EMAIL: agage@csee.usf.edu NAME: Aaron Gage TOPIC: Microcosms COPYRIGHT: I SUBMIT TO THE STANDARD RAYTRACING COMPETITION COPYRIGHT. TITLE: Mathematics of Scale COUNTRY: USA WEBPAGE: http://www.csee.usf.edu/~agage 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 VIEWING RECOMMENDATIONS: 30fps, 320x240, 2520 frames, larger resolutions OK. 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. 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.