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How Come No Two Snowflakes Are Alike?
1986 The Old Farmer's Almanac

1986 Old Farmer's Almanac
1986 Old Farmer's Almanac - RF Cafe[Index]

Reproduced here are various Mathematical Puzzles from The Old Farmer's Almanac, published continuously since 1792. All copyrights hereby acknowledged.

With less than 24 hours left in winter this year (vernal equinox is March 19th at 11:06 pm EDT), I figured I had better get this snowflake article posted now. It appeared in the 1986 edition of The Old Farmer's Almanac (OFA). That's not a publication targeting old farmers, btw. For many decades, I was a faithful purchaser and reader of the OFA, but sometime in the early 2000s, the nature of its contents changed pretty significantly and I lost my interest. I've got enough vintage issues with sunrise and sunset, moonrise and moonset, and high and low tides tables that I don't need newer versions. Besides, all that up-to-the-minute information is available online. But I digress... We've all been told about how no two snowflakes are alike - like fingerprints - and that it has something to so with static electric charges as the flake is formed. Even so, just like DNA controlling how living cells divide and reproduce with and without symmetry, the creation of a snowflake's unique symmetry is and variation is magical.

How Come No Two Snowflakes Are Alike?

How Come No Two Snowflakes Are Alike? , 1986 Old Farmer's Almanac - RF Cafe

It turns out that there is a simple but mind-boggling mathematical explanation that answers this question once and for all ... sort of ...

by Chet Raymo

illustrated by Anne Vadeboncoeur

Is there any truth to the old saw, "No two snowflakes are alike"?

Of course, the question could be laid to rest if someone succeeded in observing two identical flakes. The person who had the best opportunity for doing this was Wilson A. Bentley of Jericho, Vermont. Bentley was a farmer and amateur meteorologist. For 50 years he dedicated himself to observing flakes of snow.

Wilson Bentley was born in 1865 near Jericho. He had almost no formal schooling, but his mother had been a teacher and he acquired from her a lively curiosity and a love for nature's minutiae. Drops of water, bits of stone, the feather of a bird could equally excite his interest. But it was snow that became his lifelong passion.

On his fifteenth birthday, Bentley's mother gave him the use of an old microscope. It was snowing that day, and the boy succeeded in getting a glimpse of a six-sided snowflake with the instrument. By the age of 20, the unschooled farm boy had perfected a technique for photographing the beauty of snow. When death ended the adventure half a century later, Wilson Bentley had accumulated nearly 5,000 microphotographs of snow crystals. He had also won worldwide fame as an expert on the meteorology of snow. In his own neighborhood he was known simply as "the Snowflake Man." An editorial in the Burlington Free Press after his death said: "He saw something in the snowflakes which other men failed to see not because they could not see, but because they had not the patience and the understanding to look."

We have Bentley's word for it that no two snowflakes in his collection were alike. That fact was a source of satisfaction for him. In the simple snowflake he stood face to face with one of nature's deepest mysteries, what the Greeks called "the problem of the One and the Many": how does any form endure in the face of almost limitless possibility? The snowflake exemplified for Bentley the kaleidoscopic balance of order and disorder that is the basis of beauty in nature and in art.

But 5,000 snowflakes is a small number. If we had the patience and understanding to inspect five million snowflakes, or five billion, might we find at least one pair of twins? The answer is almost certainly no.

A single crystal of snow weighs about a millionth of a gram. In a cup of snow there are more than 10 million flakes. I estimate that something like 1022 snowflakes fall on New England in a typical snowstorm (that's 1 followed by 22 zeros). During the four-billion-year history of the earth, perhaps as many as 1034 snowflakes have fallen onto the face of the planet (add 12 more zeros). Could it really be possible that among that unimaginably large number of flakes no two were alike?

Twentieth-century physics has made substantial progress toward understanding the genesis of the snowflake's form. The hexagonal symmetry of snowflakes has its origin in the shape of the water molecule. A water molecule consists of an atom of oxygen and two atoms of hydrogen. The hydrogens are connected to the oxygen in such a way that the two hydrogen "arms" make an angle about like the arms on the side of this x. The angle of the "arms" ensures that when water molecules link together to form a crystal, the resultant symmetry will be hexagonal, just as the placement of the holes in the knobs of a Tinker Toy set determines the symmetry of the structures that can be built with the set.

Now we turn to the probabilities of combination. A deck of 52 cards can be shuffled into 1068 different combinations. A small Tinker Toy set may have something like a hundred pieces; consider, if you will, the huge number of different structures that could be built with such a set. A single snow crystal consists of something like 1018 (one quintillion) molecules of water! The number of ways that many molecules can be arranged into six-sided crystals is astronomical, vastly larger than the number of snowflakes that have ever fallen onto the face of the earth. The odds are very great indeed that no two flakes have ever been exactly identical!

Science has revealed another surprising aspect of the snowflake's form. The apparent stability of a crystal of ice, it turns out, is an illusion. On the atomic scale, the snowflake is a hubbub of activity. Electrons leap and dance. Molecules furiously wave their hydrogen "arms." Crystal imperfections jump from place to place. If you could shrink to subatomic size and enter a crystal of ice, you would think yourself caught in a hurricane of chaos. And yet somehow, in the midst of all that chaos, nature constructs and maintains a crystalline architecture of delicate beauty.

In one sense, no two snowflakes are alike; in another sense, all snowflakes are alike. The staggering diversity of snowflakes is a measure of nature's potential for novelty and change. The constancy of the snowflake's six-sided form reassures us that nature is ruled bylaw.

Wilson Bentley once wrote: "The farm folks up in this north country dread the winter, but I was always supremely happy, from the day of the first snowfall - which usually came in November - until the last one, which sometimes came as late as May." For "the Snowflake Man," snow was a continuing lesson in the way nature's beauty arises from a delicate balance of law and chaos, fixity and change.



Posted March 19, 2024

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