Why spirographs make perfect snowflakes
A snow crystal's hexagonal lattice forces it to grow six identical arms. A spirograph gets to
the same place through arithmetic: the number of arms equals
lcm(ring, wheel) ÷ wheel, so any gear pair whose petal count is a multiple of
six — like ring 144 with wheel 40, which
draws 18 arms — has genuine six-fold symmetry. Of the 72 gear combinations in the classic
set, exactly fifteen produce six-fold patterns, and the best ten of them are on this page.
Most come from ring 144, whose 2⁴·3² factorization is unusually
generous with sixes.
Printing and using the snowflakes
The Print all snowflakes button lays each design on its own page, black-on-white, sized for US Letter or A4 — ready for coloring, window decorations, or tracing onto card for cutting. The per-flake Download PNG buttons export at 4× resolution for posters or transfer prints. For classroom use, snowflakes pair naturally with the LCM lesson plan: have students predict which gear pairs will produce six-fold symmetry before drawing them — it's the same divisibility reasoning in seasonal dress.
To design your own flake, open any of these in the visualizer and raise the pen position toward 100% for sharper, icier arms — or layer two six-fold patterns from the same ring (say 144/40 over 144/60) for a crystal with both coarse and fine structure. More non-seasonal designs live on the printable patterns and mandala coloring pages.