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).