15 February 2016

Maps lie more than one usually thinks

I performed the following experiment with students of mathematics. I showed them a map of Russia on google and took a point very far to the east (Uelen, a village in the Chukotsky District next to the Bering Sea) and a point very far to the west (Venekyulya, a region in the Leningrad Oblast, at the border with Estonia). The distance between Uelen and Venekyulya is, roughly, the largest distance between two points in Russia, which is about 6000 km. I asked the students to draw a path between Uelen and Venekyulya whose length is 6000 km. Some drew the path P1 shown below which is a straight line between Uelen and Venekyulya on the google map. Others, realizing that the map is a projection of a sphere, drew path P2, slightly curved upwards.
When I showed them the actual path (see below), they were all surprised. They thought it was wrong. How can it be that the "straight line path" is so much curved upwards? Well, that's because the distortion of the map increases very rapidly when we move away from the equator.
If I had shown them the actual map, on a sphere, then they would have known what to do: they would have drawn an arc between Uelen and Venekyulya that is part of the unique great circle between them, that is, the circle centered at the Earth's center and containing Uelen and Venekyulya. This arc almost (but not quite) passes from the North Pole. To see this, we need to look at the Earth downwards from the North Pole. The picture then becomes clear and the apparent contradiction is resolved.
(Credits for this image go to https://www.jasondavies.com/maps/rotate/)
The moral of this is that we should never believe what we see. (In fact, we should never believe in anything.) Only when we have further evidence, provided by experiment, measurements, or mathematical proof should we "believe" what we see. But then the verb "believe" becomes irrelevant; at this point, we know, we do not believe. We may believe something, temporarily, until further evidence, or we may believe something because someone else we trust has done the work for us. But we should never believe something because our eyes saw it, or because a teacher once told us, or because the government or administration says so, or because a religious, for example, book writes.

One thing that bothers me with some students (and some teachers) of mathematics is that they may be comfortable with the geometry of a 5-dimensional hyperbolic space because they may have seen it in class, passed an exam on it, or doing research on it, but may be uncomfortable with down-to-Earth [sic] geometry, including knowing 2-3 proofs of the Pythagorean theorem. O tempora o mores!

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T H E B O T T O M L I N E

What measure theory is about

It's about counting, but when things get too large.
Put otherwise, it's about addition of positive numbers, but when these numbers are far too many.

The principle of dynamic programming

max_{x,y} [f(x) + g(x,y)] = max_x [f(x) + max_y g(x,y)]

The bottom line

Nuestras horas son minutos cuando esperamos saber y siglos cuando sabemos lo que se puede aprender.
(Our hours are minutes when we wait to learn and centuries when we know what is to be learnt.) --António Machado

Αγεωμέτρητος μηδείς εισίτω.
(Those who do not know geometry may not enter.) --Plato

Sapere Aude! Habe Muth, dich deines eigenen Verstandes zu bedienen!
(Dare to know! Have courage to use your own reason!) --Kant