On winter mornings, most people have seen enough snow.
But there are some people who can't get enough, including physicists who study it, mathematicians who ply its sublimely symmetrical growth, even Soldier Field maintenance men who aim snow-making machines at sled hills on days when snow clouds loom.
To them, snowflakes are beautiful and mysterious, their secrets elusive. It took until just this winter, for instance, to write a computer program that could accurately make a picture of a snowflake. For such a seemingly simple shape, experts say, snowflakes are confoundingly complex.
Scientists have studied snow crystals for nearly 400 years, but don't yet fully understand why they form as they do. The most penetrating insights conclude only when certain kinds might appear.
The biggest, prettiest snowflakes, with wide branching fingers called dendrites and sectored hexagonal plates, form when the temperature is about 5 degrees and there is moisture in the air. Lower temperatures, such as minus 20, create shapes like minuscule columns and compact plates. Snow needles form at about 22 degrees.
Warmer air makes small dendrites again. Plates form in drier air. Damp air provides feathered tendrils.
Each starts as a mote of atmospheric dust. A few molecules of water glom on, freeze, then sweep up more vapor as they float through the air. Something like a million microscopic droplets hook on in every direction, creating a six-fold symmetry rooted in molecules. Like Velcro, the branching flakes clump together and fall in clusters.
The basic snowflake shape everybody pictures is the stellar dendrite, with six serrated arms that look like a kindergartner cut them from folded paper.
But observations show that there are 34 more shapes, each distinct enough to earn a place on the widespread chart that serves as a periodic table for snowflakes.
They include solid columns and simple prisms, sheaths, scrolls on plates, triangular forms, hollow columns, cups, capped columns, bullet rosettes, radiating plates, 12-branched stars, simple needles and needle clusters, arrowhead twins, hollow plates, a mish-mash of irregular shapes and tiny nuggets called graupel.
No two are alike, scientists say, because no two are formed in precisely the same atmospheric conditions.
As for the difference between real flakes and snow-machine ice crystals, machines blow out tiny frozen water droplets -- not snow at all.
Caltech physics chairman Ken Libbrecht has devoted much of nine years and four books to snow. He has created snowflakes in chilled boxes, described their appearance and published more than 8,000 pictures of them, more or less as a hobby. No one else will fund it.
His real job -- that is, his funded one -- is studying gravitational disturbances from supernovae and acoustic waves in the sun. He has built magnetic traps for cesium atoms and pondered aspects of physics weightier than the formation of snow crystals.
But Martha Stewart mentioned him on her television show in 2006 for writing his "Field Guide to Snowflakes." Book profits underwrite his research on how ice crystals form, and he isn't tired of doing it yet.
"The more you study it, the more you get completely amazed by water," Libbrecht said. "Even though humans have watched ice grow since before recorded history, we still don't understand how. It's staring you right in the face and we don't understand how it works."
Libbrecht is self-deprecating and careful, a mental repository for man's knowledge of snow crystals. In 2005, he wrote a paper on what has and hasn't been learned, citing 140 researchers and 117 papers back to the 1600s.
Among the scientists are Johannes Kepler, who in 1611 correctly traced the motion of the planets, then dashed off a treatise about snowflakes.
And Rene Descartes made notes of several kinds of snowflakes he'd seen in 1637 -- the year he bridged algebra and geometry with an "x,y-axis" coordinate system, and uttered the famous words, "I think, therefore I am."
Later interest in crystals for metallurgy made snow a hot topic in the 1980s. Libbrecht's lab work and photos in the late-1990s led others into still more research.
With Libbrecht's pictures as a guide, mathematicians Janko Gravner and David Griffeath got attention this winter by revealing that they had a computer model that took basic weather inputs and made convincing snowflake images from them. (The best previous models made things that looked like sea urchins or fern fronds.) The program took them three years to perfect, and each image takes a full day to make.
If forced to choose a favorite, Gravner, of UC Davis, points to Figure 4 in their formal paper. It looks like six feathers with narrow plates branching off midway to their tips, and Gravner describes it as a mathematician might. It is a dendritic snowflake with plate-like extensions. The dendrites extend out along well-defined ridges. It embodies to him a snowflake's internal tension as it chooses between forming wide plates or tiny wispy branches.
"Aesthetically, it shows the interplay between chaos and order," he said.
"I'd think they were about the most beautiful thing I've ever seen," Griffeath said in a telephone interview from his office at the University of Wisconsin in Madison.
Yes, they are compelling because they are so little understood, Griffeath said. And yes, they're beautiful as both art and science.
And yes, at this time of year, they're also nearly unavoidable.
With that, Griffeath said, he had to go home and shovel.