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I love math (and have recently gotten into the beauty of the Golden Ratio) but this seems remarkably non-profound. Squaring a number that’s a little higher tha

I love math (and have recently gotten into the beauty of the Golden Ratio) but this seems remarkably non-profound. Squaring a number that’s a little higher than 3 will get you close to 10. If it were 9.999999 then it’s another story.

NegativeLatency 9y ago
Eulers identity on the other hand...
- andrepd 9y ago
  Is almost a definition.
  - gargarplex 9y ago
    Once the identity was pointed out to me and I didn't immediately grok, I realized I would never be a first-class mathematician :p
- deleted 9y ago
  [deleted]
- wyager 9y ago
  That’s not really “weird” though - it’s can be straightforwardly explained in any number of ways. You can think about it in terms of the circle group and its Lie algebra, in terms of Taylor expansions, etc. The OP is just not very interesting. Pi^2 isn’t close enough to 10 to trigger my not-a-coincidence detector. A much cooler problem of this nature is: why are musical notes generated by powers of the 12th root of 2? I remember seeing a good YouTube video about this on one of those math channels.
  - deleted 9y ago
    [deleted]
  - gowld 9y ago
    "explained in any number of ways", which involve complex numbers, geometry, trignometry, and calculus, is what makes it profound.
  - jagthebeetle 9y ago
    Isn't this just by definition of equal temperament? If you want a pitch to double after 12 steps (multiplicatively), you choose 12-TET. Other musical systems exist (e.g. quarter-tone scales), and have existed. So the musical notes thing is by human fiat, which at least pi^2 isn't. Perhaps I'm missing your meaning though?
    - thanatropism 9y ago
      Before the current "equal temperament" system there was a "just temperament system" that basically harks back to Pythagoras. The main intervals in the C scale are very close to whole fractions of low numerator and denominator. The problem is that this doesn't work very well in every key. So the emergence of modern music came with a few tries at averaging these things out until equal temperament in the log scale arose. Some of the character of older music is actually lost because of this. But hey, now all keys work the same.
    - wongarsu 9y ago
      >which at least pi^2 isn't pi^2 is 9.87. If that counts as close to 10, then half of all numbers are close to some integer. And the significance of 10 is entirely cultural. Nothing in math favours base 10, we use it because the right civilisations were dominant at the right time. Other civilisations counted using base 4, base 5, base 6 or base 12 (e.g. using the five fingers and a closed fist to count up to 6 for each hand).
    - wyager 9y ago
      The reason that powers of the 12th root of 2 are used is that these numbers are close to simple rational numbers, which the human auditory system can pick up.
  - InitialLastName 9y ago
    > A much cooler problem of this nature is: why are musical notes generated by powers of the 12th root of 2? That one isn't especially difficult. Humans like the combination of fundamental tone and first harmonic (I'm not aware that there's a culture with a musical system that doesn't respect the octave). In the western culture, we found the similarity between those tones so notable that we classify them as harmonically indistinguishable; thus, A3 (220 Hz) is harmonically tied to A4 (440 Hz). Many cultures, our western predecessors included, also appreciated the 2nd harmonic (f3, or, since octaves are now identical, f1.5) as consonant with the fundamental. Following on to that, they developed a pattern of notes at f(1.5^2), f(1.5^3) etc. Here it can be noted that the first 12 exponents of 1.5 are (renormalized to the [1:2] space as necessary): 1 , 1.5, 1.125, 1.6875, 1.265625, 1.8984375, 1.423828125, 1.06787109375, 1.601806640625, 1.20135498046875, 1.802032470703125, 1.3515243530273438 and 1.0136432647705078. Note how infuriatingly close that last is to 1; on a violin string (~.328m) that would be a 4mm difference. Human societies have wrestled with that discrepancy as far back as the Pythagoreans. Musicians and composers in western society didn't come to a compromise for the issue until ~late 16th century, when they DEFINED a semitone as the 12th root of two. This adds a little bit of error to every interval besides the octave in favor of consistency across every key.
    - wyager 9y ago
      You’re right; it’s not difficult, but it’s (IMO) a lot cooler than the OP.
- kerkeslager 9y ago
  I think what makes Euler's identity profound to me is that it is a tool for doing so many things. For example, if you want to calculate the log of a negative number: -1 = e^(i * pi) ln(-1) = i * pi ln(-x) = i * pi + ln(x) log_b(-x) = (i * pi + ln(x)) / ln(b) Now you have a formula for calculating the log of a negative number (-x) in any base.
jerf 9y ago
"seems remarkably non-profound." I'd point out the linked article doesn't particularly claim otherwise. It just provides a slightly deeper explanation but doesn't claim it's an explanation of anything particularly meaningful. In other news, if you'd like to see the Golden Ratio strut its stuff in a context not related to the supposed aesthetics of the ratio, this relatively recent (and relatively in-depth, for the channel) video and linked proof may be interesting to you: https://www.youtube.com/watch?v=FtNWzlfEQgY https://www.youtube.com/watch?v=FtNWzlfEQgY You'll seriously be asking "what was jerf talking about?" for like a third of the proof video before foom in comes phi, and it takes quite a bit of abuse before the end.
deleted 9y ago
[deleted]
montalbano 9y ago
This one's a goodun though, the significance of the closeness of 196883 and 196884: https://en.wikipedia.org/wiki/Monstrous_moonshine https://en.wikipedia.org/wiki/Monstrous_moonshine https://www.reddit.com/r/explainlikeimfive/comments/2im1l1/eli5_196884_196883_1/ https://www.reddit.com/r/explainlikeimfive/comments/2im1l1/e...

NegativeLatency 9y ago

Eulers identity on the other hand...

andrepd 9y ago
Is almost a definition.
- gargarplex 9y ago
  Once the identity was pointed out to me and I didn't immediately grok, I realized I would never be a first-class mathematician :p
deleted 9y ago
[deleted]
wyager 9y ago
That’s not really “weird” though - it’s can be straightforwardly explained in any number of ways. You can think about it in terms of the circle group and its Lie algebra, in terms of Taylor expansions, etc. The OP is just not very interesting. Pi^2 isn’t close enough to 10 to trigger my not-a-coincidence detector. A much cooler problem of this nature is: why are musical notes generated by powers of the 12th root of 2? I remember seeing a good YouTube video about this on one of those math channels.
- deleted 9y ago
  [deleted]
- gowld 9y ago
  "explained in any number of ways", which involve complex numbers, geometry, trignometry, and calculus, is what makes it profound.
- jagthebeetle 9y ago
  Isn't this just by definition of equal temperament? If you want a pitch to double after 12 steps (multiplicatively), you choose 12-TET. Other musical systems exist (e.g. quarter-tone scales), and have existed. So the musical notes thing is by human fiat, which at least pi^2 isn't. Perhaps I'm missing your meaning though?
  - thanatropism 9y ago
    Before the current "equal temperament" system there was a "just temperament system" that basically harks back to Pythagoras. The main intervals in the C scale are very close to whole fractions of low numerator and denominator. The problem is that this doesn't work very well in every key. So the emergence of modern music came with a few tries at averaging these things out until equal temperament in the log scale arose. Some of the character of older music is actually lost because of this. But hey, now all keys work the same.
  - wongarsu 9y ago
    >which at least pi^2 isn't pi^2 is 9.87. If that counts as close to 10, then half of all numbers are close to some integer. And the significance of 10 is entirely cultural. Nothing in math favours base 10, we use it because the right civilisations were dominant at the right time. Other civilisations counted using base 4, base 5, base 6 or base 12 (e.g. using the five fingers and a closed fist to count up to 6 for each hand).
  - wyager 9y ago
    The reason that powers of the 12th root of 2 are used is that these numbers are close to simple rational numbers, which the human auditory system can pick up.
- InitialLastName 9y ago
  > A much cooler problem of this nature is: why are musical notes generated by powers of the 12th root of 2? That one isn't especially difficult. Humans like the combination of fundamental tone and first harmonic (I'm not aware that there's a culture with a musical system that doesn't respect the octave). In the western culture, we found the similarity between those tones so notable that we classify them as harmonically indistinguishable; thus, A3 (220 Hz) is harmonically tied to A4 (440 Hz). Many cultures, our western predecessors included, also appreciated the 2nd harmonic (f3, or, since octaves are now identical, f1.5) as consonant with the fundamental. Following on to that, they developed a pattern of notes at f(1.5^2), f(1.5^3) etc. Here it can be noted that the first 12 exponents of 1.5 are (renormalized to the [1:2] space as necessary): 1 , 1.5, 1.125, 1.6875, 1.265625, 1.8984375, 1.423828125, 1.06787109375, 1.601806640625, 1.20135498046875, 1.802032470703125, 1.3515243530273438 and 1.0136432647705078. Note how infuriatingly close that last is to 1; on a violin string (~.328m) that would be a 4mm difference. Human societies have wrestled with that discrepancy as far back as the Pythagoreans. Musicians and composers in western society didn't come to a compromise for the issue until ~late 16th century, when they DEFINED a semitone as the 12th root of two. This adds a little bit of error to every interval besides the octave in favor of consistency across every key.
  - wyager 9y ago
    You’re right; it’s not difficult, but it’s (IMO) a lot cooler than the OP.
kerkeslager 9y ago
I think what makes Euler's identity profound to me is that it is a tool for doing so many things. For example, if you want to calculate the log of a negative number: -1 = e^(i * pi) ln(-1) = i * pi ln(-x) = i * pi + ln(x) log_b(-x) = (i * pi + ln(x)) / ln(b) Now you have a formula for calculating the log of a negative number (-x) in any base.

jerf 9y ago

"seems remarkably non-profound." I'd point out the linked article doesn't particularly claim otherwise. It just provides a slightly deeper explanation but doesn't claim it's an explanation of anything particularly meaningful. In other news, if you'd like to see the Golden Ratio strut its stuff in a context not related to the supposed aesthetics of the ratio, this relatively recent (and relatively in-depth, for the channel) video and linked proof may be interesting to you: https://www.youtube.com/watch?v=FtNWzlfEQgY https://www.youtube.com/watch?v=FtNWzlfEQgY You'll seriously be asking "what was jerf talking about?" for like a third of the proof video before foom in comes phi, and it takes quite a bit of abuse before the end.

deleted 9y ago

[deleted]

montalbano 9y ago

This one's a goodun though, the significance of the closeness of 196883 and 196884: https://en.wikipedia.org/wiki/Monstrous_moonshine https://en.wikipedia.org/wiki/Monstrous_moonshine https://www.reddit.com/r/explainlikeimfive/comments/2im1l1/eli5_196884_196883_1/ https://www.reddit.com/r/explainlikeimfive/comments/2im1l1/e...