TITLE: Patterns Are Obvious When You Know To Look For Them...
AUTHOR: Eugene Wallingford
DATE: June 13, 2016 2:47 PM
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In
Prime After Prime
(great title!), Brian Hayes boils down into two sentences the
fundamental challenge that faces people doing research:
What I find most surprising about the discovery is that no
one noticed these patterns long ago. They are certainly
conspicuous enough once you know how to look for them.
It would be so much easier to form hypotheses and run tests if
interesting hypotheses were easier to find.
Once found, though, we can all see patterns. When they can be
computed, we can all write programs to generate them! After
reading a paper about the strong correlations among pairs of
consecutive prime numbers, Hayes wrote a bunch of programs to
visualize the patterns and to see what other patterns he might
find. A lot of mathematicians did the same.
Evidently that was a common reaction. Evelyn Lamb,
writing in Nature,
quotes Soundararajan: "Every single person we've told this
ends up writing their own computer program to check it for
themselves."
Being able to program means being able to experiment with all
kinds of phenomena, even those that seemingly took genius to
discover in the first place.
Actually, though, Hayes's article gives a tour of the kind of
thinking we all can do that can yield new insights. Once he
had implemented some basic ideas from the research paper, he
let his imagination roam. He tried different moduli. He
visualized the data using heat maps. When he noticed some
symmetries in his tables, he applied a cyclic shift to the
data (which he termed a "twist") to see if some patterns were
easier to identify in the new form.
Being curious and asking questions like these are one of the
ways that researchers manage to stumble upon new patterns
that no one has noticed before. Genius may be one way to make
great discoveries, but it's not a reliable one for those of us
who aren't geniuses. Exploring variations on a theme is a
tactic we mortals can use.
Some of the heat maps that Hayes generates are quite beautiful.
The image above is a heat map of the normalized counts of
consecutive eight-digit primes, taken modulo 31. He has more
fun making images of his twists and with other kinds of primes.
I recommend reading the entire article, for its math, for its
art, and as an implicit narration of how a computational
scientist approaches a cool result.
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