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 line can be specified using one number
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.
The dimension of an object
1
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!
What is dimension?
What is dimension?
2
3
2
0
1
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 highdimensional 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 threedimensional 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
For example, with n=20 the ratio is
Ratio of volume of n-balls to
volume of containing n-cubes
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 nball 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!