Collections:
John and Barbara Neuberger
Exhibitions:
Joint Meetings of the AMS, MAA, SIAM, Baltimore, January 2002;
ICIAM July 2003 Sydney, New South Wales, Australia at the
Harbour Convention Centre;
Joint Meetings of the AMS, MAA, SIAM, Phoenix, January 2004
Provenance:
Commissioned by Professor John W. Neuberger, Mathematics,
University of North Texas. John and his colleague Robert Renka in
numerically computed the first forty
eigenfunctions on the Helge Koch fractal snowflake curve
with Neumann boundary conditions. The 6th has threefold symmetry.
Special engraving:
6th eigenfunction of Laplace-Dirichlet with a fractal snowflake
boundary ∂D,
6th eigenvalue ≈ H\>#, H for Helge,
∂F²/∂u² + ∂F²/∂v² = 0,
∂F/∂n = 0 on ∂D (in the sense of distributions)
Dimensions:
Weight:
Materials:
Solid silicon bronze, polished
Price:
Copyright:
Imagine a drum made by floating a membrane over a snowflake
fractal closed curve wave. What would it sound like? What would be its
fundamental frequencies or normal modes? This bronze image is a physical
approximation to the mathematical reality of the 6th eigenvalue, which
mathematically happens to occur near H
\>#, H for Helge.
Cf., Ivars Peterson, "Beating a Fractal Drum",SCIENCE NEWS,
Volume 146, Number 12, September 17, 1994, pages 184-185, cover
" Spiderflake Drum " reporting on the Neuberger-Renka
Dirichlet boundary condition computation. While this was a very non-trivial
numerical computation, the Neumann boundary condition required much
more sophistication, Sobelev spaces and such.
Lately, I believe some of these computations including the
Neumann boundary conditions have been redone by other
methods by student(s) of Nick Trevethen, Cambridge, UK. Time will tell.
Photo credit, Sunforge Studios
