What is the Analemma?
Well, it is certainly best understood over time, so, first, let's start with THE EQUATION OF TIME. A mathematical construct with a fantastically ambitious name, The Equation of Time is used to find the LOCATION of the sun in the sky on a specific day, at a specific time, at a specific location. It looks like this:

The Equation of Time is, more simply, the DIFFERENCE between the time on a sundial and the time on a clock. Of course the sundial came before the mechanical clock so one way to think of this difference is simply as the past trying to sync up with the future, or how much faster or slower tomorrow should be than today or yesterday.

Got it?
Not yet.

This discrepancy between solar time (sundial) and mean time (clock) is due to two primary factors -- ECCENTRICITY and OBLIQUITY. The earth does not orbit the sun in a precise circle, but instead in an ellipse. it travels faster at some points than others -- this is its eccentricity. The earth's axis does not run directly 90 degrees, but rather it is tilted (23 degrees) which causes the earth's rotation to be like a top -- this is its obliquity.
These two conditions result in the difference that is expressed by The Equation of Time as TWO COMPETING SINE WAVES, one with a period of one year, and one with half of that. The difference over the course of one year between SOLAR and MEAN time can be up to 30 minutes. The earth's eccentricity produces A SINE WAVE WITH A PERIOD OF ONE YEAR. Its OBLIQUITY produces ASECOND SINE WAVE, BUT WITH A PERIOD OF ONE HALF-YEAR.

Then.

The result of these two competing sine waves, with differing periods of repeat is is a recurring figure eight, which draws itself in the sky OVER THE COURSE OF ONE YEAR as the two sine waves fall into and out of phase -- This is the Analemma.

So. . .

If you take a photograph of the sun in the sky at the same time of day over the course of the year from a fixed camera, and composite the images, this is what you get:

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Analemma Over New Jersey

Got it?

Easily mistaken for the infinity sign, a circle or any number of more complex pretzels and knots, the Lissajous Figure is a picture of compound harmonic motion named for French physicist and mathematician Jules Antoine Lissajous (1822-1880). The shape is drawn by plotting a two-variable parametric equation as it iterates itself over time – the resulting figure is the picture of two systems falling into and out of phase.

In 1855 Lissajous constructed his "beautiful machine," devised to draw a picture of two systems superimposed and constructed in his Paris workshop of a pair of tuning forks placed facing at right angles, each with a mirror attached. The light source is focused through a lens, bouncing off the first onto the second and projecting to a large screen a few feet away. As the tuning forks are struck and tones are produced, simple vibrations begin to move the mirrors in a regular oscillating pattern. The projected image begins to form the strange and beautiful curves of a Lissajous Figure.

For his machine Lissajous was awarded the Lacaze Prize in 1873 and was exhibited at the Paris Universal Exhibition in1867. He did not otherwise distinguish himself as a scientist or mathematician. In fact, almost fifty years earlier American Nathaniel Bowditch had already produced similar figures with his harmonograph.

The simple harmonic motion which Lissajous was measuring is easily described by the motion of a clock's swinging pendulum. As the pendulum swings its speed isn't constant, but rather it accelerates and decelerates following a precisely predictable curve. If plotted over time, as the clock ticks the motion of its pendulum draws a sine wave – the so-called "pure wave" or zero-picture of a simple moving system. Ocean waves, sound waves, light waves, even average daily temperatures all produce this same oscillating sine wave pattern.

Compound harmonic motion, then, is simply the superimposition of two sine waves as they register, interfere and produce a series of overlapping waves. When juxtaposed at right angles, two sine waves recording simple harmonic motion produce the surprisingly complex figures that Lissajous identified.

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Sound Pattern 03

Sound and harmony made visible. Lissajous patterns made with 2 audio oscillators, a loud amplifier, plastic wrap, a bowl, and a laser pointer.

Lissajous Figures can be easily found today in computer graphics, in science museums, in laser light shows and, perhaps most precisely, burned into the green phosphor screen of a cathode-ray oscilloscope. A standard piece of electronic test equipment, the oscilloscope allows signal voltages to be viewed as a two-dimensional graph of potential differences, plotted as a function of time. When testing an electronic system, the phase differences between two signals form opposing sine waves on the screen of the oscilloscope connected together, constantly drawing and redrawing themselves in a precise and regular pattern.

These two varying signals produce a perpetual infinity (figuratively and literally as it will actually construct itself in the shape of the infinity sign given the right initial values). The Lissajous Figure becomes a picture of timing and sequence, registration and resonance, sound and music.

Specific shapes are produced corresponding to the resonating harmonic intervals familiar from western music (major fifth, minor third, major sixth, etc.) Any figure may be transformed into any figure and an infinite number of in-betweens as the oscillating sine waves pass in and out of harmonic resonance.

Jules Antoine Lissajous created a way to see sound (using mirrors, light and vibrating tuning forks.) But the most radical possibility of his mathematics might be in the commitment it asks of its audience. The image that Lissajous produces forms slowly right in front of your eyes — imperceptibly changing, forming, adjusting and re-aligning over time.