# “Casual” mathematical facts with practical consequences

Some mathematical facts -be them approximations or not- can be described as coincidences, without any deeper meaning in themselves, but leading to relevant practical consequences. I was thinking in these two examples:

• $2^{10} = 1024 \approx 1000 = 10^3$

This "casual" approximate equation is practically relevant and potentially dangerous (I don't know what he was thinking, that guy who invented mathematics and gave us 10 fingers!). First some engineers created the decibel and started assuming 'a 3db change means a factor of 2' - harmless enough. But then came digital computers and the convention-confusion KB=1024 bytes started.

• $\displaystyle 2^{7/12} = 1.498307\ldots \approx 3/2$

Related to the tempered system used in music since -approx- Bach, this can be seen as a fortunate or unfortunate fact. If it were an exact equation, musical tuning would be simpler and 'pure'; because it's not, the intervals we usually hear are 'impure'. On the other hand, if the approximation were just a little worse, the equal temperament would we intolerable, and to transpose music in most instruments would be a mess.

Can you think of more examples?

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About the tempered system, see also this 118-vote question with a 140-vote answer by Qiaochu: math.stackexchange.com/questions/11669/…. – joriki May 25 '11 at 15:12
I'd say the $7/12$ fact is less a "coincidence" than it is an explanation. You can instead think of it as the reason we divide the octave up into 12 steps - the fact that $19/12$ is a term in the continued fraction expansion of $log_2 3$. The previous term, incidentally, is $8/5$, and that corresponds to the pentatonic scale. The next two denominators are $41$ and $53$, and those are just not practical for a scale. – Thomas Andrews May 25 '11 at 15:18

Life as we know it in three spatial dimensions (four space-time dimensions) is only possible because a number of things coincide at this dimensionality. Planetary orbits are only stable in a space-time of not more than four dimensions. Digestion can only take place in at least three spatial dimensions, since otherwise our digestive tracts would disconnect us. So these two requirements fix the number of spatial dimensions at $3$. Coincidentally, this is an odd number, and Huygens' principle that the impulse response of the wave equation is a spherical pulse only holds in odd dimensions -- if the only feasible dimension had turned out to be an even dimension, sound wouldn't arrive all at once, and the frequency response would depend on frequency and distance. These are just some aspects that conspire at three dimensions; I suspect there are more.

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In addition, I understand that many conjectures in algebraic topology and knot theory are trivial in dimension 2 or less and dimension five or greater, but are interesting in dimension 3 or 4. For example, in 2 dimensions only the trivial knot exists (by the Jordan Curve Lemma) and in 4 dimensions any knot can be un-knotted to form the trivial knot. Dimension 3 is the only one in which the theory of knots of 1-dimensional ropes is non-trivial. – Chris Taylor May 25 '11 at 15:43
@Joriki Could you recommend me some light reading on the concept of dimensions? I've heard about them but I can't imagine them. – Voyska Aug 21 '12 at 22:33
@Gustavo: The first thing that comes to mind is Flatland. Also, I suggest not trying to imagine any more than three dimensions; I doubt that anyone is able to do that, since our minds are firmly rooted in the three-dimensional world we live in; and there's no need to do it; you can develop a good mathematical intuition for higher-dimensional spaces without imagining them. – joriki Aug 22 '12 at 6:04
@joriki Oh, really? I was feeling dumb for a long time because I couldn't imagine it. – Voyska Aug 22 '12 at 9:46
Can you comment about this: "Digestion can only take place in at least three spatial dimensions, since otherwise our digestive tracts would disconnect us."? – Voyska Aug 24 '12 at 4:36

I'm not sure whether you'd count this: The fact that an integral number of identical circles fit around a circle in the plane leads to drains being designed like this:

Assuming the aim is to have as much water throughput as possible while stopping objects below a given diameter, the solution must consist of circles of that diameter, and there happens to be this nice symmetric way of arranging six of them.

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Would it be possible to design a drain with a hole in the middle too? What purpose does the screw have? – Sputnik May 25 '11 at 14:48
@Fahad: I'm not an expert on drains, but I would suppose the screw is required because it should be possible to remove the cover in order to replace it or to remove obstructions from the pipe. – joriki May 25 '11 at 14:52
Yes sorry; a stupid question but given your criteria for drains it just made me wonder why it was there! Our kitchen sink doesn't have one... curious. – Sputnik May 25 '11 at 14:59

Even closer than the interval you have there is the perfect fourth: $2^{5/12}= 1.334839... \approx 4/3$. I do not think it is an accident; the coincidental fact that simple ratios like $4/3$ and $3/2$ are well-approximated by powers of $\sqrt[12]{2}$ probably resulted in the choice of the 12-tone equal temperament system, because the intervals represented by these ratios, respectively the perfect fourth and perfect fifth, were considered most aesthetically pleasing at the time. Had the standardisation of music occurred somewhere else, where they perhaps liked slightly more exotic intervals, we might have an entirely different system (for example, there is also a 19 equal temperament system and a 31 equal temperament system which better approximate some intervals, though these systems are virtually unheard of in the West).

Whether it is a fortunate fact that 12 happens to be the number chosen is hard to say; any system with less than 10 intervals would sound a little boring to the modern ear, but any with over 20 would seem slightly overkill. No matter what other possibilities could have occurred, we can be assured that there was always a compromise to be made between the simplicity of a musical system and the accuracy of its approximation of simple ratios.

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It's a stretch to call this "practical" I suppose, but the fibonacci numbers have several relationships that "miraculously" work well in unexpected ways. One of my favorites is the following puzzle:

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What's the puzzle, other than false visual similarity? Slope 3/8 in the left image does not match 5/13 in the right image. Close, but not the same. – Jim Garrison May 25 '11 at 20:29
The puzzle is to explain why these two differently-sized rectangles seem to be both partitioned in to the same 4 polygons. Knuth claims this was a favorite puzzle of Lewis Carroll (See "Concrete Mathematics", Graham, Knuth, Patashnik, pg 293) – Fixee May 25 '11 at 21:44
They're not, as indicated by the fact that the angles are close but not exactly congruent. They just look that way. If you were to cut them out they wouldn't fit together as drawn. – Jim Garrison May 25 '11 at 21:50
@Jim: Yup, that's why I used the word "seem". Anyway, I think it's pretty clever. – Fixee May 26 '11 at 2:02