Paper Snowflake Maker

Most paper snowflakes are wrong before the scissors come out. A square is folded in half, then again, then once more, and what unfolds has four arms or eight. It is pretty. Snow never makes that shape.

Snow makes six. The National Oceanic and Atmospheric Administration says that however a crystal’s arms branch as it grows, the six-sided shape is always maintained. The remedy for the paper version costs one unusual fold. Everything else about the craft stays the same, and the result looks like winter instead of like a doily.

Why ice counts to six

A snowflake, strictly, is a single crystal of ice large enough to fall through the air. NOAA describes how it starts: an extremely cold water droplet freezes onto a particle of pollen or dust, which makes a first tiny crystal, and water vapor then freezes onto that crystal as it falls.

Inside that crystal the water molecules are not arranged at random. Each molecule is two hydrogen atoms bonded to one oxygen atom, and when the molecules lock together in a cloud they form a lattice of hexagonal rings. Jason Benedict, a chemist at the University at Buffalo who studies crystals, draws the conclusion in one line: this crystal structure means that snowflakes are going to exhibit six-fold symmetry. Other lattices exist, Benedict notes, but only under very high pressure or extreme cold, not in the air that makes snowfall.

The link has been measured as well as argued. In 2017 a group at Tufts University led by the chemist Mary Jane Shultz studied large single crystals of ice at two scales, one visible and one molecular, on the same surface. They reported that the points of a snowflake line up with axes of the crystal lattice, and that its flat sides belong to a larger hexagon, a central water molecule ringed by six others in one layer, turned 30 degrees from the puckered ring of the lattice.

The arms come later. A young crystal is a minute hexagon, and the six arms grow from its corners, each one independently. Temperature and humidity can change within seconds as the crystal falls, and every change alters the way the arms grow. All six arms meet very nearly the same conditions at the same moments, so they tend to come out alike. Nothing passes a message from one arm to another. Six thermometers hung in the same room agree in the same way, without consulting each other.

That is also why the famous claim about no two being alike holds up. NOAA’s answer is that every snowflake follows a slightly different path from the sky to the ground. Two that matched would need the same path through the same cloud.

Honesty requires one more fact. Most snow particles are irregular in form, and empirical studies suggest that fewer than one in a thousand shows the ideal six-fold shape. The perfect star is the exception. Paper has an advantage here: folded correctly, it cannot help being symmetrical.

The fold that makes six

Every fold of a paper snowflake passes through the center of the sheet. Fold a sheet in half and the folded edge is a straight line through that center, with 180 degrees of paper on one side of it. Fold that half into three equal parts, one over the other, to make a narrow wedge. Each third is 60 degrees, and the wedge now holds six layers.

Steven Dutch, professor emeritus of natural and applied sciences at the University of Wisconsin in Green Bay, shows how to get the 60 degrees without guessing. His construction leans on the fact that the cosine of 60 degrees is one half: a corner folded over to meet a crease a quarter of the way along the edge lands at exactly the right angle. He adds a remark that most instructions leave out. The 60 degree wedge is the basis for three-fold designs. To get six points, or twelve, the wedge must be folded in half once more, down to 30 degrees and 12 layers.

That remark is the heart of the matter, and it is easy to check with a sheet of scrap paper. A cut made off-center in the 60 degree wedge appears six times in the open sheet, but three of the copies are mirror images of the other three, arranged in three pairs. The pattern repeats every 120 degrees, not every 60. Only when the cuts happen to be the same on both sides of the wedge does a true six-fold figure appear. Folding the wedge in half forces that to happen.

The common mistake is now easy to describe. Halving three times gives angles of 180, 90 and 45 degrees: eight layers and a four-fold pattern, which a symmetric cut turns into eight points. Dutch calls the eight-pointed star the more familiar one, which says something about how people fold.

Starting from a square changes little. Snowy Hydro’s classroom sheet folds an A4 page into a square, then a triangle, then the thirds, and finishes with an instruction worth noticing: cut the end off. The layers of a folded square do not end at the same distance from the center, so the ragged end is trimmed until they do. A circle of paper needs no trimming.

How one cut becomes twelve

Mathematicians have a name for the symmetry of a figure with n mirror lines through one point and n equal turns about it: the dihedral group, written Dn. As the group of symmetries of a regular figure with n sides, it has order 2n. For the 30 degree wedge n is six, so the group has twelve members, and there are exactly twelve layers under the scissors. That is not a coincidence. Each layer is the sheet as one member of the group would place it.

So a hole snipped from the middle of the wedge appears twelve times. A cut that touches a folded edge behaves differently, and more usefully. The fold is a mirror, so the two halves on either side of it join into one shape, and the cut appears six times instead of twelve, each copy twice the size of what was removed.

Here is an example with numbers, which can be reproduced in the tool above. Take a round sheet 180 mm across, folded to the 30 degree wedge. On the right folded edge, 40.5 mm from the center, cut a half circle 10.8 mm across with its flat side on the fold. Unfolded, the sheet has six round holes, each a full circle 10.8 mm in diameter, spaced 60 degrees apart on a ring 40.5 mm from the middle. Each hole has an area of about 92 square millimeters, so that one snip has removed about 550 square millimeters of paper.

Cuts that hold together

The two straight sides of the wedge are folds. The curved or ragged end is the open edge. A design holds together for one reason only: somewhere, uncut paper runs from the center to the rim while touching both folded sides often enough to join each copy to its neighbors.

Three rules follow from the geometry. A cut that runs from one folded edge all the way to the other slices the snowflake into a middle and a ring. A piece of paper that is cut free of both folded edges falls out of every layer at once, as twelve crumbs. And a strip left between two cuts is only as strong as its narrowest point, which on thin paper should not be much under two millimeters.

Thin paper is the practical secret. Twelve layers of ordinary office paper are hard to cut cleanly and the inner layers creep. Lighter paper, a smaller sheet, and scissors that are sharp right to the tip all help, and it is easier to turn the paper into the blades than to steer the scissors around a curve. Cut the big shapes first and the fine notches last, while the packet is still stiff.

A craft older than the microscope

Cutting folded paper is far older than the paper snowflake. The Guild of American Papercutters dates paper cutting in China to the centuries after paper itself was invented there, and notes that the oldest surviving cutout, from the sixth century and found in Xinjiang, is a symmetrical circle. Chinese paper cutting, made for over a thousand years mainly by women who passed it on to their children, joined UNESCO’s list of intangible cultural heritage in 2009. In Switzerland and Germany, scissor cutting is documented from the sixteenth century, with symmetry achieved by cutting the paper while folded. Most Polish wycinanki began the same way, with a sheet folded at least once. Japan has its own word for cut paper, kirigami.

The snowflake joined that tradition through science. People noticed the six points long before anyone could explain them: an anthology compiled by the Chinese scholar Han Ying in the second century B.C. contrasts the five-fold symmetry of flowers with the six-fold symmetry of snow, and Johannes Kepler tried to explain the hexagon in a treatise of 1611. Wilson Bentley of Jericho, Vermont, photographed his first snowflake in 1885 and went on to capture more than 5,000 images. Between those two, in 1864, Frances Chickering published Cloud Crystals, a snow-flake album.

A paper snowflake is still a model, and it helps to know what it leaves out. It is flat, while ice crystals have thickness and facets. Its symmetry is perfect, which real crystals rarely manage. And folding can only produce patterns with mirror lines, so a design that spirals one way cannot be cut from a folded sheet at all. The thirds, though, it gets right. Six is not a decorative choice. It is what water does.