Cantor's Dust and Transmission Errors

In the early 1960’s Mandelbrot was asked by the engineers at his company, IBM, about a perplexing problem of noise in telephone lines used to transmit information from computer to computer (Gleick, 1987:  91).  The IBM engineers thought the noise was the result of ‘external’ factors such as ‘tinkering’ by technicians of equipment (McMurray, 1995:  1308).  When Mandelbrot studied the times when the noise occurred he found long periods of error-free transmission separated by ‘chunks’ of noise (McMurray, 1995:  1308).  When Mandelbrot examined the groups of noise he found that it consisted of smaller error-free periods interspersed with smaller noisy chunks (McMurray, 1995:  1308).  Mandelbrot also noticed that the chance of the noise occurring was the same, regardless of the level of detail being observed (McMurray, 1995:  1308).  What was being observed was scaling, the probability distribution of the noise pattern as being self-similar, in that every time scale had a ratio of noisy to clean transmission and that ratio remained the same at every scale (McMurray, 1995:  1308).

Mandelbrot discovered that the noise was not external or produced by technicians but rather it was spontaneous (McMurray, 1995:  1308).  Mandelbrot had found that with any burst of errors, no matter how short, there would always be periods of totally error-free transmission (Gleick, 1987:  91).  Mandelbrot also noticed a consistent geometric relationship between the bursts of errors and the space of clean transmission (Gleick, 1987:  91-92).  At the time the practice of engineering did not have a framework from which to understand Mandelbrot’s description, but the field of mathematics did in the form of Cantor’s dust (Gleick, 1987:  92).  What Mandelbrot had done was duplicate an abstract construction known as a Cantor set, named after the nineteenth-century mathematician Georg Cantor, in this case a Cantor set arranged in time (Gleick, 1987:  92).  In essence a Cantor set is an interval of numbers from zero to one as represented by a line segment (Gleick, 1987:  92).  The next step removes the middle third which leaves two segments of which then you remove the middle third of each (Gleick, 1987:  92).  The remaining four segments have each of their middle third removed and from this the process continues infinitely (Gleick, 1987:  92).  What remains from this process is an interesting set of points or ‘dust’ that are arranged in a particular order in that they are arranged in clusters (Gleick, 1987:  92)  See Figure 2).

From this Mandelbrot was able to help the engineers by concluding that instead of increasing signal strength to drown out more noise, the engineers should settle for a modest signal and accept the inevitablitity of errors and make use of a strategy of redundancy to find and correct them (Gleick, 1987:  92).  This model changed the way engineers viewed and addressed the noise problem in transmission lines (McMurray, 1995:  1308).