Saturday, 3 January 2009
and then i was thinking about the real projective plane
...a two dimensional surface that cannot be truly built in three dimensions because it must pass through itself without having a hole in the surface... well that and a program i wrote back in the day which created Lawson Surface of Genus 4 cross hatched images with no edges and no vertices and no flat surfaces (envision a really weird looking set of donuts and 
spheres with that criteria), mind you i did this to visualize atomic bonds for silicon isomers for a chemistry research project for a prof i was a research asstistant too on a very old IBM PC with 12K memory and a only 128K diskettes to store both the program and the data tables on! In case your wondering I wrote the code in Lotus 123. Anyway we managed to come up with some pretty cool looking graphics on that old PC but ours was no where as nice as the stuff they do today, like the picture to the left.
(haha, don't even ask, i had one of those moments where i remembered something from college dayz but anyway here is a excellent explanation with pics: Real projective plane and better yet a video demonstrating RPP in 3 space.) and it led me to these really cool pics are about tesselations which is when you completely cover a plane surface with tiles. I thought it was interesting to note that equilateral triangles are one of the three regular polygons that tile a plane. the other two being the square and hexagon because that made me remember something neat i wanted to share with everyone. (Funny how that works heh...) And that is fractals and Fibonacci numbers.
Ah the common spinach leaf, so unassuming so good to eat! What is so special about this leaf? If you take a look at the leaves of your plants, you will notice that they are symmetrical! and based on a very special set of numbers!
Fibonacci Numbers give us a mathematical basis for how leaves grow on a plant, which is called Phyllotaxis. This and Leaf Geometry help the plant maximize its exposure to the vital sun light which fuels its growth. 92% of all plant leaf growth is based on Fibonacci numbers. Math, not just for kids and its nutritious!
Coxeter, in his "Introduction to Geometry, Wiley 1961" said, "it should be frankly admitted that in some plants the numbers do not belong to the sequence of f's [Fibonacci numbers] but to the sequence of g's [Lucas numbers] or even to the still more anomalous sequences" hmm, gee, i wonder what that anomalous sequence might be... oh yeah its a fractal!
Fractals are fun! A fractal is generally "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole," a property called self-similarity.
Approximate fractals are easily found in nature. These objects display self-similar structure over an extended, but finite, scale range. Examples include clouds, snow flakes, crystals, mountain ranges, lightning, river networks, cauliflower or broccoli, and systems of blood vessels and pulmonary vessels.
I don't know about you but that number looks good enough to eat!
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