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Posts tagged ‘yay math’

Of coffee rings and ballistic deposition

A friend mine shared a gorgeous video show how coffee rings form at a microscopic level.

Apparently he found the video through a impressive blog called Empirical Zeal that’s maintained by a Physics grad student. The blog post, entitled “The Universal laws behind growth patterns, or what Tetris can teach us about coffee stains,” does a nice job of introducing the notion of universality classes and has several links to some excellent articles on recent mathematical advances in the field.

The mathematics centers on the KPZ (Kardar-Parisi-Zhang) equation, which is well-described in this expository mathematical article by Ivan Corwin:  “The Kardar-Parisi-Zhang Equation and Universality Class.”

Apparently there's a coffee stain font by Mark Mustaine! (This image is taken from the Empirical Zeal post. Go read it!)

Apparently there’s a coffee stain font by Mark Mustaine! (This image is taken from the Empirical Zeal post. Go read it!)

In celebration of pi

pi_backgroundIf you happen to be in a bar and you need to quickly calculate pi, head over to a dartboard.  Draw a square that perfectly encloses the dartboard. (The length of its sides should be equal to the dartboard’s diameter.)  Start throwing darts randomly at the square.

If you manage to distribute the darts uniformly at random over the square then the fraction of the total darts that land in the dartboard will equal the area of the dartboard divided by the area of the square.  If the radius of the dartboard is r, then this fraction will be

displaystyle frac{text{Area of Circle}}{text{Area of Square}} = frac{pi r^2}{ (2r)^2} = frac{pi}{4}.

Notice that the ratio is independent of the value of r!

Therefore if you take the number of darts that land in the dartboard (and not in the surrounding portions of the square), divide by the number of darts thrown and then multiply by four … you’ve roughly approximated the value of pi!

(But remember … it only works if you’re _bad_ at darts.)

 

While we’re at it, here’s an oldie but goodie. (Not sure of the original source.)

pi_animation

Note that the radius of the circle is frac{1}{2} so that the circumference comes out to pi.

But, of course, nowadays I feel an obligation to nod toward the well-reasoned exhortations of the 2 pi crowd:  The Tau Manifesto.

MORE LINKS!

Visualizing pi.

—  A lo-fi page on formulas that compute pi using the Fibonacci numbers. (link)

— Some pi jokes, if you need them.

be-rational-get-real