Comments by "Lawrence D’Oliveiro" (@lawrencedoliveiro9104) on "Numberphile2" channel.

The “intuitive” way to think about rotations in 3D might be to express them in terms of latitude and longitude on the surface of a sphere. Starting from long 0° and lat 0°, you might move x° east or west, and y° north or south. This works OK starting from (0°, 0°), but it doesn’t work so well when you start from, say, the North Pole. Because every direction from there is south, how do you define how far east or west you want to go, and from what point? This phenomenon is called “gimbal lock”. It is a phenomenon well known in computer graphics if you try to specify rotations in terms of “Euler angles” (as the longitude/latitude angles are known). It was also a known hazard during the Apollo moon landings, as it imposed limits on how their spacecraft could maneouvre without confusing the orientation gyroscopes. Quaternions are really nice because they avoid gimbal lock. Related to this, they also interpolate nicely between any two orientations, which is handy for animation purposes in CG. Trouble is, many people working in CG are afraid of quaternions, because “math is hard”. This includes some developers of CG software.
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4:23 I think people discovered the consequences of changing the Fifth Postulate before Lobachevsky and Bolyai, but they were seen as “absurd” (really just “counter-intuitive”), and that was sufficient excuse to not pursue the matter further. What L&B did was stick to the courage of their convictions, and say that, since alternatives to the Fifth Postulate did not create any (new) contradictions in the mathematical theory, they must be considered to be just as valid as Euclid’s version. In other words, this was a change in the meaning of mathematics, from thinking about things that had to be “intuitive”, to things that had to be “logical”.
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5:33 I believe there is a society which has the toast “To mathematics! And may it be of no damn use to anybody!” Of course, “pure” maths often turns out to have surprising applications...
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6:24 The other “axioms” are more about definitions of terms. If you change them, I guess you don’t end up with geometry at all. Instead, you probably create something more like topology.
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Have you thought of generalizing the problem to more than 2 dimensions? You just have to define what a “queen move” would be in higher dimensions. The generalization of movements along “rank” and “file” is easy: it just becomes a move parallel to any axis. The generalization of a “diagonal move” I came up with is: all unequal coordinates (there must be at least 2) of the beginning and ending position differ by the same absolute magnitude. Thus, if only two coordinates change by the same absolute amount, you have a diagonal in 2D; if three coordinates change, that’s a 3D diagonal, and so on.
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