math – Crasstalk

California’s New $9.00 Minimum Wage in Historical Perspective

$9.00 per hour! Why when I was a kid walking to school up hill both ways in the snow we only made $0.23 per hour and an hour lasted ten times longer than it does now. Can you believe these whippersnappers wanting to get $9.00 per hour?

Or, how exactly does California’s newly passed $9.00 per hour minimum wage, the highest state minimum in the nation, compare in constant dollars to the minimum wage when your grandpappy was only knee high to a grasshopper? Let this handy chart explain it to you. Continue reading →

This Is Why You’re Still Fat

If you’ve ever tried to lose weight, you know how frustrating it is to hear the simplistic ‘eat less, exercise more’ advice. Your personal experience flies in the face of this seemingly straightforward equation. Certainly in a vacuum, eating less and exercising more works. However, our bodies are an open, interactive systems, and rarely cooperate with simple math. Continue reading →

The Hairy Ball Theorem

If you have the type of not too short hair that completely succumbs to gravity, you will have a cowlick somewhere on your head.** It’s a direct consequence of a mathematical theorem called the hairy ball theorem. Yes, it’s called that and it’s a bona fide mathematical theorem with a real but difficult proof that requires you to get a fancy-schmancy degree in math to really understand it.

But if we just want to feel like we understand it, we can certainly do that. It’s the Internet, after all.

Let’s comb through some informal ways of describing this theorem. One is

You can’t comb a hairy ball flat without creating a cowlick.

or closely related

You can’t comb the hair on a coconut.

A perhaps more useful, less frivolous description of the theorem is:

A cyclonic system must always exist on a planet with an atmosphere.

In other words, somewhere on Earth, at any given moment, there will be a cyclone (in the most general sense).*** I should note that, in certain ways, this is a supremely useless theorem because it doesn’t tell you where the hell the cyclone is. But it is 100% absolutely sure that there is one.

The reason that people built analogies with hair and wind is that the “hairs” on the ball correspond to what are called vector fields for systems that have stuff that moves like a fluid, like wind or piss streaming over someone’s hairy ball. The hair is mostly a visualization tool for particles that move continuously in a closed environment, like the surface of a sphere.

Imagine you could track some representative particles in a fluid. Say, particularly big balls of dust are floating around. Or fireflies are caught in a dust devil at night. And you snap a picture of this with not too short exposure. The balls of dust or the fireflies would come out as blurry streaks that move in thrall to the air. The blurry streaks would correspond approximately to vector fields. So a fairly accurate way of looking at vector fields is to understand it as a concise description of the motion of particles at every single point at a given moment in whatever system you are looking at.

And what do cowlicks and cyclones have to do with this? Basically, the CENTER of the cowlicks/cyclones are points where there is no motion. It’s the eye of the storm you always hear about where everything is calm. These are called the zeros of the vector fields. So if you were to drop something lightweight at the exact center of where the cyclone occurs, that thing you dropped wouldn’t move because there’s nothing moving around it.****

If you’re looking at a stream or a river, you see eddies, the watery analog of cyclones, all the time. But would you still see these if the water were flowing in a completely smooth, straight, artificial indoor stream? Probably not. But now let’s say there’s a sphere and there’s water coursing over the surface. And imagine that it’s completely self-contained. No water drips from it and no water drops on to it. The kicker in this hypothetical situation is this: there will be at least one eddy (i.e. a “cyclone” in our previous parlance or a “zero” in our more technical one) no matter how smooth we make the surface of the sphere. The mathematics of the hairy ball theorem guarantees this.

If you’re curious how the hairy ball theorem is stated in technical terms, it goes like this:

There does not exist an everywhere nonzero tangent vector field on the 2-sphere.

**A technical caveat. This doesn’t apply if you’re wearing your hair slicked back or in a ponytail or in some way that makes it defy gravity. It’s mostly applicable if you’ve let your hair loose so that it looks like it’s cascading down your head.

***This ties in with something called the Poincare-Hopf theorem which is even more general than the hairy ball theorem. Because a cyclone is a “cowlick” (i.e. not just any zero, but a special type of zero called a sink of a vector field) with index one, there must be a second cyclone. This more general theorem predicts not one but two cyclones.

****Not in reality, of course. It’s a thought experiment, so bear with me.

Are You Smarter Than a Football Player?

I have spent much of the past two days in front of the television watching sturdy and well-formed young men run around an empty stadium in Indianapolis dressed in nothing but skin-tight biker shorts and muscle shirts. I haven”t seen so many bulges since that morning at the Provincetown Pride Parade. But that’s a story for another day. No, I’m not watching Spring Break on the Logo Channel, it’s the NFL Network. That can only mean one thing, America. It’s Combine Week in the NFL.

The NFL Combine is basically an audition for hundreds of prospective NFL football players. These are incoming players, eligible for the draft for the first time. They run, throw, block, hop, skip, jump and do everything but sing show tunes in front of scouts and coaches from all 32 NFL teams and a national television audience. Off the field, players are examined by doctors, interviewed by general managers and taken apart mentally by staff psychologists. Last year in a notorious Combine moment, wide receiver Dez Bryant of Oklahoma State was asked by Miami Dolphins General Manager Jeff Ireland if it was, in fact, true that his mother was a prostitute. Ireland, who later apologized, was either looking for a quick lay after the session ended or just wanted to see how Bryant reacted to sudden stress.

For the average football fan who is not actually going to make any selections in the upcoming draft , the Combine is at once as dull as watching articial turf grow and as fascinating as brain surgery. Hour after hour, day after day, 300-lb linemen defy several laws of physics and run 40 in 5 seconds, 200-lb running backs lift twice their weight 18 consecutive times and quarterbacks throw perfect spirals all the way to Chicago.

But the most intriguing part of the Combine is, sadly, not on television. The good stuff takes place behind the scenes in closely guarded classrooms, where players are given a raw intelligence test called the Wonderlic Cognitive Ability Test, written, scored and evaluated by the Wonderlic Corporation of Vernon Hills, IL.

In today’s ultra-competitive business environment, thousands of employees in a variety of professions have at some point taken a Wonderlic-type test as a condition of employment. It makes sense if the applicant is a CPA or a HVAC repair-person. A business wants to know it’s not hiring Paris Hilton. But what does an IQ-style test have to do with the physical ability and the commensurate willingness to remove a quarterback’s head from his shoulders in the hopes of gaining a Wild Card berth or gain one extra yard only to have James Harrison drive his rocket-fuel-propelled body into your chest, sending you hurtling into the club seats. The heart and motivation to sacrifice self for team cannot be measured on a 50 item multiple-choice test.

Indeed, a major academic study completed at the University of North Carolina in 2007 concluded there was no correlation between high Wonderlic scores and success in the NFL. So why take this test at all? Who knows–why take the SAT? Perhaps the problem is that football is just asking the wrong types of intelligence questions. The Wonderlic’s questions are fairly straightforward, by-the-book IQ type questions, such as:

A train travels 20 feet in 1/5 second. At this same speed, how many feet will it travel in three seconds?
When rope is selling at $.10 a foot, how many feet can you buy for sixty cents?
The ninth month of the year is…

You can take an entire sample test here.

Despite its reputation as a Neanderthal ThugFest, football at its highest levels is by far our brainiest and most intellectual game. It’s true that baseball players are frequently required to both scratch their balls and spit tobacco at the same time. And then guess curveball. But that’s pure hand-eye coordination–as is having the innate ability to consistently miss your shoe with your spit.

Basketball and hockey are mostly athleticism, grit and instinct. And yes, the very best players see the game as it will be three seconds from now not as it is in our reality. That is more than raw intelligence for a sport–it is a gift of timing, intuition and physical creativity. Kinisthetic and athletic geniuses like Michael Jordan, Wayne Gretzky and Kobe Bryant literally play a different game than we do.

But football is on a whole other level when it comes to complexity. A typical play call on offfense might be:

“Scatter-Two Bunch Right-Zip-Fire Right-273-Pivot-F Flat.”
“Duece Right 19 Slot on 1 and Dice Right Ice Cream Alert 654 Jose”
“Trips Right 255 X Block Slant H Disco Alert 12 Trap”

If you are a quarterback, a position which only a kind of football savant can play, not only do you have to know your play and know where all your backs, linemen and wideouts will be and when they will be there, but you also must determine what the defense is doing. And this despite the defense doing everything it can to disguise its intentions.

And the incoming and data must be processed and communicated to ten co-workers in less than 30 seconds, again and again, under extreme physical and mental pressure in a setting where the decibel level is often as high as at a Clash concert. All this and it suddenly seems a 50 question IQ test is not enough. If you’re going to be my quarterback, I’m going to want transcripts, letters of recommendation from your sophomore English professor and that research paper you did on the Civil War. Oh, and I’ll need your Calc final.

Most of all, I’m gonna want to see how you react when I ask if it’s true you’re sister’s a whore while we’re both standing on train tracks at high noon with the Acela bearing down from the east, Ray Lewis coming from the west, and Larry Fitzgerald open down the road for six easy points. And right now you’re down to four seconds to figure out how long ago the Acela left Detroit. Do all that correctly and a hundred million dollar, seven-year deal is yours. And I’ll tell you I was just kidding about your sister.

Photo here.