this is Connes' best lecture series to understand his model of reality.
at 58 minutes into lecture 2 he goes into his music "very simple example" - they have the same frequency spectrum and thus the same area but not the same geometric shape.
"due to Chapman" -
citing
This example was constructed by S. J. Chapman. Notice that both polygons have the same area and perimeter. "Can One Hear the Shape of a Drum?"
ALAIN CONNES:
You can compute this spectrum, it has some multiplicity....it's the square root of some number... and these numbers they fall in two, three classes. A fractional 1/4, integers, and fractional 1/2. We can make a piano out of this. ...The color of each note corresponds to [one of the three classes of notes, the details of the spectrum are not given here]....
I can distinguish these two spaces by which chords you can play in each of them - So I don't havenot only the scale (the piano) but which cords I can play in one [space] and which I can play in the other...
Blue-red [two note chord] is not possible for shape 2 but Blue-red is possible for shape one...each of these notes corresponds to an eigenfunction, a vibration on the disk [or drum]. There is a chord at the point if two of these eigenfunctions do not vanish. ...So it's a very naive simple statement....I have a chord possible at the point - if the two eigenfunctions do not vanish....
So then you get a four-dimensional time-frequency "chord" of two two-note noncommutative chords that is quantum nonlocal to create each point of spacetime.
Alain Connes:
The ear is sensitive to multiplication by 2 and the ear is sensitive to multiplication by 3...
no power of two is equal to power of three, because you have a unique decomposition into prime factors...when you take 2 to the power of 19...it is almost 3 to the power of 12....this is telling you that the 12th root of 2 is almost the 19th root of 3. [Noncommutative!] When you compute them...by doing the continuous fraction expansion....it's not difficult....then you realize that in order to have good music...you see the digits are almost the same.. you have to take the spectrum...We are looking for space by knowing its spectrum...I will tell you what the space is. This will be the end of my lecture. So what is the spectrum? The spectrum are the powers of this number. If you look at a guitar...they are identical to this one... [fret board is identical to the 12 root of 2/19th root of 3]...Now we can ask a mathematical question [shows slide of spectrum of sphere as square root of j x (j plus 1) greater than or equal to zero]...Is there a natural space which has this spectrum....The sphere can not work...This spectrum is a set of powers so it grows exponentially - the sphere did not grow exponentially....what does it tell us that it grows exponentially? It tells us something very profound, that the corresponding space has to be smaller than epsilon [sum] for any epsilon.... Amazingly there is a beautiful answer which is the quantum sphere [meaning variable to the exponential minus variable to the noncommutative inverse exponential]. It will fit within the framework of noncommutative geometry....For the quantum sphere it's a noncommutative algebra.
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