Wednesday, January 8, 2014

Bull

The earliest stories we have of the minotaur, from the Minoan civilization on Crete, are quite different from the story told by Ovid. Even in Ovid's version, the minotaur is simply part bull and part man; the bull-headed man was not a settled image until the Rennaisance. In older stories he was a man with the powerful shoulders, back, and neck of a bull-- a strong giant of a man. He was also said to have been companions with a bull, though unlike Ovid's version the relationship seems to have been one of friendship, similar to the relationship of Gilgamesh and Enkidu. The bull was described as "sky-colored" which has traditionally been interpreted as white, but could also mean blue, as in this sculpture.
The minotaur was a traditional Greek hero similar to Herakles, accomplishing various feats. Hecataeus of Miletus, when describing the ocean as a river encircling the world, recounts that the minotaur traveled on a raft around this river. In another tale, it is said that his words were frozen into solid form as he spoke them, perhaps a reference to the invention of writing.
In Minoan imagery, the minotaur is always associated with a two-headed axe. The island of Crete is largely bare of trees; the minotaur is said to be responsible for clearing the island with his axe.
The Romans called the minotaur "Paulus," meaning small or humble-- an ironic name similar to "Little John" in the Robin Hood myth. The minotaur was usually depicted wearing cloth with a red and black pattern, similar to a checkerboard or a scottish tartan.

Monday, January 6, 2014

Beginning Calculus for Pirates

Let's say you want to download a perfectly legal, uncopyrighted 1.2GB file using bittorrent. There is a graph at the bottom of the screen in your downloading software that shows the number of MB per second downloaded. When the download is going well, the graph line is up high, meaning you're downloading a high number of megabytes every second. When the line is near the bottom of the screen, that means you're not getting very much data each second and it's gonna be a long time before that file is downloaded.
With me so far?
Here's the big question: what is it about this graph that shows how much, total, has been downloaded so far?
It's not the height of the graph at any one point; that's showing how fast data is coming down the pipe.
Suppose you were getting 5MB per second for 100 seconds. Then you would have 500MB downloaded. Or you could get 1MB per second for 500 seconds, and still end up with 500MB total downloaded. A little bit per second for a long time, or a lot per second for a short time. You just multiply the height (MB per second) by the length of the graph (seconds) to get the total downloaded (in MB).
In other words, you calculate the area under the graph to get the total amount downloaded.

The area under the graph of download rate gives the total amount downloaded.

The area under the graph of the rate gives the total.

That sentence is what calculus is all about. "Taking the integral" just means calculating what the total area under the curve is at each moment in time. Newton's big insight was that the graph of the rate and taking integrals were related in just this way-- If you want a graph of how much total has been downloaded so far and all you have is a graph of how much is being downloaded each second, you can "take the integral."
All of the rest of first year calculus is just tricks and methods to calculate this more quickly, or to go in the opposite direction and get the graph of MB per second from the graph of how much total has been downloaded at each point. That's called "taking the derivative."