Can one hear the sound of a simplicial complex?
Can you hear the sound of a simplicial complex?
(See also this blog entry on quantum calculus)
A simplicial complex is a finite set of sets invariant under the process of taking non-empty subsets. The connection Laplacian
L(x,y) is 1 if x and y intersect and 0 else. We see here the Barycentric refinement graphs. It has the sets of G as vertices
and connects two if one is contained in the other. Studying the geometry of simplicial complexes is so closely related to
Graph geometry.
The
energy theorem
tells that the sum of the matrix entries g(x,y) of the inverse g of L is the Euler characteristic of G
which is the number of sets with odd number vertices minus the number of sets with even number of vertices.
This is also the number of positive eigenvalues minus the number of negative eigenvalues of L. The sum of the eigenvalues
is the trace of L which is the number of simplices in the complex. We currently still don't have found two isospectral
simplicial complexes! It is still possible that one can hear a simplicial complex.
CLICK AND HOLD THE MOUSE BUTTON DOWN TO HEAR G
You hear the eigenvalues of the connection matrix L. Here are direct links to the
MP3's:
1D circle,
Dodecahedron,
Homology sphere,
1-simplex,
Tetrahedron,
Klein bottle,
Projective plane,
Tesseract,
3 sphere,
Torus,
Cube,
Dunce Hat,
Icosahedron,
triangle,
4-simplex,
Octahedron,
Star graph,
Twenty four cell.
Oliver Knill, October 22, 2017.
