A polyhedron in three dimensions is defined as the set of points that satisfy a system of linear inequalities.

The octahedron with defining equations

Polyhedra

Given this algebraic description, there is no reason to restrict to three dimensions! A polyhedron in is defined as the set of points that satisfy linear inequalities

These higher-dimensional geometric objects are essential in linear programming.

Cubes in higher dimensions

Cube Hypercube

What can we say of the hypercube in higher dimensions?

Combinatorial structure

A three-dimensional polyhedron has v vertices, e edges, and f facets. These numbers satisfy the Euler relation (verify this on examples!)

v-e+f=2

An n-dimensional polyhedron also has faces: these are the points where a fixed set of the defining inequalities are equalities!

The faces of an n-dimensional polyhedron can be of dimensions 0 (vertices) to n-1 (facets) and n (the polyhedron itself).

Combinatorial structure

The combinatorial structure of a polyhedron describes the relationship among the faces.

• Every vertex is contained in three edges• Every edge is contained in two facets• Every facet has four edges• Every edge has two vertices

Combinatorial structure

The combinatorial structure of a polyhedron describes the relationship among the faces. For the square:

• The square has four edges• Every edge has two vertices• Every vertex has is in two edges

Schlegel diagrams

Schlegel diagrams are a tried-and-tested method of seeing four (and sometimes higher) dimensional polyhedra.

Schlegel diagrams

If we label the vertices of the cube by 1,2,3…, the corresponding edges by 12, 23, … and the facets by 1234, …,

The complete combinatorial structure can be read off these diagrams!

The hypercube

What does the Schlegel diagram of a 4D hypercube look like?

What we see is the projection onto a three-dimensional face of the 4D hypercube. All the combinatorics of this object can be derived from this projection!•16 vertices•Each vertex incident to 4 edges•12 edges•8 facets (the seven “regions” we see in the picture + the projection facet)

4D Rubik’s Cube

Mathematical structure•16 vertices

•8 facets/colours (each a 3D cube)

•Each facet has 27 small cubes

•There are 24 ways of rotating each facet (the orientation preserving symmetries of the cube)

Homework: find out what happens to the other cubes when rotating the blue cube.

4D Rubik’s Cube

Volumes in higher dimensions

In one, two and three dimensions we have the notion of length, area, and volume.

Volumes in higher dimensions are the subject of measure theory.

Computing volumes and areas

Volumes and areas can be computed using integrals and symmetry

Computing volumes and areas

…or simply using the combinatorial structure of the object

The sphere and cube

In two and three dimensions we can embed a unit sphere in a cube of side length 2, with the volume ratios given below.

The hypersphere and hypercube

The n-dimensional ball of radius r is defined by

The n-dimensional sphere of radius r is defined by

The n-dimensional hypercube with length 2r is the set

The hypersphere and hypercube

The volumes of these sets can be computed in the same way as in the three dimensional case:

where is the Gamma function, and

Let’s see how these two volume functions behave as n increases.

The hypersphere and hypercube

Volume of n-balls Ratio of volume of n-balls to volume of containing n-cubes

For example, with n=20 the ratio is

Surprise 1

If the ratio of volumes between a hypersphere of diameter 2 and a hypercube of diameter 2 is in dimension 20, this means that only about of the mass (almost all!) of the hypercube is outside the unit ball, concentrated in the corners!

Boundaries of n-balls

Shells of unit n-balls of width r are defined as the outer boundaries of the ball

How much of the mass of a ball is near its boundary?

Surprise 2

If r=0.01 (1/100) and n=500, then more than 99% of the mass of the n-ball will be in a shell of width 1/100th of the radius of the sphere, that is, almost on the boundary!

Surprise 3 (Concentration of Measure)

For example, in dimension n>100, more than 90% of the mass will be concentrated in a tiny neighbourhood of any equator!

Thanks!