Take the Fibonacci numbers modulo m — keep only the remainder after dividing by m — and the sequence repeats. The length of that repeat is the Pisano period, written π(m), and it is different for every modulus in a way the arithmetic does not advertise: π(10) is 60, π(11) is 10. Draw the residues instead of listing them and the repeat becomes a figure. Every one here is generated by pisano gallery; for the interactive version — the same designs drawn live in a shell compiled to WebAssembly — see the desktop.
The number line, moduli 1–9
The reference case. Reduced mod m the naturals cycle through every remainder in order, so the chords close up into a regular polygon every time. Everything interesting below is a departure from this.
A period of 60, so far more chords than its neighbors, and still symmetrical.
m=10 · π=60 · 4z · sym
Fibonacci moduli 8, 21, 55
Near-duplicates of one another. All three moduli are Fibonacci numbers, all three periods hold two zeros, none of the three is symmetrical.
m=8 · π=12 · 2zm=21 · π=16 · 2zm=55 · π=20 · 2z
Fibonacci moduli 13, 34, 89
The other family, alternating with the first as you count down the Fibonacci numbers. Four zeros each, symmetrical each — one family looks like the other copied and folded onto itself.
Fibonacci, Lucas, triangular, the naturals, and the primes. Fibonacci is the ingredient, not the recipe. The primes are the odd one out: their residues never repeat, so there is no loop to close and the figure is an open path — mostly landing on 1, 3, 7 and 9.
Multiplying the sequence through by 2 makes the plain designs reappear at every 2th modulus, with new ones interleaved between them. That is the "design m/2" labeling — m is still an integer modulus, and there is no such thing as mod m/2.
Multiplying the sequence through by 3 makes the plain designs reappear at every 3th modulus, with new ones interleaved between them. That is the "design m/3" labeling — m is still an integer modulus, and there is no such thing as mod m/3.
Multiplying the sequence through by 4 makes the plain designs reappear at every 4th modulus, with new ones interleaved between them. That is the "design m/4" labeling — m is still an integer modulus, and there is no such thing as mod m/4.
Odd turns left and steps, even turns right and steps, zero does neither. The naturals alternate odd and even and make a staircase. Fibonacci runs odd, odd, even, so it turns left, left, right — and closes into something that looks like a plus sign. The triangular numbers cancel two lefts against two rights and drift instead.
Now the remainders drive the turns, so a residue of zero also stops it. Some paths close on themselves; others repeat a motif off to infinity, drawn here for three passes. Which one happens is decided by the net turn over a single pass, and nothing else.
These moduli trace the same figure as each other up to where it sits and which way it faces: 6, 12, 16, 18, 24, 36 and 48 all draw one shape, and 14, 28, 32, 42 and 46 another.