Fans of the Simpsons know the
term “floor pie” – a notable episode showed pie obsessed Homer being caught in
Bart’s diabolical floor pie trap.
Hilarity then ensued.
Pi ~ (2L*n)/T*h,
where
L=length
of needles
n=total
number of needles dropped
T=distance
between lines
H=number
of needles that crossed lines
In real life, that would mean
something like scattering nails a hardwood floor and counting those that cross
the lines, a pretty tedious process. In
computer life, though, a relatively short bit of code in a computing language,
or an excel model, should do the trick.
I chose the latter.
Basically, my excel spreadsheet
just created a large number of virtual needles of length L, centered at random
x and y co-ordinates, within a given area having spacing T between lines, and
tested to see which of them had a top or bottom above or below the lines, after
a random rotation.
As long as excel’s random number
generator is actually reasonably random (sometimes I think it is a little fat
in the tails), this should be a reliable simulation. There are some sine and cosine calls, so
floating point cut-offs would eventually limit the accuracy of the simulation. I am not sure when these factors would kick
in, but given the slow convergence of my Monte Carlo, it could be a moot point.
Of course a physical experiment
would have its own uncertainties, such as whether a given needle actually
crossed a line or not. If one attempted
to use this method to calculate pi to arbitrary accuracy, surely there would be
ambiguous cases, which would limit the accuracy of the experiment.
My results showed that the process does approximate pi, but the convergence is slow, to say the least. Even with 10,000 throws, a typical simulation only came within about 0.1% of pi. Here are some results. The numbers across the top show the number of needles cast in the simulation, and the numbers along the side show different runs of the simulation. So, for example, this table shows 10 trials for each count of needles cast (10 needles, 100 needles, 1000 needles, 10,000 needles) :
10
|
100
|
1000
|
10000
|
|
1
|
0.00000
|
3.52941
|
3.31492
|
3.147954
|
2
|
3.00000
|
3.75000
|
3.27869
|
3.12989
|
3
|
3.00000
|
3.15789
|
3.01508
|
3.09598
|
4
|
6.00000
|
4.00000
|
2.63158
|
3.25556
|
5
|
3.00000
|
2.50000
|
3.14136
|
3.06748
|
6
|
6.00000
|
2.60870
|
3.29670
|
3.15956
|
7
|
3.00000
|
4.00000
|
2.84360
|
3.05033
|
8
|
3.00000
|
3.33333
|
3.19149
|
3.12826
|
9
|
3.00000
|
3.52941
|
3.09278
|
3.14795
|
10
|
6.00000
|
2.14286
|
3.48837
|
3.15956
|
3.60000
|
3.25516
|
3.12946
|
3.13425
|
|
Diff from Pi
|
0.45841
|
0.11357
|
-0.01214
|
-0.00734
|
Pct Diff
|
14.6%
|
3.6%
|
-0.4%
|
-0.2%
|
As you can see the accuracy of
the estimate of pi tends to go up with the number of needles cast. For the 10,000 needle Monte Carlo, the 10
trial overall average for the estimate of pi is accurate to within 0.2%. That’s still not very close, though.
The Monte Carlo is set up so that
the length of the needle compared to the distance between the lines can be
varied. There appears to be some
evidence that the simulation works best if the length of the needle is about
half the spacing between the lines. I am
not sure if that’s a real effect, though.
It might just be normal statistical variation. It’s hard to see why that should matter in a
computer simulation, though it might make a physical simulation easier to
perform, in practice.
L/T
|
Est
|
Pi
|
Diff
|
Pct Diff
|
0.1
|
3.144704
|
3.141593
|
0.003112
|
0.10%
|
0.2
|
3.135268
|
3.141593
|
-0.00632
|
-0.20%
|
0.3
|
3.187073
|
3.141593
|
0.04548
|
1.45%
|
0.4
|
3.141697
|
3.141593
|
0.000104
|
0.00%
|
0.5
|
3.140686
|
3.141593
|
-0.00091
|
-0.03%
|
0.6
|
3.143699
|
3.141593
|
0.002107
|
0.07%
|
0.7
|
3.14176
|
3.141593
|
0.000168
|
0.01%
|
0.8
|
3.131017
|
3.141593
|
-0.01058
|
-0.34%
|
0.9
|
3.148454
|
3.141593
|
0.006861
|
0.22%
|
An Italian mathematician, Mario
Lazzarini, claimed to have performed the experiment in 1901. He got pi to an amazing accuracy, 355/113,
which is accurate to 5 decimal places. To be charitable, he was awfully lucky. There is a lot of doubt about whether the
experiment was actually performed, and if so, whether the results were cooked
ahead of time. The last link below has a
detailed analysis of his results, including some examination of whether his
results were “too good”.
Sources:
The Pleasures of Probability, Richard
Issac, Springer
---------------------------------------------------------------------------------------------------------------
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.”
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