Dominos as logic gates
And the Domino half-adder
Category Archives: Math
Knot Too Shabby
One of the reasons knots have given mathematicians fits is that the same knot can appear in very different guises. Tug here, tug there, and soon a knot will become unrecognizable, but remain fundamentally unchanged. To allow a knotted string to wiggle around without danger of untying, mathematicians seal its two ends together, making it a knotted circle. The first question mathematicians have to answer is simply, when are two knots really, secretly the same?
The dream is to create a sort of machine: Send in one of these looped knots, and out pops some result that would be the same regardless of the particular configuration of the knot. Because the answer wouldn’t vary with the arrangement of the knot, such a machine is called a “knot invariant.” And indeed, in 1927, mathematician J.W. Alexander created just such a “machine,” a method that produces a polynomial (an expression like 3×2 + 4x + 1) from any knot. The good news is that Alexander’s method always gives the same polynomial for a particular knot, even if the knot has been wiggled around to look very different. The bad news is that it can also give the identical answer for knots that really are different. For example, the granny knot and the square knot have identical Alexander polynomials.
Going Beyond Schrödinger
Erwin may or may not have killed a cat. (His wife’s cat) But he didn’t humiliate it.
What’s cuter than a platonic solid? A cat dressed up as a platonic solid for Halloween!
How Does He Do That?
Hint: it’s not magic.
You probably want to use your mouse rather than your finger.
Not Hyperbole
There's Mathness to This Method
A gravity-powered binary calculator. Made of wood. (and therefore a witch)
via Ovablastic
Are You Eyeballing Me?
The game works by showing you a series of geometries that need to be adjusted a little bit to make them right. A square highlights the point that needs to be moved or adjusted. Use the mouse to drag the blue square or arrowhead where you feel it is ‘right’. Once you let go of the mouse, the computer evaluates your move, so don’t let up on the mouse button until you are sure. The ‘correct’ geometry is also shown in green. To avoid the need for extra mouse clicks, a mouse button up counts as the move being finished, so be careful.
You will be presented with each challenge three times. The table to the right shows how you did on each challenge each time.
An Abbie-Someone Distribution
The Lake Wobegon Distribution at The Universe of Discourse
[T]he remark reminded me of how many people do seem to believe that most distributions are normal. More than once on internet mailing lists I have encountered people who ridiculed others for asserting that “nearly all x are above [or below] average”. This is a recurring joke on Prairie Home Companion, broadcast from the fictional town of Lake Wobegon, where “all the women are strong, all the men are good looking, and all the children are above average.” And indeed, they can’t all be above average. But they could nearly all be above average. And this is actually an extremely common situation.
To take my favorite example: nearly everyone has an above-average number of legs.
The post goes on to use some baseball statistics, in a way that probably won’t give Chad apoplexy, arguing that professional baseball players shouldn’t follow a normal distribution, because they are not selected at random from the population. They should represent the part of the distribution several standard deviations above the average.
One flaw in the reasoning is that not all highly skilled athletes with the right abilities become baseball players, but I think the basic argument is sound.
Of course, college students probably aren’t a normal distribution. Schools screen their applicants, and there can be further skewing within that population; students drop out of classes, and not all courses are created equal. Take physics, for example — there are typically different levels of introductory physics: a so-called physics-for-poets class, a class that require just algebra, and one that requires calculus. Generally speaking your physics ability would correlate somewhat with the class you are taking. Even if the physics-taking population as a whole comprised a normal distribution, each individual class should not: the easiest class should be deficient in students at the high end, and the hardest class should be missing the low end.
The Value of Sports Statistics
Chad’s got a post up about how Baseball Statistics Are Crap. I’ve got a different beef.
(There are, certainly, a lot of dubious statistics in baseball. I just don’t agree that things are as bad as Chad says but maybe it’s just that I’m used to the idiosyncrasies. I do understand the infield fly rule, after all. If that weirdness makes sense, maybe the weird statistics do, too.)
Anyway, my objection is that even with these simplified statistics, the sportscasters and writers read too much into them. They don’t understand what the statistics are saying, and the value of statistics is to be able to compare players. In baseball it’s not so bad — even if the stats are flawed, a player hitting .356 is objectively a better hitter, by this measure than one who is hitting .290. But what does “by this measure” mean? In baseball, you can hit for average or for power — there are different skills and abilities useful to the team, and you want to find the statistic that is appropriate to the skill you are trying to quantify.
In this regard, I think, football is an example where the reporters are a great abuser of statistics. And this goes beyond saying “turnover ratio” when “differential” is meant (one thing that’s gotten better over the years). The main abuse, I think, is saying that accuracy is measured by completion percentage, and this seemingly happens all the time.
Accuracy is your ability to hit a target, and if you want to compare apples-to-apples, the target should be the same one. A stationary target at 10 yards is easier to hit than a moving one at 40 yards. A better receiver, who can get open, is easier to hit, and also affects the ability for other receivers to get open. You can have a receiver who drops the ball even though it’s “right at the numbers,” or one who catches everything thrown his way. When nobody’s open and he’s trapped, a quarterback can take a sack or throw the ball away, giving him an incompletion. All of that affects completion percentage, and none of it reflects accuracy.
Chad Pennington is touted as an “accurate quarterback” by many sports journalists, who, in the next breath, mention he has a weak arm and dumps the ball off quite often. Short passes. Connection? I think so!
My favorite example is Donovan McNabb. When Terrell Owens was about to join the team, analysts were all cautious about how Owens would tolerate the inaccurate McNabb, who had never completed 59% of his passes. Until that year, when he completed 64%, and everyone was saying how accurate he had become. Owens leaves, and the completions percentage drops back down. (It’s up again this season, and last — he’s got better receivers, and he dumps the ball off to Westbrook when he has to)
1magine
via Cap’n Refsmmat’s Blog of Doom
Yes, I’ve been up to no good again. Together with my band, now named The Quirk Side of the Moon, I’ve created yet another parody song, this time of John Lennon’s Imagine.
Imagine you were in a world with no math…