As everyone no doubt knows, today is Pi Day. That is, March 14 is Pi Day (it’s the
fourteenth day of the third month, which is sometimes written as 3.14, the value
of the transcendental number Pi, to two decimal places).
Speaking of Pi, here’s a bit from The Witches’ Stones
Book 1: Igniting the Blaze that actually references Pi:
“I think I can get us into orbit
before the Organization ship catches us,” Steph said. “That might buy us some time, if I can keep
the planet between us and the hounds’ ship.
We used to call it playing orbital tag back in training. It’s a matter of matching speed and
maneuver.”
Steph entered some commands into
the navigation computer. “It’s pretty
simple, really. When the distance
between us and that ship is the same as pi times the distance between us and
the planet, we break off into a circular orbit.
The hounds will have to do the same – they’ll have too much momentum to
alter course and try to cut in front of our bow as we finished the first
orbit. Besides, even if they could do
that, we’d see what they were up to right away and be in the better firing
position to take them out with our bow-chaser laser cannon. In the olden days, I think they called it
having the weather gauge on the opponent’s ship.”
Basically, he is ensuring that he can maneuver his ship in
such a way that he can get the Planet of the Amartos between himself and the
better armed pursuing enemy ship, thus screening his ship from his enemy’s fire. Then,
as long as he can keep the planet between them, his ship is safe, orbit after
orbit (adjusting his orbit as necessary, when he detects that his opponent is
doing so). So, a bit of simple Pi-based geometry still comes in handy in the 30th
century.
Anyway, that’s the Pi-related book plug. Now onto other interesting astrophysical and Dodecahedron
Books related matters involving pi.
Naturally, astrophysics is full of matters that involve Pi
(via circles, spheres, spirals, etc):
·
The fundamental forces, such as gravity and
electromagnetism are 1/R**2 laws, that fall off equally in all directions (i.e.
with spherical symmetry). That’s the
underlying cause of a lot of the things mentioned below.
·
Many orbits are circular, or close to it.
·
Same with ring structures, such as the rings of
Saturn.
·
Stars, planets and other large bodies tend to
become spherical if they have enough mass.
Indeed that’s one of the defining features of a planet, rather than an
asteroid.
·
The rotation of astrophysical bodies around
their spin axes is usually some variation of circular motion.
·
Galaxies often have a beautiful spiral symmetry.
·
The coordinate systems used to locate object in
the sky are based on something called the celestial sphere, so Pi comes in
there too.
·
Not surprisingly, much of the math used to
analyse astrophysical data involves Pi, from geometry to trigonometry to
advanced techniques like Fourier analysis and power spectrum analysis.
But what about Pi and the dodecahedron, you ask? Is there a connection? That’s a good question for a Dodecahedron
Books blog on Pi Day, which we will address below.
Dodecahedron Books is named after the three dimensional
geometric shape, the dodecahedron. This
is composed of 12 equal pentagons, which are folded up together into a
beautiful characteristic shape. A
pentagon, of course is the two dimensional shape that is made of 5 equal sides,
which is itself redolent of fascinating bits of mathematical arcana, mostly
connected with a number known as Phi, or the golden mean. There is a fair bit of lore, ancient and
modern, about these shapes, which we will explore in a later blog. For now, though, let’s focus on Pi, rather
than Phi, and see if we can find a connection between Pi and the pentagon
and/or dodecahedron.
You can inscribe a polygon (a regular two dimensional shape)
in a circle, then inscribe that circle into another polygon. Using the example of the pentagon, that process
looks something like the picture below:
As you can see, the perimeter of the inner pentagon will be
less than the circumference of the circle in which it is inscribed, while the
perimeter of the outer pentagon will be larger than that same circle. So, we would expect that the average of those
two pentagon’s perimeters should approximate the circumference of the circle,
more or less.
Let’s assume that the radius of the circle is equal to
exactly 1 unit. With a little high
school geometry and trigonometry, we can see that the perimeter of the inner pentagon
is given by:
2 X sin(36 degrees) = 2 X .58799 = 1.775571
There are five sides to a pentagon so the perimeter is
= 5.877853
A little more high school geometry and trigonometry shows us that the perimeter of the outer pentagon is given by:
2 X tan(36 degrees) = 2 X .72654 = 1.453085
There are 5 sides to the outer pentagon, so that gives a
total perimeter of 7.265425.
If we average those two, we get 6.571639.
The value of the perimeter of the circle is of course twice
the radius times Pi, or:
2 X 3.1459 = 6.283185.
2 X 3.1459 = 6.283185.
So, sadly our estimate isn’t really very close, differing
from Pi by about 5%. Oh, well let’s try
something with more sides than dodecahedron, say an octagon (8 sides). If we go through the same process, adjusting
the size of the angles accordingly, we get an estimate for Pi of 3.18759. So, that is considerably better.
Using a spreadsheet to carry this on, we get the following
table for various numbers of sides for our polygons:
|
Number of Sides
|
Pi Estimate
|
Diff from Pi
|
Pct Diff
|
|
3
|
3.897114
|
0.755521663
|
24.049001%
|
|
4
|
3.414214
|
0.272620909
|
8.677793%
|
|
5
|
3.285819
|
0.144226797
|
4.590882%
|
|
6
|
3.232051
|
0.090458154
|
2.879372%
|
|
7
|
3.204104
|
0.062511599
|
1.989806%
|
|
8
|
3.187588
|
0.045995325
|
1.464077%
|
|
9
|
3.176957
|
0.035364046
|
1.125673%
|
|
10
|
3.169683
|
0.028090799
|
0.894158%
|
|
20
|
3.148189
|
0.006596400
|
0.209970%
|
|
50
|
3.142630
|
0.000330092
|
0.010507%
|
|
100
|
3.141851
|
0.000258602
|
0.008232%
|
|
500
|
3.141603
|
0.000010336
|
0.000329%
|
|
1000
|
3.141595
|
0.000002584
|
0.000082%
|
|
5000
|
3.141593
|
0.000000103
|
0.000003%
|
|
10000
|
3.141593
|
0.000000026
|
0.000001%
|
So, as you can see, the estimate gets better and better as
we add more sides to our polygon. By the
time we have arrived at the 500-gon (100 times our original pentagon, which we
could call the hecto-pentagon, because that sounds cool), the estimate for Pi
that we get is almost indistinguishable from 3.14159, at 3.14160. We are within one part in three million.
Interestingly, a graph of the above shows a function that
looks a lot like our old friend, the power function. It is interesting how that comes up in all
sorts of places. Of course, a
mathematician might say that is a result of the fact that the trig functions
can be approximated by power functions.
Indeed, you might say in this case, that Pi is baked right into the pie.
If we used this estimate of Pi for some simple geometry on
astrophysical scales:
·
At the distance of the Earth to the Sun (one
astronomical unit, about 93 million miles or about 150 million kilometres) we
would be out by about 30 miles or 50 kilometres.
·
At the distance of four light years (a trip to
the nearest star besides the sun), we would be out about eight million miles,
or about twelve kilometres.
·
At the distance from the Earth to the center of
the galaxy, we would be out about 40 billion miles or 60 billion kilometres.
·
At the distance to the Andromeda galaxy, we
would be out about 4 trillion miles, or about half a light year.
So, if you are planning a really long trip, perhaps you
should think of approximating Pi with a little more accuracy than the 500-gon
can give you.
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