How was this infinite sequence of numbers found? (non-commutative geometry )

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    Laplace equation
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Discussion Overview

The discussion revolves around the exploration of an infinite sequence of numbers related to non-commutative geometry, specifically in the context of frequencies associated with isospectral drums as presented in Alain Connes' work. Participants inquire about the derivation of this sequence and its connection to the solutions of the Laplacian equation.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • The original poster (OP) references a sequence of numbers (5/4, 2, 5/2, 13/4, ...) and suggests they represent solutions to the Laplacian equation related to drum shapes and their frequencies.
  • Another participant mentions Connes' example of two isospectral drums, highlighting the shapes involved (a triangle and a square, and a rectangle with a different triangle) and seeks information on the frequencies associated with triangular drums.
  • The OP later finds and shares a link that purportedly contains the answer to their inquiry about the frequencies of triangular drums.

Areas of Agreement / Disagreement

Participants do not express disagreement, but the discussion remains focused on the OP's initial question and subsequent findings without a broader consensus on the derivation of the infinite sequence.

Contextual Notes

The discussion includes references to external resources for further exploration of the topic, but does not resolve the mathematical steps involved in deriving the infinite sequence or the specific frequencies of triangular drums.

Who May Find This Useful

Readers interested in non-commutative geometry, mathematical physics, and the study of vibrations in different geometrical shapes may find this discussion relevant.

Heidi
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Hi Pfs,
I read these slides:
https://indico.math.cnrs.fr/event/782/attachments/1851/1997/Connes.pdf
It is about non commutative geometry (Alain Connes)
After Shapes II, you see a the plots of the square roots of a sequence of numbers given below:
5/4, 2, 5/2, 13/4 ....
I think that they are the solutions of laplacian equation related to the shape of the "drums" above. that is to say the frequencies one can get when hitting this drums.
How to retrieve this infinite sequence?
It is at page 50
thanks
 
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In this article Connes gives an example of two isospectral drums (which are non connex).
One drum with a triangle and a quare and the other with a rectangle and a different triangle.
I found this
https://math.libretexts.org/Bookshe...ons/6.01:_Vibrations_of_Rectangular_Membranes
explaining how to get the possible frequencies emitted by a square or rectangular drum.
Do you know what are the frequencies ommitted by a trangular drum (half a square)?
 
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