Mathematical duality

by Neil Rickert

There are many dualities that mathematicians study.  In this post, I shall discuss two of them, in order to illustrate some of the ideas involved.

Line, point duality

As a simple example of mathematical duality, consider the duality of points and lines in Euclidean geometry.  And recall from your school days, that when we talk of a line in Euclidean geometry, we are talking of an infinite straight line extended in both directions.  The short “lines” that we actually draw can be called “line segments” so as to distinguish them from the extended lines.

Given any line, there are infinitely many points on that line.  Given any point, there are infinitely many lines that pass through that point.  Given any two distinct points (i.e. points that do not coincide in their location), there is a unique line that passes through both points.  Given any two non-parallel lines, there is a unique point of intersection of those two lines.

You can already see why we call that a duality.  There’s a kind of symmetry between describing lines in terms of points, and points in terms of lines.

The duality, in effect, shows that there are two different ways of describing the same thing.  We can say that points are primary, and define lines in terms of points.  Or we can say that lines are primary, and define points in terms of lines.

The line/point duality is a simple example.  Let’s move on to a more important duality.

Time, frequency duality

We can describe sound as pressure changes over time in a medium such as air, so that we would be describing sound in terms of pressure as a function of time.  Or we can describe sound as tones (or frequencies or wavelengths), so that we describe the sound in terms of the intensity of the tones as a function of their frequency.  It will be convenient to assume one dimension, such as is approximated by the column of air in the ear canal.  That keeps things simpler than if we go to three dimensions.

Again, we have two different ways of describing the same thing.  If we describe in terms of pressure as a function of time, I shall say that we are describing in the time domain.  If we describe as the intensity of tones as a function of their frequency, I shall say that we are describing in the frequency domain.  The Fourier Transform can transform between a time domain description and a frequency domain description.

The relation between a time domain description and a frequency domain description is quite complex.  A single pure tone (or a single frequency) looks like a point, thus like something very localized, when described in the frequency domain.  But, when described in the time domain, it looks like an infinite sequence of pressure oscillations so is not at all localized.  Similarly, a short pulse in pressure looks like a simple localized event when described in the time domain.  If we take its Fourier transform, we see that when described in the frequency domain it shows up as infinitely many frequencies (or tones), so it is not all localized in a frequency domain description.  More generally, what we might describe as a logical reduction in one domain can appear to be a case of an emergent phenomenon in the other domain.

As a case in point, consider the rich tone from a violin.  As described in the frequency domain, we see a combination of many frequencies, all multiples of a base frequency.  These are usually referred to as harmonics.  So that gives us a kind of reduction of the rich tone into more elementary pure tones (or single frequencies).  Seen in the time domain, we have a complex train of pressure changes from which the rich tone appears to emerge.

Our music recording methods are based on using the time domain.  A microphone converts pressure to electrical signals, and our analog recording systems then record those electrical signals.  The playback equipment converts those electrical signals to pressure variations, so that we recreate a close approximation to the original time-varying air pressure.  With a digital recording system, digital sampling of the electrical signal generates a sequence of numbers that can be used to reconstitute an approximation to the signal.

Ironically, analog methods directly produce a pretty good approximation to the original pressure variation that constitutes the sound as described in the time domain.  Digital methods produce a poorer approximation, due to the chopping up caused by digitization.  However, when we take a Fourier transform, and look at that in the frequency domain, we see that the difference between what we get from analog methods and what we get from digital methods, involves only very high frequencies to which the human ear is not sensitive.  And that’s why digital methods work so well.  Typically, when sound is reconstituted from a digital recording, the resulting signal is sent through a high pass filter that removes those very high frequencies.  The way that a high pass filter works is better described in the frequency domain than in the time domain.

When we look at the way the human ear works, we see that is better described in terms of the frequency domain.  It is perhaps even better to use a hybrid account that has combinations of frequencies that are varying over time.

Final comment

I have described two examples of mathematical duality.  I have, I hope, illustrated why philosophical reductionism might be too simplistic a way of looking at complex phenomena.

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