Pi
Day 2024 –A Really Big PI-related near integer
There is a really good Youtube channel done by a math prof that I
follow on Youtube, Richard E. Borcherds. I will have a little more to
say about him later, but first I will discuss the Pi-related content
in one of his videos.
https://www.youtube.com/watch?v=a9k_QmZbwX8
I won’t attempt to get into any high-level explanation, as that
would involve very advanced abstract algebra, which is far above the
pay grade of a lowly retired statistician. That said, I did take one
university-level abstract algebra course long ago and can confidently
say that this type of math is very abstract indeed. Stats degrees
are usually heavy on calculus, linear algebra and probability, not so
much on abstract algebra. Arguably though, these subjects are all
inter-related, though most of us can only learn so much math in one
lifetime.
Anyway, the screenshot above shows a couple of very big numbers:
As you can see, the second number is quite close to the first number,
up until the last 3 digits before the decimal point. They differ by
almost 144.
And here is the Pi Day angle: that number is e raised to the
power Pi times the square root of 163, and it nearly comes out to an
integer, with the first 12 places after the decimal point being 9s.
So, that’s a transcendental number taken to the power of another
transcendental number multiplied by an irrational number. Given that
these are numbers that are non-repeating and non-ending, that seems
pretty weird.
The professor notes that a normal pocket calculator will actually
show an integer after doing the calculation.
Even better, a Windows-based spreadsheet shows the result to 20
places past the decimal, as an integer.
And here is what the stats language R does. It just rounds the
number, long before getting to the decimal. There are some R
packages that handle really big numbers, but I couldn’t get any of
them to work, so we will have to leave it at that.
Professor Borcherds recommends a particular online calculator, called
free42, which emulates a particular high-level HP calculator on the
PC. It looks interesting.
Apparently Martin Gardner (who
wrote a recreational math column for Scientific American back when it
was good) claimed that this was an integer, though Ramanujan
had an accurate calculation of this number around a century ago,
showing that it was not an integer.
The professor goes on to show a few more numbers involving e to
the Pi times the square root of some other numbers, that end in a
remarkable run of 9s, though not so spectacular as the first one.
You are probably now getting the feeling that something is going on
here that is much more than a coincidence. And you would be right.
From this point, the explanation gets into elliptic modular function
theory and other highly complicated areas of math.
You can see that this all involves mathematics with complex numbers,
which is where the factor of Pi sneaks in (note the presence of e to
the 2 Pi in some of those formulas).
The video then segues into a number of areas, touching on some more
of Ramanujan’s work, as well as Euler’s. These involves number
theory (primes), as well as group theory. Eventually, the aptly
named “Monster Group” makes an appearance. This object exists in
198663 dimensions, and has order 10 to the 53. Here's a SF comic book that makes use of the concept (though with some slight errors).
This all seems rather esoteric, but these matters do come up in
“sort-of” practical endeavors such as particular physics (group
theory) and cosmology (string theory in 26 dimensions).
It turns out that Richard Borcherds actually was a co-winner of the
Fields Medal, based on work related to the monster group. I didn’t
know that when I stumbled over his Youtube channel and watched a
bunch of videos on complex analysis and group theory. Then I picked
up a book that I have about the Monster (just a recreational math
book) and there he was! For those who don’t know, the Fields Medal
is similar in status to the Nobel prizes, maybe more so, as the
Fields are only handed out once every four years.
The professor’s videos are quite compelling and often rather oddly
amusing. So, I would definitely recommend his channel if you are
interested in math, even if the videos are probably something of a
stretch for most of us.
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So, now that you have dutifully read some Pi Day math, you should
read a science fiction book, or even better, a whole series. Book 1
of the Witches’ Stones series even includes a reference to pi.:
Kati of Terra
How about trying Kati of Terra, the 3-novel story of a feisty young
Earth woman, making her way in that big, bad, beautiful universe out
there.
So, go on a romantic adventure, a sort of spaceship and planet-side
road trip, while escaping from dangerous slavers, with our gal Kati
of Terra and her companion Mikal (and, of course her ever helpful but
cantankerous Granda node):
http://www.amazon.com/gp/product/B00811WVXO
http://www.amazon.co.uk/gp/product/B00811WVXO
The Witches’
Stones
Or, you might prefer, the trilogy of the Witches’ Stones (they’re
psychic aliens, not actual witches), which follows the interactions
of a future Earth confederation, an opposing galactic power, and the
Witches of Kordea. It features Sarah Mackenzie, another feisty young
Earth woman).
https://www.amazon.com/dp/B008PNIRP4
https://www.amazon.co.uk/dp/B008PNIRP4
The Magnetic
Anomaly: A Science Fiction Story
“A geophysical crew went into the Canadian north. There were some
regrettable accidents among a few ex-military who had become
geophysical contractors after their service in the forces. A young
man and young woman went temporarily mad from the stress of seeing
that. They imagined things, terrible things. But both are known to
have vivid imaginations; we have childhood records to verify that. It
was all very sad. That’s the official story.”
A short story of about 6000 works.
https://www.amazon.com/dp/B0176H22B4
https://www.amazon.co.uk/dp/B0176H22B4
The Zoo
Hypothesis or The News of the World: A Science Fiction Story
In the field known as Astrobiology, there is a research program
called SETI, The Search for Extraterrestrial Intelligence. At the
heart of SETI, there is a mystery known as The Great Silence, or The
Fermi Paradox, named after the famous physicist Enrico Fermi.
Essentially, he asked “If they exist, where are they?”.
Some
quite cogent arguments maintain that if there was extraterrestrial
intelligence, they should have visited the Earth by now. This story,
a bit tongue in cheek, gives a fictional account of one explanation
for The Great Silence, known as The Zoo Hypothesis. Are we a
protected species, in a Cosmic Zoo? If so, how did this come about?
Read on, for one possible solution to The Fermi Paradox.
The
short story is about 6300 words, or about half an hour at typical
reading speeds.
https://www.amazon.com/dp/B076RR1PGD
https://www.amazon.co.uk/dp/B076RR1PGD
