TITLE: Mathematics of Scale
NAME: Aaron Gage
COUNTRY: USA
EMAIL: agage@csee.usf.edu
WEBPAGE: http://www.csee.usf.edu/~agage
TOPIC: Microcosms
COPYRIGHT: I SUBMIT TO THE STANDARD RAYTRACING COMPETITION COPYRIGHT.
MPGFILE: amgmos.mpg
ZIPFILE: amgmos.zip
RENDERER USED:
POVray 3.0 for Linux
TOOLS USED:
Terrain Maker, povchem 1.0 for DNA strand, mpeg_encode
CREATION TIME:
Five weeks or more (render time unknown)
HARDWARE USED:
i486DX2/66 and Pentium Overdrive 83, 32MB each, Linux 2.0.33
ANIMATION DESCRIPTION:
One of the interesting aspects of nature is the recurrence of patterns
at all scales. The shape of a circle or sphere appears everywhere, due to
the low-energy state that it provides in physical systems. The spiral appears
in many places, from hurricanes to sea shells to water running down the drain,
to the horns of a ram. This animation attempts to show how looking closer
and closer into the fabric of the world only exposes more of these patterns
at each level.
In order to clarify the segments, here are some clues. In the
derivation segments (the ones with numbers) the objects and numbers are
linked by color. That is, a green line and a green number refer to the
same quantity. For Pi, this shows the theoretical value of Pi and how the
correct value is approached as the number of sides on the polygon increases.
For the Golden Ratio, this shows the various relationships that produce this
fascinating number.
VIEWING RECOMMENDATIONS:
30fps, 320x240, 2520 frames, larger resolutions OK.
DESCRIPTION OF HOW THIS ANIMATION WAS CREATED:
The nice thing about POVray's scene description language is that
it made many parts of this animation very easy. The derivation of Pi,
for instance, was done by writing a loop to create a polygon (out of
cylinders) of any number of sides, calculate its circumference and
diameter, and compute and render the result. So I managed to let POVray
do most of the dirty work by just automating this process.
All of the golden spirals that were used required a loop that
would automatically place the individual points, and deriving this loop
took some experimentation. These calculations should be available in the
archive.
The planet that is zoomed into was taken directly from the
POVray tutorial, since it looked good and required very little modification
to use. The hurricane shape is a collection of seven spirals (made out
of semi-transparent spheres of decreasing radius). The island in the
eye of the hurricane was needed in order to place the shell somewhere;
this was a height field with some texturing. The movement of the ocean
was done by moving different normal patterns over time, since I really
did not want to model real water (or have it remain static over time).
At the molecular level, I tried to tie back to the polygons used
to approximate Pi in the first segment. I made what should pass for a
methane molecule (which is also triangular on four faces). The cube
molecule is something I made up -- I don't know if anything like that
actually exists, but I wanted to have the square. I skipped the pentagon
shape, since I think I could have found it as a facet of a huge molecule,
but decided that it would not be worth the effort. The six-sided molecule
is cyclohexane, which I pulled off of the povchem web page. Finally,
the double-helix of human DNA was borrowed from the Protein Data Bank (PDB)
archives (it is protein 149d) and converted to POV format with povchem.
As for the spiral shape being found in the DNA molecule, I actually found
evidence that this is the case (at least approximately).
With regards to the length of the animation, I realize that it
is perhaps longer than was strictly necessary, but with the new 5MB size
limit, I really wanted to take my time getting my point across. Besides,
the idea of pushing the record for the longest submission to date is
always a factor.
Here's a brief description of the numbers involved in the first
couple of segments:
Pi, the ratio of the circumference (distance around) a circle its diameter
(distance across): 3.14159265358979323846264338327950288419716939937510582...
Approximation of Pi. The arctangent of 1 is equal to Pi/4, so an approximation
of Pi using an approximation of the arctangent function is:
4 - 4/3 + 4/5 - 4/7 + 4/9 - 4/11 + ...
This is actually a very slow way of calculating digits of Pi, since it
converges very very slowly. There are methods which are much faster.
It is the regularity of this pattern that I found interesting.
Golden Ratio, Phi, a number that is common in natural shapes and patterns.
The value of this ratio is equal to 2*Cosine(Pi/5) = (1 + sqrt(5))/2.
This is the value of x that satisfies the equation (x - 1) = (1/x).
Its conjugate, (1 - sqrt(5))/2, is also very interesting. Here are the
numeric values:
Golden Ratio: 1.618033988749894848204586834365638117720309179805762862135...
Conjugate of GR: -.618033988749894848204586834365638117720309179805762862135...
You may notice that if you invert either number, you get the other (though
the sign may be different). This property alone is pretty cool. It gets
more interesting, however.
Fibbonacci numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...
Most people learn this series as a simple mathematical progression. Each
new value is equal to the sum of the two previous numbers. This, along
with the factorial function, are often used to teach the concept of recursion.
So why did I include this series? Well, let's go back to the Golden Ratio (GR)
and the Conjugate of the Golden Ratio (CGR). What if you want to calculate the
Nth digit of the Fibbonacci series without having to calculate any of the
previous ones? Try this:
F(N) = (GR^N - CGR^N)/sqrt(5)
The Fibbonacci numbers appear in nature, perhaps more than anyone notices.
The next time you cut open an apple, or a bell pepper, or tomato across the
middle (instead of from top to bottom) take a moment to count how many
compartments you find in the cross section. I am suggesting it will tend
to be 3, 5, or 8. The arrangement of sunflower seeds, pinecone points,
buds on asparagus, etc., all exhibit these patterns.
Another neat thing about the golden ratio is that it can be used to plot
the golden spiral, which (as the animation shows) appears in a number of
natural shapes. The golden rectangle often shows up in architecture
and paintings, as it has a number of visually pleasant properties. And,
of course, there is the mystical shape of the five-pointed star (which was
turned into the pentagram by the cult followers of Pythagorus). If you
take the five-pointed star apart (nevermind the circle, they added that
for other reasons), you can actually calculate the golden ratio a large
number of ways (between ten and fifty, I think, perhaps a lot more).
This animation has been a very educational experience for me. I derived
the Golden Ratio (twice) on my own, and verified it after the fact from
a textbook. The shapes that contain the spiral have captivated me, and
some of the history involved (like where the shape of the pentagram came
from before its meaning was distorted by superstition)
has been rather illuminating. I am only hoping that the viewer feels
some fraction of the intricacy of these quantities in the natural world.
All of this started when I saw "Pi: the movie" when it was being shown
during the summer of 1998. I strongly suggest that anybody who finds
this stuff interesting to visit http://www.pithemovie.com -- not only
is it a well-done site, representing a very interesting movie, but it
also has links to numerous mathematics sites.
If you actually took the time to read this, the animation should have
had time to download :) Thanks for your patience.