3 ms·

I’m pretty sure people were aware of trig functions before they became aware of Euler’s identity, and became aware of the general idea of “oscillation” before t

by drdeca 4y ago

I’m pretty sure people were aware of trig functions before they became aware of Euler’s identity, and became aware of the general idea of “oscillation” before that.

motohagiography 4y ago
Toward Gauss' goal of engraving a beacon into Siberia to signal our civilization to extraterrestrials, what is more representative of our current evolution and intelligent intent, an elementary theorem from 3,000 years ago written in megalithic land art with rocks and sticks, or, our most succinct representation of mathematical beauty, transmitted with complete consistency over all spectra of sound, radio waves, visible light, and even x-rays? What's unique about this function is that it is an oscillation of reals, but between integers. It's the effect of representing the integers in terms of the real, imaginary, and transcendentals. That oscilating sequence is such that if they have discovered Euler's identity, they would be able to see that we have as well.
- drdeca 4y ago
  My point is that if another human person came up with this sequence themselves, this would not by itself convince me that that person is aware of Euler’s identity. There are many other ways to produce this sequence. For example, ceiling(n/2) times (-1)^n . Or just “it flips between negative and positive, and increases in magnitude by 1 every other step (starting with -1)”. Like, yes, it shows that they are familiar with the concept of something oscillating. It does not demonstrate that they are familiar with any trig functions nor that they are familiar with complex numbers.
  - motohagiography 4y ago
    I'd agree we don't need complex numbers to produce this, because it's right there in the function (cosx(x)/2pi) without them. However, by way of intent, if someone says "hey check out these integers in order, but both positve and negative simultaneously, in the shape of a universal function over reals - as though they thought it was important to show they could produce those integers from the reals. Of what is that an expression?" What other identity or function of reals yields all integers? I don't actually know, but I'm betting being wrong on the internet that the artist behind this song in this context would pick this specific one for both aesthetic and practical reasons. However, let's say I'm wrong but wanted transmit Euler's Identity in a universal physical signal anyway, what function would we use? In the piece, there is a piano accompanyment of triads I haven't done the analysis of to see how they are linked to functional melody notes - but given the composer states his music is an expression of geometric principles, there may be more information in that, and if I persist in my folly instead of posing the half baked version in a comment, I will become wise. Thank you for indulging, the friction refines the idea faster.
    - drdeca 4y ago
      Maybe I should have included this small note previously, (I had figured it was a small typo or something) but, x cos(x) / (2pi) does not actually produce integers for integer values of x (other than for x=0) . I imagine what you mean is more along the lines of x cos(pi x) ? However, this does not produce the sequence -1,1,-2,2,... either. Rather, it produces -1,2,-3,4,... Which, again, can be produced with (-1)^x • x , not using any trig functions. I suppose one might object that (-1)^x doesn’t have real values for non-integer values of x, but I’m not sure why that should be super relevant to the “what does this sequence of numbers indicate”. If you want a formula for -1,1,-2,2,-3,3,... then, well, there’s the one I posted earlier, using the floor or ceiling function, or if one wanted one could come up with something for it using multiple uses of trig functions, but it would be inelegant.

by drdeca 4y ago

I’m pretty sure people were aware of trig functions before they became aware of Euler’s identity, and became aware of the general idea of “oscillation” before that.

motohagiography 4y ago

Toward Gauss' goal of engraving a beacon into Siberia to signal our civilization to extraterrestrials, what is more representative of our current evolution and intelligent intent, an elementary theorem from 3,000 years ago written in megalithic land art with rocks and sticks, or, our most succinct representation of mathematical beauty, transmitted with complete consistency over all spectra of sound, radio waves, visible light, and even x-rays? What's unique about this function is that it is an oscillation of reals, but between integers. It's the effect of representing the integers in terms of the real, imaginary, and transcendentals. That oscilating sequence is such that if they have discovered Euler's identity, they would be able to see that we have as well.

drdeca 4y ago
My point is that if another human person came up with this sequence themselves, this would not by itself convince me that that person is aware of Euler’s identity. There are many other ways to produce this sequence. For example, ceiling(n/2) times (-1)^n . Or just “it flips between negative and positive, and increases in magnitude by 1 every other step (starting with -1)”. Like, yes, it shows that they are familiar with the concept of something oscillating. It does not demonstrate that they are familiar with any trig functions nor that they are familiar with complex numbers.
- motohagiography 4y ago
  I'd agree we don't need complex numbers to produce this, because it's right there in the function (cosx(x)/2pi) without them. However, by way of intent, if someone says "hey check out these integers in order, but both positve and negative simultaneously, in the shape of a universal function over reals - as though they thought it was important to show they could produce those integers from the reals. Of what is that an expression?" What other identity or function of reals yields all integers? I don't actually know, but I'm betting being wrong on the internet that the artist behind this song in this context would pick this specific one for both aesthetic and practical reasons. However, let's say I'm wrong but wanted transmit Euler's Identity in a universal physical signal anyway, what function would we use? In the piece, there is a piano accompanyment of triads I haven't done the analysis of to see how they are linked to functional melody notes - but given the composer states his music is an expression of geometric principles, there may be more information in that, and if I persist in my folly instead of posing the half baked version in a comment, I will become wise. Thank you for indulging, the friction refines the idea faster.
  - drdeca 4y ago
    Maybe I should have included this small note previously, (I had figured it was a small typo or something) but, x cos(x) / (2pi) does not actually produce integers for integer values of x (other than for x=0) . I imagine what you mean is more along the lines of x cos(pi x) ? However, this does not produce the sequence -1,1,-2,2,... either. Rather, it produces -1,2,-3,4,... Which, again, can be produced with (-1)^x • x , not using any trig functions. I suppose one might object that (-1)^x doesn’t have real values for non-integer values of x, but I’m not sure why that should be super relevant to the “what does this sequence of numbers indicate”. If you want a formula for -1,1,-2,2,-3,3,... then, well, there’s the one I posted earlier, using the floor or ceiling function, or if one wanted one could come up with something for it using multiple uses of trig functions, but it would be inelegant.