Pi Day 2019 –Shooting Arrows at a Target to Calculate PI
It can be
amusing to produce simple Monte Carlo simulations in Excel, and such
simulations can be useful for many things, such as numerical solutions for
integrals of functions. Excel is
probably not the most elegant or efficient program to use, but it is quick and
easy to see what is going on, without needing any actual programming per se.
One such
simulation can be used to estimate the value of Pi. It is a simple matter:
· Create a random set of X and Y coordinates between
-1 and +1.
· Calculate the Euclidean distance of each set of
coordinates from the origin (i.e. use the Pythagorean Theorem of D=sqrt(X*X +
Y*Y).
· Determine whether the point is within the unit
circle (D<=1), or outside of the unit circle. Count the cases for each.
· Determine (in/(in + out)), multiply that by 4 and
that’s your estimate of Pi. In other
words, this will be the ratio of a unit circle that is inscribed in a unit
square.
Here is a
graph of one run of the simulation, using 1000 points, along with some results.
Granted,
this 1000 point run resulted in a pretty rough estimate of Pi (3.18 vs 3.14, or
about a 1% error). However, if you had a
computer with enough memory, you could estimate pi to as many decimal places as
you liked with this method, by increasing the number of points used. Be warned though;
it converges very slowly, so you would need an awesome computer and/or a lot of
time.
If we do
100 runs, using this technique, we get the following picture. As you can see, the majority of the estimates
are within 1% of the true value of Pi, though some are very close, and some are
quite far.
Looking
at a histogram of the above results we see that the mean value is now much closer
to PI, a result expected from the Central Limit Theorem (though a lot more runs
of the experiment are obviously needed before the graph takes on the classic Gaussian
shape - i.e. Bell Curve).
If we
step up the number of points in the simulation, the estimate of Pi improves
quite substantially. Using about a
million points, the error was generally within about .005%, as the graph below
shows.
If we do
a histogram of these 100 runs of 1 million points, we get the picture below.
Stepping
up our 100 run simulation from 1000 points to 1,000,000 points per run
increased our accuracy of PI by about 100 times, if we take the mean value for
each experiment (from .094% to .001%).
That’s more than I would have expected – my initial thought was that the
accuracy should go up by about 30 times (the square root of 1 million over the
square root of one thousand). There’s
probably a good explanation for this, but I’m in a hurry to finish the blog by
Pi Day, so that may remain a mystery for the reader to ponder.
Using our
million point estimate of PI, our average estimate is 3.14163, which is close
to the usual approximation of Pi that most of us can actually remember,
3.14159. So, that’s good to within one
part of one hundred thousand. For most engineering
purposes, that’s probably good enough.
For example, if you were digging a 1 km tunnel, and needed an accurate value
of Pi to calculate an angle of attack, that would give you an error of only 1
centimeter over that distance.
Here’s what
Herbert Schubert, a math prof, said in 1889 about the actual need for accuracy
in determining the value of Pi, and the practical futility of calculating Pi to
tremendous accuracy:
Conceive a sphere constructed with the earth at its
center, and imagine its surface to pass through Sirius, which is 8.8 light
years distant from the earth [that is, light, traveling at a velocity of
186,000 miles per second, takes 8.8 years to cover this distance]. Then imagine
this enormous sphere to be so packed with microbes that in every cubic
millimeter millions of millions of these diminutive animalcula are present. Now
conceive these microbes to be unpacked and so distributed singly along a
straight line that every two microbes are as far distant from each other as
Sirius from us, 8.8 light years. Conceive the long line thus fixed by all the
microbes as the diameter of a circle, and imagine its circumference to be
calculated by multiplying its diameter by π to 100 decimal places. Then, in the
case of a circle of this enormous magnitude even, the circumference so
calculated would not vary from the real circumference by a millionth part of a
millimeter. This example will suffice to show that the calculation of π to 100
or 500 decimal places is wholly useless.
Beckmann, Petr. A History of Pi (p. 101). St.
Martin's Press. Kindle Edition.
That being
said, the estimate from this method is still pathetic compared to what can be given
by other methods, especially numeric methods, such as the many convergent
infinite series, or even the Archimedean technique of inscribing nesting polygons
within circles.
Still,
this is a method that could have been used in the ancient world (I am not
saying it was, it’s just an interesting thought experiment). One could imagine a pharaoh directing his
priests to perform such an experiment, perhaps by getting archers to shoot at a
distant square target, and counting hits within the inscribed circle versus
outside of the circle.
The
archers would have to be instructed to not aim at the center, of course, as
that would bias the results (maybe a “rain of arrows”, as at Agincourt, would
be best). And, the 1000 arrow experiment
repeated 100 times is feasible – but even a powerful pharaoh would be hard
pressed to do the million arrow experiment.
Though they did some amazing things, like building vast pyramids, when
properly motivated. With a sufficiently mathematically inclined god to
propitiate, you never know. Perhaps
there is a planet out there, where this has taken place.
And here
are a couple of other Pi Day posts that I have done, with similar Pi-related experiments:
Calculating
Pi from Probability Theory (Buffon’s Needle): https://dodecahedronbooks.blogspot.ca/2017/03/pi-day-floor-pie-and-floor-pi.html
Calculating
Pi via Nested Polygons:
And as
always (courtesy of my wife's excellent pumpkin pies, who also wrote the Kati of Terra and Witches' Stones series, shown below, which you should consider reading, too):
Sources:
Beckmann,
Petr. A History of Pi (p. 101). St. Martin's Press. Kindle Edition.
So, now
that you have done some 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.
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 (they’re the most interesting type – the novelist who wrote
the books is pretty feisty, too).
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.”
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.
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.
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