[squeakdev] [offtopic] Fourier transform
Eliot Miranda
eliot.miranda at gmail.com
Thu Aug 4 18:15:40 UTC 2022
On Thu, Aug 4, 2022 at 9:55 AM Yoshiki Ohshima <Yoshiki.Ohshima at acm.org>
wrote:
> It's fun indeed.
>
> I remember making some Etoys to draw some regular waveforms...
>
> [image: e.jpg]
>
<3 <3 <3
Relating sine waves to square waves, triangle waves, etc, is super valuable
math. Interference also relates to e.g. the uncertainty principle. An old
friend of mine, who shall remain nameless, and who was actually a
computational physicist, not a physicist, was a tutor at the open
university in the summer when he wasn't at rutherford lab/cern. And he
taught the uncertainty principle completely wrongly because he'd never
understood the idea of the interference of superposed sines of different
frequencies. He taught that one can not know the energy and the position
of a particle at the same time. This is wrong.
If a particle is a wave, then its energy is determined by the particle's
frequency. The higher the frequency the higher the energy. if it is
composed of a single sine wave then we know exactly what its energy is, but
we have no idea where it is sine the sine wave exists from infinite to
+infinity. If one interposes an infinite range of frequencies of sines of
the same amplitude and phase they constructively interfere at one point
only and destructively interfere everywhere else, the so called delta
function. [Imagine two sines of different frequencies generating the sum
and difference frequency beats, think bent guitar double stops, then add a
third wave of a different frequency, now the beats are longer, etc, etc.
This proves for example that any function can be expressed as a sum of
sines, since any function can be constructed from an infinite set of delta
functions]. So if a particle is a wave, then once we have a delta function
we know exactly where the particle is, but we know nothing about its energy
since it is composed of an infinite spread of frequencies. However,
between those two extremes there are an infinite variety of possible wave
packets of differing widths. The more constrained in space the particle is
the more frequencies in the packet there are. The less constrained the
fewer the frequencies. So we can know to some degree both the position and
energy of a particule, but the more we know one the more uncertain the
other becomes. And the principle falls out precisely from wave
superposition.
> On Thu, Aug 4, 2022 at 7:58 AM Vanessa Freudenberg <vanessa at codefrau.net>
> wrote:
>
>> On Thu, Aug 4, 2022 at 01:15 Stéphane Rollandin <lecteur at zogotounga.net>
>> wrote:
>>
>>>
>>> > I'm unable to think algebraically very effectively but can
>>> > think visually (for example I didn't understand the fourier transform
>>> > algebraically (the double integral formulation), but understand it
>>> > perfectly well as an infinite set of infinite integrals of the
>>> products
>>> > of a sine wave with an arbitrary waveform (itself composed of sine
>>> > waves)).
>>>
>>> As a visual person myself, Fourier transform did only really click with
>>> me intuitively when I saw it related to epicycles. See Mathologer's
>>> video here:
>>>
>>> https://www.youtube.com/watch?v=qS4H6PEcCCA
>>>
>>>
>>> Stef
>>
>>
>> Thank you for that video! Really enjoyable – I knew the epicyclic
>> explanation for how Fourier synthesis can generate a curve, but never
>> understood Fourier analysis, how to find the factors for a given curve. I
>> had a light bulb moment in the last part of the video where all the
>> integrals in the infinite sum become zero except for one particular term.
>> Beautiful!
>>
>> Vanessa
>>
>>>
>>
>
> 
>  Yoshiki
>
>
>

_,,,^..^,,,_
best, Eliot
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