18
M. R. SCHROEDER
many topics that had to be counted out from this brief account are
1. The application of continued fractions to electrical network problems
which, incredible, led to the construction of the "squared square" [7] -
long considered impossible [8]. (The squared square is a square with
integer sides, completely covered, without overlap, by smaller
incongruent integer squares.
2. Heat conduction in thin tours: the solution is based on the
representation of integers by the sum of two squares [9].
3. The eigenvalue distribution of normal modes in cubical (and near
cubical)resonators,which depends on therepresentationof integers by
the sum of three squares [7]. (The author's first encounter with number
theory, in his Ph.D. thesis on concert hall acoustics.)
4. Search algorithms, game strategies [10] and countless other applications
based on Fibonacci numbers.
5. Certain unexpected properties of the zeroes of Riemann's zeta-function,
found by A. Odlyzko [11], and their possible relation to the Wigner
distribution function (which governs the distribution of energy levels in
the atomic nucleus and eigenfrequencies in complex vibrational
systems).
6. And, most recently, the elucidation of the structure of quasi-crystals, a
new state of matter combining "forbidden" 5-fold rotational symmetry
and sharp, crystal-like, diffraction patterns [12].
What riddle will be solved next by number theory? Is this effectiveness of
the higher arithmetic completely unreasonable? Or are we witnessing here a
"pre-established harmony" $L la Leibniz between mathematics and the real
world?
REFERENCES
[1] M. R. Schroeder, "Diffuse sound reflection by maximum-length
sequences," J. Acoust. Am. vol. 57, 140-150 (1975). See also: M. R.
Schroeder * Toward better acoustics for concert halls," Physics Today,
24-30, Oct. 1980.
[2] M. R. Schroeder, "Constant-amplitude antenna arrays with beam
patterns whose lobes have equal magnitudes," Archiv fiir Electronic
und Uebertragungstechnik (Electronics and Communication) vol. 34,
165-168(1980).
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