I've always hated this fact with passion but it's like hating gravity. You can't fight it. Personally, I would have gone with one's compliment and a modified equality but that's just my OCD speaking.
Tetrahedra almost tessellate but don't quite do it: https://www.pnas.org/content/pnas/103/28/10612/F1.large.jpg
In music, the frequency ratio of a semitone is ideally 2^(1/12), but without some tiny fudging (called tuning), you can't make harmonies as the frequencies almost but don't quite line up right. I forget exactly how this one works so I may have something off.
Other coincidences that drive me wild: speed of light is almost but not quite 3.0E8m/s And the fine structure constant being almost but not exactly 1/137.
On that subject, if you decided to make your base unit of length the distance light travels in a nanosecond it'd almost be a foot but not quite.
Similarly the earth-sun distance is 8 light-minutes.These feel right, like measuring mass in stone for people, kilos for sugar, carat for diamonds, electron-volts for particles etc.
IIRC correctly, it's not just harmonies; the range of a piano is big enough that if you tune each octave exactly based on that ratio, you'll end up with the first and last octaves sounding off from each other.
Hypothetically, or on an electronic instrument, you could. But if you did all 2^(1/12) ratios, your octaves wouldn't be in tune. Strings on a piano do not behave like an ideal string. Their overtones are not 2X, 3X, 4X, 5X, etc. times the fundamental frequency. Instead, the actual overtones are higher than the ideal frequencies. This is called inharmonicity (https://en.wikipedia.org/wiki/Inharmonicity).
So when tuning a piano, you have to tailor the way you tune it to each different piano if you want that piano's lower strings to be in tune with its higher strings.
I think I've been hearing this for a long time but didn't realize it was real so questioned my perceptual system.
Thank you for the info!
Some of it, of course, sounds like cats screeching but some of it is genuinely astonishing.
Now that you mention it, that convention does make me mad. Not because I have to write an extra character, but due to operations getting mixed-up based on implied context. For example:
Unary + is implicit: 3 = +3
Addition is written as juxtaposition: +2+5 = add(+2, +5) = +7. The first number can have an implicit sign, e.g. 2+5 = +2+5
"Subtraction" is a redundant operation; it's just addition involving a negative number: 8-3 = +8-3 = add(+8, -3) = +5.
The nice thing about this perspective is that subtraction commutes: +8-3 = -3+8.
What's annoying is that we also take juxtaposition to mean multiplication, and this flip-flops depending on implicit characters like unary '+'. For example:
-8-3 = add(-8, -3)
-8+3 = add(-8, +3)
ab = multiply(a, b)
2a = multiply(2, a)
a2 = multiply(2, a) (non-idiomatic)
-2a = multiply(-2, a)
a-2 = add(a, -2)
(1)(3) = multiply(1, 3)
(-1)(-3) = multiply(-1, -3)
-1-3 = add(-1, -3)
-1(-3) = multiply(-1, -3)
(-1)-3 = add(-1, -3)
This seems weirder than the case of addition/subtraction, probably because we already have the horizontal-bar notation for division, which seems even nicer. Note that the "÷" character itself is simply an inline approximation of the horizontal-bar notation (with placeholders above and below); the "foo/bar" notation is a more direct inline approximation (no placeholders, just 'tipping' the bar). Interestingly we don't use "bar\foo" to mean the same thing.
This gets you an extra bit of precision when multiplying two signed numbers, because you no longer need to leave room for two sign bits in the product.