The Rhythm of Numbers · Physics · added
Butterfly of a Number
An electron on a crystal lattice in a magnetic field can only have certain energies. Which ones depends on a single number: the magnetic flux through one cell of the lattice. Set the flux to a fraction p/q and the energies split into q bands. Follow a number down its continued fraction and the bands split level by level, in the same counts as its rhythm.
How to use it
- Choose a number from the buttons, or type the terms of any continued fraction.
- Look at the butterfly. Each green row is one fraction. The gold rows are your number’s own fractions, and the dashed line is the number itself.
- Step through the levels under the picture. For pi, the 113 bands of 16/113 fall into seven groups of 16, with one of 17 in the middle.
- Press Play to hear the bands from low energy to high. Each band is a note and each gap is a rest.
The butterfly, with this number's levels marked
Across: energy, from −4 to 4. Up: flux from 0 to 1. Each green row is one fraction p/q with q up to 40. Gold rows are this number's own fractions; the dashed line is the number itself.
One level at a time
All levels of this number
| fraction | bands | grouped by the widest gaps | total width |
|---|
Total width is the combined length of all bands. It shrinks like 9.33/q, so for an irrational flux nothing solid is left: the allowed energies form a dust, like the Cantor sets of the fractals chapter.
What is known and what is ours. The spectrum is Hofstadter's (1976); the splitting of bands by the continued fraction, and the whole numbers attached to the gaps (they set the measured Hall conductance), are established physics, seen in graphene superlattices in 2013. Reading the levels as the bars, phrases and sections of a number's rhythm is this project's addition.
Read more
Large and small, in plain words →Kepler’s Figures →The paper, Chapter 18 →