Mathematicians are trying to ‘hear’ shapes -- and reach higher dimensions

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EXCERPTS: More than 50 years ago Polish-American mathematician Mark Kac popularized a zany but mathematically deep question in his 1966 paper “Can One Hear the Shape of a Drum?

[...] More than 20 years after Kac’s paper, three mathematicians proved that you can’t actually hear the shape of a drum. The team was able to produce multiple examples of drums with different geometries that created the same frequencies of sounds.

[...] In the past several decades, researchers have solved a host of problems about “hearing” the sounds of shapes.

It turns out you can hear the shape of a triangle, a result first proved in Catherine Durso’s 1988 doctoral thesis for the Massachusetts Institute of Technology...

[...] You can also hear the shape of a truncated cone—that is, a cone that has its pointy tip cut off, researchers reported in the December 2021 issue of Physical Review E. Also in 2021 Rowlett and her colleagues showed that you can discern the shape of a trapezoid from sounds if it isn’t obtuse.

Yet among all the individual results about hearing shapes, a different team of researchers pointed out a gaping unsettled idea: it remains to be seen whether it is generally true that you will be able to discern the outline of a given type of shape or surface from its sounds.

The question of the relationship between a shape and its associated set of frequencies “is far from being closed, from both theoretical and practical perspectives,” researchers wrote in a 2018 paper presented at the IEEE/CVF Conference on Computer Vision and Pattern Recognition. “Specifically, it is not yet certain whether the counterexamples,” such as the case of the drum, “are the rule or the exception. So far, everything points towards the latter.”

Some of the questions concerning “hearing” shapes have taken researchers to places that are challenging to even picture: higher dimensions... (MORE - missing details)

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