All is number, all is frequency

Two vibrations at right angles draw Lissajous figures. Place them in the lambdoma and waveforms, numbers and star polygons lie in the same places.

Since Aristotle the Pythagorean creed has been summed up in the formula: all is number. One can add: all is frequency. The bridge between the two is the lambdoma, which orders ratios of numbers as ratios of tones.

Lissajous figures

Let two vibrations meet at right angles, for example two tones on the two axes of an oscilloscope, and the point traces a Lissajous figure. Its shape depends on the ratio of the frequencies, the phase and the strength of the two vibrations. Two equal frequencies give a circle at equal strength and the right phase, corresponding to a full revolution in the geometry of polygons.

RatioPhase

The Lissajous figure for the most important intervals. The second slider shifts the phase.

Waveforms in the lambdoma

Place the Lissajous figures in the cells of the lambdoma, numerator horizontally, denominator vertically, and it turns out that the waveforms match the number-theoretical and geometric character of their cell exactly. Particularly striking are the crossings of the curves, reminiscent of the star polygons.

Lissajous figures in the lambdoma up to 6 × 6. Gold: the diagonal of equal frequencies, here circles. Reducible fractions such as 2/4 show the same figure as 1/2.

Only a quarter matches

One reservation must be made: star figures exist only as long as the fraction is greater than 2. Only in this quarter of the table do figure and number coincide. Beyond this bound there are no more star figures. But this quarter is also responsible for the “decimal code” of the tetractys, and it is enough to make the distribution of the primes visible. For the question of the primes only arises above the bound of 2.

This is, however, also a matter of point of view. One can swap the number series of numerator and denominator, and then the two halves of the lambdoma exchange their reciprocals. The cells “divisible” then become cells “not divisible, but with a common divisor”, and some of these in turn become “divisible”. Only where neither number divides the other does the type stay the same. What stays symmetric is the pattern of new, coprime cells and repeated cells. It shows most clearly when the lambdoma is shown in the form that gives it its name, tilted by 45 degrees like the letter lambda.

Light and colour

Another field is the arrangement of light waves in the lambdoma: the octaves of electromagnetic waves, additive and subtractive colour mixing, the colour cube and a hexagram of colours. The idea is that colours, like tones, have a harmonic order as ratios of frequencies. This is worked out on the page The octaves of light. It also shows that complementary colours follow no fixed interval.

The four great fundamental quantities

Taken together: the relationship of wavelength and frequency is that of space and time. Complete it into a fourness and you get the tetractys of energy, space, time and matter, simply as the image 1 + 2 + 3 + 4 = 10 (expressly meant as a metaphor, not as a sum). This fits an often-quoted idea that Max Planck, the founder of quantum physics, expressed in a lecture in Florence in 1944: that what is real is not visible, perishable matter, but the invisible spirit without which matter would not exist at all.

Sources and further reading

Sources

Jules Antoine Lissajous, Mémoire sur l’étude optique des mouvements vibratoires, Annales de chimie et de physique 51, 1857
Max Planck, Über das Gesetz der Energieverteilung im Normalspectrum, Annalen der Physik 4, 1901
Max Planck, Das Wesen der Materie (The nature of matter), lecture in Florence 1944, typescript in the Archives of the Max Planck Society