Category Archives: Math

Renaissance Wrestling

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Via the Giant’s Shoulders #16, I found Arcsecond: The Renaissance Man Uniform Gravitational Acceleration SMACKDOWN

The post is interestig enough, but what really got me was the following pictoral representation of perfect squares:

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If you keep adding up the odd numbers, you get the next perfect square (i.e. sum the quantity (2i-1, from 1 to k, and you get k^2). You see this by adding a new “L” of dots to the previous square, which always has 2 more dots that the previous one, i.e. it’s the next odd number in the sequence, and it makes a new square.

That is so cool! If I had previously known this, I had forgotten it. And I can easily imagine this being taught to me ages ago, and not making quite the same impression because I couldn’t fully appreciate the elegance of it.

Yummy, Tasty, Packingfraction-Ohs

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Volume Packing of Breakfast Cereal

With Raisin Bran, I tend to fill the bowl with cereal, then add milk, and when I finish the cereal, there’s only a small amount of milk left. With Cheerios, on the other hand, after I finish all the cereal from a full bowl plus milk, there’s still rather a lot of milk left. I generally put in another half-bowl (maybe two-thirds) worth of cereal, and finish that, too.

Being a physicist (and, as noted earlier, a gigantic dork), it occurs to me that this can probably be explained by the different volume packing factors for the different shapes. Raisin Bran is mostly flat flakes, which Cheerios are little toroids. Those two shapes will fill space very differently.

Touch My Monkey

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The story of the Gömböc

To give it its full mathematical description, a Gömböc is a three-dimensional, convex and homogeneous object with exactly one stable point of equilibrium and one unstable point of equilibrium. Requiring it to be homogeneous amounts to saying that you’re not allowed to cheat: the material from which the Gömböc is made has to be uniform throughout, so you’re not allowed to use weights, as those found in roly-poly toys, or other irregularities to get the Gömböc to self-right. Convexity means that the Gömböc is not allowed to bulge inwards, in other words, the straight line connecting any two points on the Gömböc has to lie entirely within the Gömböc. It’s easy to create a non-convex shape with one stable and one unstable equilibrium point, hence the restriction to convexity.

I have this mental image of Dieter describing the Gömböc. I don’t know why.

Your three-dimensional, convex homogeneity has grown tiresome. Now is the time on Sprockets vhen ve dance!

Math Goes to the Movies

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The mysterious equilibrium of zombies

In any decade there are really only a handful of movies about math (“Proof” comes to mind, as well as “A Beautiful Mind”), but a surprising number of movies that end up embodying math, even if it’s accidental. “Six Degrees of Separation” is based on the math of social networks. Thrillers have a special propensity for edgy twists on game theory. And what is a disease-outbreak movie if not an illustration of mathematical epidemiology, with puffy suits? To see movies through their math, sometimes, is to watch a whole different drama.

New and Improved. Now With More Math!

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What I Would Do With This: Groceries

I’ve known this for years: the express line isn’t necessarily the fastest lane at the grocery store, or fastest per item, because of the overhead of the transaction (paying, getting change, etc). I knew this even before Apu spilled the beans (Mrs. Simpson, the express line is the fastest line not always. That old man up front, he is starved for attention. He will talk the cashier’s head off.) but now someone has actual data and done a real analysis.

Check is slower than credit which is slower than cash. Students are sometimes surprised that cash is faster than credit. From my observations, the fastest cash transaction will outpace the fastest credit transaction by a wide margin but there is also huge variance in credit transactions. I mean, some people have absolutely no idea what they are doing with that thing. The same can’t really be said of cash.

I’m secretly amazed every time someone behaves like it’s the first time they’ve ever swiped a credit card at a checkout line, and it’s rocket science to figure it out. Hint: you can swipe the card before the clerk finishes scanning them!

When figuring the transaction overhead, there is a huge penalty for a non-tech-savvy shopper paying with credit. Of course, there is a large overlap with the cash paying “Oh, I have exact change. Let me get my coin purse!” customer, often a senior senior citizen. (That’s not age discrimination, it’s profiling)

Is Degrees Squared a Unit?

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Physics Buzz: Six degrees of Paul Erdős

Some famous names have low Erdős numbers— Bill Gates has an Erdős number of 4, Steven Chu’s is 7, and Albert Einstein is 2.

If Chu’s is 7, mine is no greater than 10; I can trivially trace a path through my thesis advisor to his, to Chu.

And I’ve already mentioned that I sort of have a Bacon number of 3. Consequently, I’d like to popularize the notion of combining the two by adding them, making my Bacon-Erdős number, or Berdős number, 13 (or perhaps adding in quadrature, making my Berdős number 10.44)

Meanwhile, Chad asks Who Is the Erdos of Physics? Maybe we can make this three dimensional.

(Update: I’ve found that my Erdős number is no larger than 9, and I may be able to bring it down to 6 fairly easily)

Schrödinger's Quarter

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The Coin Flip: A Fundamentally Unfair Proposition?

The physics, and statistics, of flipping a coin.

The 50-50 proposition is actually more of a 51-49 proposition, if not worse. The sacred coin flip exhibits (at minimum) a whopping 1% bias, and possibly much more. 1% may not sound like a lot, but it’s more than the typical casino edge in a game of blackjack or slots. What’s more, you can take advantage of this little-known fact to give yourself an edge in all future coin-flip battles.