Geometry in (Very) High Dimensions

Renato FeresWashington University in St. Louis

Department of Mathematics

1 / 20

The RGB color cube

2 / 20

The space of musical chords (Dmitry Tymoczko)

3-dimensional space of three-note chords, with major and

minor triads found near the center. (An orbifold, much like

a 3-dim Moebius band.) Tymoczko is composer and professor

of music at Princeton University.

3 / 20

Primary faces

The 6 basic emotions

Based on research by Paul Ekman

Illustration by Scott McCloud

From Making Comics by McCloud (2006)

Other facial expressions of emotions are combinations

of these 6, just as colors are combinations of R, B, G

(a 6-dimensional space.)

4 / 20

Combinations of primary faces

5 / 20

Thermodynamic space

To specify the kinetic state of one mole of gas we need:

3 velocity components for each of 6.022 × 1023 molecules.

So we need more than 18 × 1023 dimensions.

6 / 20

Distance in Cartesian n-dimensional space Rn

Space Rn consists of all n-tuples of real numbers: x = (x1, . . . , xn).

Distance between x = (x1, . . . , xn) and y = (y1, . . . , yn):

d(x, y) =√(y1 − x1)2 +⋯+ (yn − xn)2

Euclid (c. 300 BC) Descartes (17th century)

7 / 20

Cubes and Balls in Cartesian n-space

The n-cube C(r) of side length 2r: set of x such that −r ≤ xi ≤ r.

It is also the Cartesian product: Cn(r) = [−r, r] ×⋯× [−r, r] (n times.)

8 / 20

n-dimensional balls

The pythagorean formula in dimension n is

∣x∣ =√x2

1+⋯+ x2

n= distance from x to origin (0, . . . ,0).

The n-ball of radius r is the set of x at distance ≤ r from the origin:

Bn(r) = {x ∈ Rn ∶ ∣x∣ ≤ r}.

9 / 20

Testing our geometric intuition ...

10 / 20

Similar arrangement in dimension 3

11 / 20

What happens to the red ball as n goes to infinity?

Now, do the same in arbitrary dimensions:

Begin with a cube of dimension n and side length 1.

Pack 2n balls inside, each of radius 1

4.

Add a (red) ball at the center that touches all the other balls.

Let rn be the radius of the red ball.

What happens to rn when n goes to infinity?

12 / 20

rn →∞

In dimension 2,

r2 =√(14)2 + (1

4)2 − 1

4= 1

4(√2 − 1) .

In dimension n,

rn =√(14)2 +⋯+ (1

4)2 − 1

4= 1

4(√n − 1)

13 / 20

Volumes

The fundamental volume is that of the cube. We define it to be

Vol(Cn(a/2)) = Vol(n-cube of side length a) = an

By filling the ball with small cubes and taking a limit,

Vol (Bn(r)) = Vol (Bn(1)) rn.Using multivariate calculus and induction in the dimension,

Vol (Bn(1)) = πn/2

Γ (n2+ 1) ∼ 1√

nπ

⎛⎝√

2πe

n

⎞⎠n

.

This is very small for big n.

14 / 20

Example: dimension 100

Volume of cube = 1Volume of inscribed ball = 2.37 × 10−40

15 / 20

How are volumes distributed? (1. Shells)

Fraction of volume of n-ball contained in the outer shell of thickness a:

Vol(Bn(r)) −Vol(Bn(r − a))Vol(Bn(r)) = rn − (r − a)n

rn= 1 − (1 − a

r)n .

Therefore, for arbitrarily small a, as n→∞,

Vol (shell of thickness a)Vol (Bn(r)) → 1.

The volume inside a ball (for big n) is mostly near the surface.

16 / 20

How are volumes distributed? (2. Central slabs)

Consider the n-dimensional ball of fixed volume equal to 1.

It can be shown that:

radius rn ∼√

n

2πe

about 96% of the volume lies in the (relatively very thin) slab

{x ∈ Bn(rn) ∶ −12≤ x1 ≤ 1

2} .

This is true for the slab centered on any (n− 1)-dimensional subspace!17 / 20

Hyper-area of slices

Associated to the n-dimensional cube of side 1, define:

e = 1√n(1, . . . ,1), a vector of unit length;

Vn−1(t) = hyper-area of slice perpendicular to e above t.

Hyper-area of slice above t (big n)

t

a

0

0.4

0.8

1.2

-1.5 -1 -0.5

00.5 1 1.5

Theorem (CLT)

Vn−1(t) = Voln−1 ({x ∶ ∣x1 + ⋅ ⋅ ⋅ + xn√n

∣ = t}) ∼√

6

πe−6t

2

18 / 20

If we identify Probability⇔ Volume ...

Theorem (Central Limit Theorem)

If X1,X2, . . . are random numbers (uniformly distributed) on [− 1

2, 12],

limn→∞

Prob(∣X1 +⋯+Xn√n

∣ ≤ x) = ∫ x

−x

√6

πe−6t

2

dt

0-1.5 -1 -0.5 0.5 1 1.5

0

100

200

300

400

500

600 Probability experiment:

Compute 5000 sample values of the random number

then plot the distribution of values of Y (histogram)

19 / 20

Morals to draw from the story

Concepts in dimension 2,3 often generalize to n > 3;But peculiar things can happen for big n.

Insight is regained by thinking probabilistically.

Probability theory can be “interpreted” geometrically.

Algebra, analysis, combinatorics, modern physical theories, music,and many other subjects use geometric ideas to represent conceptsthat may not seem geometric at first.

This is reflected in the many flavors of modern geometry:

Riemannian and pseudo-Riemannian geometries;

algebraic geometry;

symplectic and non-commutative geometry;

information geometry, etc., etc., etc.

20 / 20