Acoustical Society of America
141st Meeting Lay Language Papers

Sequences from Number Theory for Physics, Signal Processing, and Art

Manfred R. Schroeder-
Drittes Physikalisches Institut
Universitt Gttingen
Brgerstr. 42-44 - 37073
Gttingen - Germany
(My Homepage)

Popular version of paper 3aSP1
Presented Wednesday Morning, June 6, 2001
141st ASA Meeting, Chicago, IL

Number theory has been considered since time immemorial to be the very paradigm of pure some would say useless mathematics. Number theory is the queen of mathematics, according to Carl Friedrich Gauss, the lifelong Wunderkind of arithmetic. What could be more beautiful than a deep, satisfying relationship between the whole numbers---0, 1, 2, 3, and so on. (One is almost tempted to call them wholesome numbers.) In fact, it is hard to come up with a more appropriate designation than their learned name: the integers meaning the "untouched ones". How high they rank, in the realm of pure thought and aesthetics, above their lesser brethren: the real and complex numbers---e.g.2.88 and the square root of -1.

Yet, as we shall see in this talk, number theory can provide totally unexpected answers to real-world problems in physics and engineering, and generate attractive art. Specifically, I will focus on the application of sequences of numbers that have found wide use in concert hall acoustics, deep-ocean monitoring of global warming, error-correcting codes for the Internet, X-ray astronomy, speech synthesis, and precise radar ranging of interplanetary distances for checking Einstein's theory of general relativity. Other applications are in graphic design and the creation of appealing melodies.

How can mere sequences of numbers have so many interesting applications? The basic reason is that the sequences considered here have unique correlation properties, i.e., they have extremely weak relationships or "correlations" with themselves when the numbers are shifted even slightly.. This minimal correlation implies maximal distance, just the property needed for constructing good error-correcting codes. Maximal distance means that two sequences, or code words, are as different as possible to facilitate the unambiguous correction of any errors.

The sequences can be represented as many different sine waves added together (a Fourier transform), to create a visual or graphical pattern that corresponds to the sequence: the number 1 represents a spike, and 0 represents a node (a region of zero displacement). Minimal correlation means that the sequences have a flat power spectrum. A flat power spectrum means that the contribution of each of the sine waves is equal. For wall panels known as "diffusors" that have geometric patterns based on such number sequences, this produces wide-angle diffusion --the spreading of sound waves at large angles-- which is just what's needed for optimum concert hall acoustics.

Most importantly, the sequences allow the construction of constant-power radar, sonar, or lidar signals with a flat power spectrum. These signals have thus constant power over time and a wide range of frequency values.

Constant power in time means maximal power output (from a radar, say) with a peak power limitation. Constant power in the frequency spectrum means (according to Heisenberg's uncertainty principle, which specifies the a level of precision in which certain pairs of variables can be measured) maximum timing accuracy. Thus, these signals allow precision delay measurements in very unfavorable signal-to-noise environments. In this manner the delay of radar echoes from the planets Venus and Mercury in superior conjunction (when they are behind the sun) have been measured with an accuracy of a few microseconds, thereby confirming Einstein's theory of general relativity (which predicts not only the bending of light near the sun but also the slowing of electromagnetic radiation near massive bodies). This accuracy has been achieved in the face of the fact that as little as 10-27, one billion billion billionth, of the outgoing radar energy is returned to earth.

In another application, the delay of sonar pulses transmitted from the Indian Ocean were clearly received in Greenland, a distance exceeding 25 000km. The delay is a function of the average ocean temperature in the deep-ocean sound channel, itself an indicator of global warming.

The basic mathematics underlying these feats is actually quite simple. For a binary sequence of period-length 7, for example, one starts with, say, three 1s, namely 1,1,1. One now adds the leftmost two terms modulo 2 yielding 1+1=0, which is appended to the sequence already in hand: 1,1,1,0. Continuing this process of adding (modulo 2) the two terms two and three places from the right yields 1,1,1,0,0,1,0,1,1,1,..., which, as can be seen, repeats after seven bits.

For error-correcting applications, the first three bits (1,1,1 in our example) are the information bits and the next four bits (0,0,1,0) are the check bits. This is the fundamental "simplex" error-correcting code of length seven. Elaborations of this scheme (ReedMuller codes, for example) are ubiquitous in error correcting schemes for transmissions from and to space vehicles, computers and the Internet.

The "duals" of the simplex codes (in which information and check bits are interchanged) are the famous Hamming error-correcting codes. For code length seven, the Hamming code adds three check bits to four information bits and can correct precisely a single error (or signal that no error is present).

For other applications, the 0s in the original binary sequences are converted to 1s and the 1s are replaced by -1s. This yields -1, -1, -1, 1, 1, -1, 1, -1, -1, -1, ... which has the already mentioned correlation property: shift the sequence by any amount (other than a multiple of 7), multiply the original and the shifted sequence and add seven consecutive products. This always yields the same low value -1 (compared to multiplying without shifting which yields the high value +7). This correlation property leads directly to the flat-spectrum property that is crucial in many applications.

As an example for the application of number theory in graphic design, see the accompanying illustration "Prime Spectrum" by the author, which shows a pattern directly related to the distribution of prime numbers. To wit: if two whole numbers do not share a common divisor (such as 33 and 35, for example) they are called coprime. Now consider all coprimes from the one million pairs of numbers selected from the range 1 to 1000 and perform a Fourier transform of the coprime data. Fourier transforms extract periodic behavior from any given data and the bright stars illustrate the fact that the distribution of coprime numbers is periodic in two dimensions. The two dimensions here are simply the two dimensions seen in the illustration, corresponding to the two numbers, each running from 1 to 1000 (for more information "Number Theory in Science and Communication."), But the actual result of the computation shown here exceeded the author's expectation.

The included musical example (courtesy Lars Kinderman/Robin Whitehead) shows that simple sequences of "Baroque Integers" of numbers can lead to attractive melodies. The sequence, devised by the author, demonstrated here is derived from nothing (0) by appending one more than nothing (1) to give 01. Then 1 is added to each term and appended to the two terms we already have yielding 0112. This process of "add 1 and append" is repeated over and over again: 0112 1223 1223 2334 ... . Next every nth term of this sequence is selected, yielding, for n=3, 1 2 1 3 3 3. As a final step the numbers are converted to musical notes in the C-major scale, say: 1=C, 2=D, 3=E etc. Repeated notes are not repeated but simply held longer, thereby imparting a certain rhythm to the tune, which has an appealing baroque, Scarlatti-like quality. This is in effect an example of fractal music; for more information, see the author's book "Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise."

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