Today’s π day. March 14 is 3/14 and π is 3.14…. 3.1415926535 was as far as I cared to memorize, and even that was a waste of time.
This is a perfect day to share any π-related stories. My favorite is the day I learned about radians as an angle measurement and that a circle was 2π radians. 11th grade, I think. That had a profound effect on my future development as an adult.
Additional reading: piday.org is supposed to be the “official” π day site (at least, it’s the first site that showed up in Google).
March 14th, 2008
9 comments
Posted by Donnie
Filed under Mathematics
In college football (American), a ton of people start calling every year for a playoff system to replace the BCS. (Quick primer: usually, a champion of a sports league is determined by a playoff between teams who did the best during the season. Instead, the BCS determines the top two teams by a combination of human and computer polls, essentially creating a 1 game playoff.)
What people don’t think about is that a playoff system is not very likely to crown the best team in the country as champion.
Let’s take an 8-team playoff, for example, and have University of A to be “objectively” the best team in the country. And let’s assume the UA team is so good, it will defeat other elite ( top 8 ) teams 75% of the time. UA will have to win three games in a row, which it has a 42% chance of doing. This speaks highly of UA, but their chances of being recognized as champions are still worse than a coin flip.
The 75% number is higher than can be expected. 60% may be closer to realistic, giving the best team a 21% chance of winning the crown.
If that’s the system people want, then by all means they can clamor for it. As long as they recognize it’s unlikely that the best team in the country will be known as the champions. (College basketball is even worse for determining the #1 team, but everyone likes gambling on March Madness so no one brings it up).
My biggest beefs with the BCS system are:
1. Human pollsters, who cannot have knowledge of every single game played, cannot possibly hope to compare all teams adequately, and have ingrained biases, are given greater weight than the computers.
2. The computer polls are forced not to considered margin of victory in their calculations. This can throw them out of whack, as this story on Jeff Sagarin’s rankings indicates (at one point, North Dakota State was ranked in the top 20). I guess for some reason, the BCS thought that maybe these programmers wouldn’t have developed algorithms that made sure a 52-point victory didn’t mean more than a 35-point victory.
In other words, the BCS has to give computers less weight, because their own rules make computers less accurate in predicting the top two teams.
If it were up to me, I’d eliminate computer poll restrictions, and completely ignore the humans. The only check would be at the end of the year, a council would get together that could veto the computers’ selection if 75% agreed, at which point the human polls would be used to determine the game.
This is mostly off-the-cuff, so blast away with holes in this thinking.
January 8th, 2008
8 comments
Posted by Donnie
Filed under Mathematics, Sports
Dr. Gene Ray, Cubic continues to enlarge the quotation rotation with more hits:
“ONEism is demonic Death Math. I have so much to teach you, but you ignore me you evil asses.”
“Your opposite eyes were moved to 1 corner to overlay for single perspective, but that corrupts your Opposite Brain. Hey how about making a donation to Gene Ray, 2580 Highland Pointe Dr. Cumming, Georga 30041.”
October 23rd, 2007
2 comments
Posted by Donnie
Filed under Mathematics, Science/Technology
So now we need to be told more food means more calories? Whoever created the chart accompanying the story must have decided Americans are that bad at math. (To be fair, the story itself seems fine.)
October 19th, 2007
3 comments
Posted by Donnie
Filed under Mathematics
Probably a fair portion of you have heard the name John Conway in conjunction with a game called Life. It’s amazing how the few simple rules can generate tons of interest, research, and analysis.
Playing Life:
This isn’t a game where you take turns to try to obtain an objective; think of it more as a simulator. You start with a grid of cells, a universe (really small for this example). Fill in the cells however you wish. Empty cells are dead, while the grey cells are alive:
A game of Life, turn 0
Each cell has eight neighbor cells surrounding it. The following rules determine whether a cell will be alive on the following turn:
- A dead cell with 3 living neighbors gains life.
- A live cell has 2 or 3 living neighbors stays alive.
- A live cell with 0 or 1 living neighbors dies of loneliness.
- A live cell with 4 or more living neighbors dies of overcrowding.
So taking our starting position above:
What will happen for turn 1?
The green cells are dead cells that will live next turn, and the red cells will be killed.
The next few turns you can see here:
It’s still far from certain what the fate of this universe will be. Sometimes, all the cells will eventually die. Othertimes, the universe will remain stable, or continually expand. There’s no good way to predict how it will end up for an arbitrary starting position.
Another complexity is how we treat the edges of the universe. You could have the universe “wrap around”, so the cells on the far left and far right edges (as well as top and bottom) would actually be neighbors.
I could hack up a basic implementation of Life pretty quickly, but it’s been done so many times, I’ll just link you Johan Bontes’ program. According to the website it’s an awesome program (plus it’s free).
Or, if you have time and graph paper, I suppose you could do it by hand.
September 15th, 2007
5 comments
Posted by Donnie
Filed under Mathematics
