Basically just visual demonstrations of how awesome math is.
Life as an Algorithm
I just found an entire Tumblr dedicated to GIFs created from Conway’s Game of Life algorithm. If you know about Conway’s GoL, then you understand why this is so cool. If you don’t know about it, well …
It starts with a simple question: How do you model a collective? That could mean the population of a species in its habitat, or the dynamics of a swarm of bees, or the pattern of bacterial growth as a colony senses a food source. These are (relatively) simple problems to observe, but very difficult to predict or model.
Back in the 1970’s, John Conway was able to use these new-fangled things called “computers” to create self-replicating “cells” within a digital world. The user supplies a few rules (such as: how a cell survives or grows depending on its neighbors, the initial number of groups, the shape of the world, the starting density) and lets the algorithm do the rest.
It creates a dizzying array of interesting population dynamics, with implications from evolution to synthetic biology.
It’s self-replicating digital worlds such as these that have inspired the new science of swarms, in which scientists hope to unlock how groups lacking individual intelligence (fish schools, bird flocks, locust swarms) can act as superorganisms and exhibit complex behaviors.
An algorithm of emergence. Check out Game of Life Algorithms to see more.
If you roll a circle inside one 3 times its size, it will actually trace out a 4 pointed star shape called an Astroid (this shape is traced out in the animation in orange). But what if inside the smaller circle, there is an even smaller one tracing out a smaller Astroid? This animation shows the intricate shape that is generated by adding the effects of all the Astroids. [code] [also]
Unravel by Kevin Weber
“Long exposure study of wheels on cars passing by, except not following the cars as usual. Turned into a uniquely abstract and colorful album.”
For each natural number n, we draw a periodic curve starting from the origin, intersecting the x-axis at n and its multiples. The prime numbers are those that have been intersected by only two curves: the prime number itself and one.
Next Summer Blockbuster: Mandelbrot 3D!!
Pure math, pure beauty
Man, I love fractals. Fractals are an example of an infinitely repeating set of numbers, really nothing more than a mathematical description of symmetry. But when their patterns are rendered into light, our eyes convert shapes to nerve impulses, and our brains make sense of it all, why do we find them so attractive? Especially when to most people, including me, the math is pretty much a mystery.
These incredible sculptures cycle on down to infinity, their points, bulbs and peaks are the same no matter how close or far you magnify them. Above is a couple of 3-D fractal renderings from the Mandelbulb Project. Whereas most fractals are 2-D representations, these more complex renderings grow like living ice sculptures or spiny corals. It’s no coincidence that we associate their shapes with living things. These fractals show up in places as unexpected as broccoli.
Visit the Mandelbulb Project website if you’re into things like “awesome videos” and “holy crap that’s amazing” or “whooooooooa”.
Then answer this: Why does something that’s purely mathematical strike us as so purely attractive?
(Thanks to this post at Bad Astronomy for the inspiration)
Ferrous Wheels and Quantum Corrals
I was inspired by a very bad pun I heard this morning. Hidden inside of it is an important lesson on the quantum nature of electrons.
What you’re looking at is a famous experiment in which iron atoms were manipulated by the atomic needle tip of a scanning tunneling microscope until they formed a perfect circle on a copper surface (the bottom one is an artist’s interpretation, however). That’s already cool, manipulating individual atoms to form a sort of “ferrous wheel” of their own … but it gets cooler!
See the waves in the center of the corral? I’m reminded of the ripples on the surface of a pond when a pebble is thrown in. They form on the atomic scale due to the quantum nature of electrons. If you’ve religiously read everything I’ve ever posted, you know that atoms are mostly empty space and that electrons exist like waves rather than tiny points flying around. Copper’s electrons exist in a very particular pattern of orbitals that is represented like this:
When the iron atoms in that corral up there form their circle, the electron waves in the copper surface interact and intersect in such a way that a standing wave is formed. You are literally looking at the quantum nature of electrons! Each electron in an orbital can only exist in a certain wave, otherwise the other electrons in that atom would interfere with it.
Let Brian Cox tell you more about those standing waves with a little help from his friends.