No Two Alike
Every crystal here grows from one frozen cell and one local rule, and comes out with six arms that match to the last bit of floating point: the symmetry error reads exactly zero. Nothing copies one arm onto the others. Change the temperature and humidity and the same rule grows plates, sectored plates or ferns; add a little turbulence and the arms start to disagree, the way real snowflakes do.
A snowflake is a single crystal of ice that grew while it fell. It starts as a tiny hexagonal prism — hexagonal because of how water molecules lock together — and then the corners, which stick out furthest into the damp air, catch vapour slightly faster than the faces between them. Faster growth makes them stick out further still, and they run away into arms. The arms sprout side branches for the same reason, and those sprout their own.
The puzzle is the matching. Six arms, each a little tree of branches, and the trees are copies of one another — yet there is no blueprint in the crystal, no signal passing from one arm to the next. The answer is that nothing has to coordinate them. Each arm grows by the same local rule from the same air, and the crystal is small enough, a few millimetres, that the air is the same on every side. Change the air as the crystal falls and all six arms feel it at once, which is why the rings and branches line up. That is the stat on this page: the symmetry error compares every cell with its rotated and mirrored images, and with no turbulence it reads exactly zero.
In the 1930s the physicist Ukichiro Nakaya grew the first artificial snow crystals on a rabbit hair in a cold room in Sapporo and mapped what came out. Shape depended on two things: temperature and how much excess vapour the air held. Plates near 0 °C, needles and columns around −5 °C, plates again below −10 °C, and the big feathery stellar dendrites in a narrow band near −15 °C when the air is wet. Nakaya called snow crystals letters sent from the sky: read the shape and you can tell the weather in the cloud it grew in.
No two are alike because no two fall through quite the same air. Turn on Turbulence and every cell gets a slightly different, seeded share of vapour: the arms stay recognisably siblings but stop being twins, and the symmetry error climbs off zero. Real snowflakes sit somewhere in between — usually close enough to symmetric to fool the eye, rarely perfect.
A silk curtain billowing in a breeze. Grab it, throw a ball through it, cut it with scissors and watch the torn edges flutter.