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Topological data analysis hearing the shapes of drums and bells

Mark Kac asked a famous question in 1966 entitled Can one hear the shape of a drum?, a spectral geometry problem that has intrigued mathematicians for the last six decades and is important to many other fields, such as architectural acoustics, audio forensics, pattern recognition, radiology, and imaging science. A related question is how to hear the shape of a drum. We show that the answer was given in the set of 65 Zenghouyi chime bells dated back to 475-433 B.C. in China. The set of chime bells gradually varies their sizes and weights to enable melodies, intervals, and temperaments. The same design principle was used in many other musical instruments, such as xylophones, pan flutes, pianos, etc. We reveal that there is a fascinating connection between the progression pattern of many musical instruments and filtration (or spectral sequence) in topological data analysis (TDA). We argue that filtration-induced evolutionary de Rham-Hodge theory provides a new mathematical foundation for musical instruments. Its discrete counterpart, persistent Laplacians and many other persistent topological Laplacians, including persistent sheaf Laplacians and persistent path Laplacians are briefly discussed.