Everybody knows that

a² + b² = c²

(referring to the sides of a right triangle). That formula tells you the straight-line distance on a flat plane. Say I walked three blocks east in a grid-style city and then four blocks north. Then I’ve traveled

sqrt( 3² + 4²)

blocks i.e. 5 blocks, as the crow flies. I could also go one block down to the sub-sub-…-sub-basement and then the same rule would apply in 3-D, as the mole burrows

sqrt( 3² + 4² + 1²) = sqrt(26)

blocks north-east-down.

You see the same formula—weird, right?—in statistical calculations of standard deviation:

sqrt( dev&sub1; ²  +  dev&sub2; ²  +  dev&sub3; ² + ...) = ST DEV

Each of the |dev|’s is a data point’s deviation from the center (middle, barycenter, mean) of all such observations.

So standard deviation is a Pythagorean conceptin a way, situating your n data points on the corner of an n-dimensional box and then calculating a hypotenuse. If that doesn’t make you wary of standard deviation as the only, absolute measure of variability … well, it should.

But so, what’s up with all the 2’s? Higher mathematics doesn’t use numbers! Replace each of the ^2’s with a ^p for any power, and you’ve got ∞ new, valid, measures of distance, called L_p norms. This symbolic change gives rise to some mind-expanding imagination weapons, including the much-hyped non-Euclidean geometry (p ≠ 2, Cthulhu fans).

p=1 corresponds to city-block distance — (not in Boston, Prague, or Edinburgh because the streets don’t connect squarely – in some imaginary flat, square Lattice City).

p=2 corresponds to Navy distance on a flat lake. If you sail or fly across the Atlantic, the curvature of the Earth starts to make a difference and p ≠ 2.

p=4 could measure financial volatility in a way that penalizes kurtosis. (It’s still a two-way measure, though. Better would be p=5 or another high, odd number, to get a quasimetric.)

These new conceptions of “distance” make more sense of physical reality:

  • General relativity (which gave us GPS’s and satellites) requires Riemannian geometry, since spacetime is curved.
  • Euclidean geometry (p=2) fails if you’re Magellan, since the Earth’s surface is curved.
  • And we wouldn’t have the atom bomb without the even more brain-wrinkling geometry of nuclear physics.
  • If you want to tolerate noise or slight differences among data, Lp norms let you treat similar things as the same while still maintaining différence among quite different things.

If you want to use mathematics on things besides physics – like text mining, psychology, chess, racism, self-versus-other, financial time series, cluster analysis, terms of trade, marketing data, voting, bargaining, morality, functional spaces, utility theory, strategic arms races – you’ve got to be aware of the distance measure in your relevant space.


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