Math Encounters -- Patterns and Disorder

If a set of numbers is large, does it have to contain any patterns? Just how large does a set need to be, to contain interesting configurations? Do specific sets of integers, such as the primes, contain hidden patterns? Come explore all this and more with mathematician Bryna Kra, as she discusses the difference between structure and randomness.

Math Encounters is a public presentation series celebrating the spectacular world of mathematics and presented by the Simons Foundation and the National Museum of Mathematics.

[davidjames]

In "randomness" you can search for patterns and find them (such as your phone number in the digits of pi), but true randomness cannot be compressed because compression looks for general patterns and tries to encode them in less data space. So for example, if our task was to compress 100 random bits (0s and/or 1s), in some cases it may appear that we can compress the 100 bits (such as all 0s or all 1s), but in the general case we cannot. Therefore, those types of random bitstrings can be said to have no "useful" patterns that can be used to compress it.

[keniangervo]

A really interesting talk indeed. I haven't yet fully watched it but I'm interested about how she made that fine presentation. If someone could answer me I would appreciate it (what software did she use to make that presentation?) Thanks in advance!

[WriteRightMathNation]

Here's a trouble with current discussions of ``patterns'' that are ``observed'' among finitely many objects:  I claim that for any finite set of objects, in any arrangement whatsoever, there is a ``pattern'' that extends to a countably infinite number of objects containing the original finitely many, and in some ``natural'' sense, the ``infinite pattern''  so described is unique.  Before I make these notions precise and indicate how to prove a theorem that should convince you of the veracity of my claim, let us note that we can always denote the original arrangement as a finite sequence of rational numbers, by numbering the finitely many objects using nonzero rational numbers, and numbering the ``possible or potential locations'' of the objects within some intended realm in which the ``pattern'' is to be perceived, using the natural numbers.  This yields the original  ``observed arrangement'' as sequence that lists the objects that are in the locations in order, beginning with location 0, and ending with the last occupied location, writing a 0 in any unoccupied location.  Thus for example, if I have five objects labeled 1, 2, 3, 4, 5, and I place them on a (3x3) chessboard, with labels 0, 1, 2, 3, 4, 5, 6, 7, 8, written in row major order, then the finite sequence (1, 5, 0, 3, 2, 4, 0, 0, 0) represents the following arrangement:

1 5 0
3 2 4
0 0 0

More generally, the arrangement

x0 x1 x2
x3 x4 x5
x6 x7 x8

is coded as (x0, x1, x2, x3, x4, x5, x6, x7, x8).  Now, let y be any positive integer.  There is a unique ``pattern'', meeting a certain condition I shall give shortly, such that both the finite sequence (x0, x1, x2, x3, x4, x5, x6, x7, x8) and (x0, x1, x2, x3, x4, x5, x6, x7, x8, y) satisfy.  Here's the theorem:

Theorem:  Let x be a finite sequence of positive integers, of length n, and let y be any positive integer.  The there is a unique polynomial function f from the complex numbers to the complex numbers, of least degree, such that f(0)=x0, f(1)=x1, ..., f(n-1)=x_{n-1}, and f(n)=y. 


The proof of the above theorem is a simple application of Vandermonde matrices.  I shall leave the details to the reader.  Note then that as a consequence, ANY choice of a way to extend the original sequence x fits a ``pattern''.  For this reason, it is unreasonable to teach third graders to answer questions like the following with ``expected'' answers and to count wrong some answer one does not like: 

``Find the next number in the sequence:  2, 4, 6, 8...''

The child who writes 3 for their answer is no less correct than the child who writes 10 for their answer.  In fact, I would want to meet the child who writes 3 for his or her answer and ask the reason.  It's guaranteed to be at least as good a reason as ``because the teacher wrote that on the board three days ago in class and I remembered it''.  In fact, the justification Johnny may give for saying that 3 is his answer is likely to be much more interesting than any answer given by a child who just happens to remember that it was the answer someone wanted them to put.

[WriteRightMathNation]

Morse-Hedlund is about periodicity (or lack thereof) in infinite sequences.  But people claim to see ``patterns'' in finite sequences, or to recognize ``randomness'' in finite sequences.  Morse-Hedlund helps not at all to understand why one would refer to some finite string as a ``pattern'' or ``random''.  As I indicated with my argument, the claims about ``patterns'' or ``randomness'' for finite sets of observations are meaningless.

[WriteRightMathNation]

She asks the question rhetorically:  ``Can you create something that has no pattern in it?''  Based upon my argument that I gave a few minutes ago, my answer is ``No''.  But that's because both the notion of ``pattern'' and the notion of ``random'' are meaningless.  Everything that anyone says is random has a pattern according to someone else, and everything that has a pattern is to someone else random.

[davidjames]

It appears the guy that introduced her and claims to have known her since childhood decided he needed to take a nap during her lecture.