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.
The Pythagorean creed was: 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 only on the ratio of the frequencies and on the phase. Two equal frequencies give a circle at the right phase, corresponding to a full revolution in the geometry of polygons.
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. Above it 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.
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 field is still largely unexplored.
The four great constants
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 of Max Planck, the founder of quantum physics: that what is real is not visible, perishable matter, but the invisible spirit without which matter would not exist at all.