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From:
James Fischer <[log in to unmask]>
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Date:
Wed, 28 May 2003 14:52:32 -0400
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Rick Green said:

> Nature has many examples of six-sided structures.
> Honeybees and other insects make six-sided cells...
> The mathematics of six sides is special in nature.

Given that bees have brains no bigger than a pinhead,
hexagons are a very elegant approach, and an amazing
feat of engineering for an insect.

But one can do much better, and at a very fundamental
level, "nature" does much better all the time, as it
will show anyone with some soap and a bottle.

Hexagons are NOT "magic", are not "special", and are not
even close to the most efficient solution to the problem
at hand.

What you are working on is the "Kepler Conjecture".
It has to do finding the densest packing of various
solids, like spheres, cubes, or hexagons.  This is
often called "sphere packing" as a generic name.

Tiles and tiling are really the same exact game, but
in 2 dimensions.  Check out "Penrose Tiles" and the
art of MC Escher for some good examples.

A hexagon is not an "answer" to the Kepler Problem.
A hexagon is not "special", nor is it "favored" in
nature.  No one has a perfect proof solving the Kepler
Problem yet, but a number of people have gotten close
enough to convince everyone that they solved it for a
while.  Entire lives have been wasted on this problem.
Don't try solving it at home.

If you put some soap and water into a bottle and shake,
you can fill the bottle with a foam of bubbles, and see
many of them form 14-sided regular solids all by
themselves, as if by magic.

Sorry to burst your bubble, but you just did a quick
experiment that proves that a hexagon is NOT the most
efficient way to utilize space. It is "close", but
not very close.

Lots of metals are "packed" into this same 14-sided
shape.  It is called a "tetrakaidecahedra".  If any
solid has a claim on being "nature's way", then it
is this shape, and not the hexagon. But even it is
not "perfect" in any way, and it has no claim to any
magical properties.

Lord Kelvin must have stared long and hard at the head
on his beer back in the 1800s, and was the first to notice
that this 14-sided shape was a possible solution to the
Kepler Problem.

In 1993, Weaire and Phelan improved on Kelvin's solution
by a whopping 0.3% by using two different shapes in sets
of 8 (two dodecahedra,  and six weird 14-sided things
that I'm not sure even have a name.)

But using two different solids is considered "cheating"
by some, so we are left with the tetrakaidecahedra.

The hexagon is a very good engineering compromise, but there
is nothing special about it, and no basic truths about nature
or the universe emerge from it.

(For those who took advanced math, yes, it is true that in 3-D,
face-centered cubic close packing and hexagonal close packing
both give the same "n" of 74 point something percent, but bees
make hexagonal LATTICES, which are nowhere near as dense as
"close-packed hexagons".)

So, Plato started us off with his mystical mumbo-jumbo about
"perfect spheres", but it turned out that ellipses are the
way the universe draws "circles".  Likewise, Plato drew
hexagons, but tetrakaidecahedra are really the most common
and best space-filling shape one can find in nature (if one
uses a microscope).

There's a fine line between "applied topology" of this sort
and mysticism.  Plato fell into the trap, and Buckminster Fuller
jumped in with both feet, but their rantings were never correct,
and we should NOT mislead our kids that Platonic solids are "special".
They are just "simple shapes".

The universe is messy and complex.
We don't really understand very much of it.
Bees don't understand it either.
But bees CAN make hexagons without a ruler, and most of us can't.
Isn't that amazing enough?

                jim (who is out of a job if the
                 universe runs on magic)

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