[Index]
Reproduced here are various Mathematical Puzzles from
The Old Farmer's Almanac,
published continuously since 1792. All copyrights hereby acknowledged.
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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](images/snowflakes-old-farmers-almanac-1986_small.jpg)
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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