Mark Kac, a Polish-American mathematician, popularized a question in his 1966 paper, "Can One Hear the Shapes of a Drum?" Can you figure out the shape of the drum that made the sounds if you hear someone beat it? Can there be more than one drum shape?
The first person to pose this or related questions was Kac. He received the Chauvenet Prize in 1968 for his 1966 paper. A mathematician at a university in Sweden says that it is well written and easy to understand.
The work of Kac pushed these problems into a mathematical field called isospectral geometry, which inspired researchers to ask similar questions for different shapes and surfaces. An area of research that was sparked by their work is growing.
The mathematicians proved that you can't hear the shape of a drum after 20 years. Multiple examples of drums with different geometries were produced by the team.
One of the mathematicians, Carolyn Gordon, was on a short visit to Europe. The Institute of Oberwolfach is located in the Black Forest. Gordon says that her time at Oberwolfach was just the week when things were falling into place for the research on hearing shapes.
For a long time, she was working on related problems. She studied whether two shapes presented in a sort of abstract manner are the same. She used the other research question to work on the drum problem.
Visitors weren't set up for easy access to the outside world. Gordon says that there was a phone that you could use at certain hours. It was difficult to connect, but it was fun. One of the mathematicians on her team is married to Gordon. The question had been open for a long time and we were eager to get something written up in print.
When the researchers realized that they needed to show two different drums that sounded the same, a turning point occurred. We came up with ideas for other pairs that were more complex. She says they were trying to smash the drums with the paper constructions. The mathematicians discovered that the paper they created didn't work. We went back to the original pair and realized it was fine.
Their work answered a question that was thought to be intractable. To find out the different tunes sent out by a vibrating system is a problem which may or may not besolvable in certain special cases, but it would be hard for the most skilled mathematician to solve the inverse problem.
Many unanswered questions were left after the discovery.
Researchers have solved a lot of problems about hearing shapes.
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Catherine Durso found that you can hear the shape of a triangle in her PhD thesis. According to a 2015 paper, you can hear the shape of parallelograms. The shapes produce different sounds. Additional interesting findings were found in that paper.
She suggests that you make four straight edges for your drums. It would be possible to hear a square one. It sounds special. An equilateral triangle drum would sound different than any of the other ones. You would always be able to hear it among the other people. I like to think that it would sound pleasant.
In the December 2021 issue of Physical Review E, researchers showed that you can hear the shape of a truncated cone.
The idea that you will be able to discern the outline of a given type of shape or surface from its sounds is something that has not been proven.
The question of the relationship between a shape and its associated set of frequencies is not closed from both theoretical and practical perspectives. The case of the drum is not certain if it is the rule or the exception. Everything points toward the latter.
Researchers have taken questions about hearing shapes to places that are difficult to picture.
A problem that was solved way back in 1964 is connected to a recent preprint paper by Rowlett. It involves going beyond the three dimensions of space to a mathematical realm of 16 dimensions.
We're thinking aboutflat tori. She says that a torus is just a circle. In three dimensions, mathematicians refer to tori as having the shape of a glazed doughnuts, but they don't refer to its doughy insides.
When you listen to the shapes of 16-dimensional tori, what happens? The shape of tori can't be heard in 16 dimensions.
There are practical reasons for jumping to the 16th dimensions. The more dimensions you have, the more ways you can change it. It was easy to see the differences in this case.
It inspired Kac to a great extent. That was a big part of getting this field to grow. There is a question of whether one can hear the shape of flat tori. Is 15-dimensional or 14-dimensional a possibility? Rowlet asked.
A desire to discover "the tipping point" between when you can and can't hear the shape of a flat torus was the motivation for Rowlett's recent preprint paper. She says that the shape of tori can only be heard in three dimensions.
The team took a circuitous path to reach that answer. The question had been answered by her students. The problem was solved by Alexander Schiemann in the 90s.
The connection between Schiemann's work and the question Rowlett was pondering was muffled by mathematical differences. The answer to the question was published using number theory language. It wasn't mentioned that key words like "isospectral" were not mentioned. She notes that the paper that proves this doesn't mention the word "torus".
In their not yet published paper, Rowlett, Nilsson and Rydell give three mathematical perspectives on the problem Schiemann studied, building bridges that connect the technical aspects of understanding his results from the three mathematical views.
The people who are interested in these types of problems can get access to the tools from different fields. She thinks that when a different team needs to pull out a related result, they won't have to dig so much.
In the late 1800s, microphones were a new technology and were used to determine the shape of a bell. A team of researchers used microphones that could have shocked Schuster. They used them to demonstrate that you can hear the shapes of rooms.
The researchers used a few microphones in an arbitrary setup to create a 3D geometry of the room. Problems in architectural acoustics, virtual reality, audio forensics, and more could be solved by the scientists.
Since Schuster's time, the landscape of research has changed considerably. Who knows what new sounds and shapes will be explored in the future with the continued meeting of mathematicians from different fields.
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