Papilla

Papilla's wings are made of allowed energies. Turn up the magnetic field and her pattern repeats inside itself, forever.

The Physics

Put an electron on a 2-D lattice in a magnetic field and its allowed energies depend on the flux through each cell. When the flux ratio is a fraction p/q, the band splits into q subbands; when it is irrational, the spectrum becomes a Cantor set. Plotted against the flux, the energies form a self-similar butterfly.

ψ_{n+1} + ψ_{n−1} + 2 cos(2πnα − ν) ψ_n = E ψ_n, α = Φ/Φ₀

• ψ_n: wavefunction amplitude on row n (dimensionless)
• α: flux per cell in flux quanta (dimensionless)
• Φ: magnetic flux through one cell (Wb)
• Φ₀: flux quantum h/e (Wb)
• ν: phase (rad)
• E: energy in units of the hopping energy t (dimensionless)

Douglas Hofstadter computed the spectrum in 1976; in 2013 graphene moiré superlattices showed its fingerprints in experiments.

Honest caveat: Those experiments saw butterfly features, not the whole fractal.

On the shirt

Front: Papilla, the pixel-art character, with the title. Back: Harper's equation, the p/q rule, the Cantor-set limit and the 1976 and 2013 sources. Printed on a unisex tee.