Surprises in high dimensions Martin Lotz Galois Group, April 22, 2015.
-
Upload
kelley-weaver -
Category
Documents
-
view
216 -
download
2
Transcript of Surprises in high dimensions Martin Lotz Galois Group, April 22, 2015.
![Page 1: Surprises in high dimensions Martin Lotz Galois Group, April 22, 2015.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ea75503460f94ba974a/html5/thumbnails/1.jpg)
Surprises in high dimensions
Martin Lotz
Galois Group, April 22, 2015
![Page 2: Surprises in high dimensions Martin Lotz Galois Group, April 22, 2015.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ea75503460f94ba974a/html5/thumbnails/2.jpg)
Life in 2D
Ladd Ehlinger Jr. (dir). Flatland, 2007.
![Page 3: Surprises in high dimensions Martin Lotz Galois Group, April 22, 2015.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ea75503460f94ba974a/html5/thumbnails/3.jpg)
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?
![Page 4: Surprises in high dimensions Martin Lotz Galois Group, April 22, 2015.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ea75503460f94ba974a/html5/thumbnails/4.jpg)
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
![Page 5: Surprises in high dimensions Martin Lotz Galois Group, April 22, 2015.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ea75503460f94ba974a/html5/thumbnails/5.jpg)
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
![Page 6: Surprises in high dimensions Martin Lotz Galois Group, April 22, 2015.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ea75503460f94ba974a/html5/thumbnails/6.jpg)
What is dimension?
![Page 7: Surprises in high dimensions Martin Lotz Galois Group, April 22, 2015.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ea75503460f94ba974a/html5/thumbnails/7.jpg)
What is dimension?
11
22
00223
3
Requires a more
sophisticated notion of
dimension!
![Page 8: Surprises in high dimensions Martin Lotz Galois Group, April 22, 2015.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ea75503460f94ba974a/html5/thumbnails/8.jpg)
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!
![Page 9: Surprises in high dimensions Martin Lotz Galois Group, April 22, 2015.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ea75503460f94ba974a/html5/thumbnails/9.jpg)
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.
![Page 10: Surprises in high dimensions Martin Lotz Galois Group, April 22, 2015.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ea75503460f94ba974a/html5/thumbnails/10.jpg)
Polyhedra
The Platonic solids
![Page 11: Surprises in high dimensions Martin Lotz Galois Group, April 22, 2015.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ea75503460f94ba974a/html5/thumbnails/11.jpg)
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
![Page 12: Surprises in high dimensions Martin Lotz Galois Group, April 22, 2015.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ea75503460f94ba974a/html5/thumbnails/12.jpg)
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.
![Page 13: Surprises in high dimensions Martin Lotz Galois Group, April 22, 2015.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ea75503460f94ba974a/html5/thumbnails/13.jpg)
Cubes in higher dimensions
Cube Hypercube
What can we say of the hypercube in higher dimensions?
![Page 14: Surprises in high dimensions Martin Lotz Galois Group, April 22, 2015.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ea75503460f94ba974a/html5/thumbnails/14.jpg)
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).
![Page 15: Surprises in high dimensions Martin Lotz Galois Group, April 22, 2015.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ea75503460f94ba974a/html5/thumbnails/15.jpg)
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
![Page 16: Surprises in high dimensions Martin Lotz Galois Group, April 22, 2015.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ea75503460f94ba974a/html5/thumbnails/16.jpg)
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
![Page 17: Surprises in high dimensions Martin Lotz Galois Group, April 22, 2015.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ea75503460f94ba974a/html5/thumbnails/17.jpg)
Schlegel diagrams
Schlegel diagrams are a tried-and-tested method of seeing four (and sometimes higher) dimensional polyhedra.
![Page 18: Surprises in high dimensions Martin Lotz Galois Group, April 22, 2015.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ea75503460f94ba974a/html5/thumbnails/18.jpg)
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!
![Page 19: Surprises in high dimensions Martin Lotz Galois Group, April 22, 2015.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ea75503460f94ba974a/html5/thumbnails/19.jpg)
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)
![Page 20: Surprises in high dimensions Martin Lotz Galois Group, April 22, 2015.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ea75503460f94ba974a/html5/thumbnails/20.jpg)
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.
![Page 21: Surprises in high dimensions Martin Lotz Galois Group, April 22, 2015.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ea75503460f94ba974a/html5/thumbnails/21.jpg)
4D Rubik’s Cube
![Page 22: Surprises in high dimensions Martin Lotz Galois Group, April 22, 2015.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ea75503460f94ba974a/html5/thumbnails/22.jpg)
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.
![Page 23: Surprises in high dimensions Martin Lotz Galois Group, April 22, 2015.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ea75503460f94ba974a/html5/thumbnails/23.jpg)
Computing volumes and areas
Volumes and areas can be computed using integrals and symmetry
![Page 24: Surprises in high dimensions Martin Lotz Galois Group, April 22, 2015.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ea75503460f94ba974a/html5/thumbnails/24.jpg)
Computing volumes and areas
…or simply using the combinatorial structure of the object
![Page 25: Surprises in high dimensions Martin Lotz Galois Group, April 22, 2015.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ea75503460f94ba974a/html5/thumbnails/25.jpg)
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.
![Page 26: Surprises in high dimensions Martin Lotz Galois Group, April 22, 2015.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ea75503460f94ba974a/html5/thumbnails/26.jpg)
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
![Page 27: Surprises in high dimensions Martin Lotz Galois Group, April 22, 2015.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ea75503460f94ba974a/html5/thumbnails/27.jpg)
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.
![Page 28: Surprises in high dimensions Martin Lotz Galois Group, April 22, 2015.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ea75503460f94ba974a/html5/thumbnails/28.jpg)
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
![Page 29: Surprises in high dimensions Martin Lotz Galois Group, April 22, 2015.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ea75503460f94ba974a/html5/thumbnails/29.jpg)
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!
![Page 30: Surprises in high dimensions Martin Lotz Galois Group, April 22, 2015.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ea75503460f94ba974a/html5/thumbnails/30.jpg)
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?
![Page 31: Surprises in high dimensions Martin Lotz Galois Group, April 22, 2015.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ea75503460f94ba974a/html5/thumbnails/31.jpg)
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!
![Page 32: Surprises in high dimensions Martin Lotz Galois Group, April 22, 2015.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ea75503460f94ba974a/html5/thumbnails/32.jpg)
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!
![Page 33: Surprises in high dimensions Martin Lotz Galois Group, April 22, 2015.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649ea75503460f94ba974a/html5/thumbnails/33.jpg)
Thanks!