Hexagons
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Hexagons are polytopes with six edges and six vertices.
Hexagons are polytopes with six edges and six vertices.

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Originally shared by ****
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The Power of Hexagons: How to animate the rotation of a block
A cube is an illusion, drawn in 2D to appear 3D. One consequence of this is that a two-dimensional rotation will not automagically make the cube appear naturally rotated on a three-dimensional plane. This is because no account is taken of perspective in two...
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This paper is a relevant and insightful contribution to the emerging field of ethnohexagonalism:

"Metaphors of the snake/tree. . . and the turtle/hexagon (possibly representing the powerful and efficient "packing" of divine energy as matter . . . think honeycomb!) are perhaps the deepest totemic meanings in the human psyche as well as the most basic representations of geometric connectivity. . . . In my own geometric odyssey, I was surprised - and then again, not - that hexagonal (six-sided) simulacra at all scales in nature (from patterns in the stars to turtles, honeycombs, and diamonds) are so often chosen, however unconsciously, to symbolize vast questions and choices of human existence. "
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Hexagonal honeycomb patterns often appear in nature, where strength, rigidity and lightness are called for.
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Packing spheres

What's the densest way to pack spheres?  Here are two equally good ways.

In fact there are infinitely many equally good ways!  We start by laying spheres on the plane in a hexagonal arrangement, as tightly as we can.  Then we put a second layer like this on top, with the new spheres resting in the gaps between the old ones.  Then we put on a third layer.  But now there are 2 really different ways to do it!

The spheres in the third layer can be directly above the spheres in the first layer - that's the picture at right.  Or they can be not directly above - shown at left.

As we continue, we keep getting more choices.

One systematic choice is to make the layers alternate like ABABAB.....  That's called the hexagonal close packing, and that's how crystals of magnesium work.

Another systematic choice makes every third layer be the same, like ABCABC...  That's called the cubic close packing or face-centered cubic, and that's how crystals of lead work.

(Why "cubic?"  Because - even though it's not obvious! - you can also get this pattern by putting a sphere at each corner and each face of a cubical lattice. Trying to visualize this in your head is a great way to build your brain power.)

There are also uncountably many unsystematic ways to choose how to put down the layers of spheres, like ABACBCAC....  You just can't use the same letter twice in a row.

In 1611, the famous astronomer Kepler conjectured that sphere packings of this sort were the densest possible.  They fill up

π / 3 √2  =  0.740480489...

of the space, and he claimed you can't do better.

Proving this turned out to be very, very hard.  Wu-Yi Hsiang claimed to have a proof in 1993.   It was 92 pages long.  Experts said it had gaps (pardon the pun).  Hsiang has never admitted there's a problem.

Thomas Hales claimed to have a proof in 1998.   His proof took 250 pages... together with 3 gigabytes of computer programs, data and results!

The famous journal Annals of Mathematics agreed to check his proof with a board of 12 referees.   In 2003, after four years of work, the referees accepted his paper.  But they didn't exactly say it was correct.  They said they were "99% certain" it was right - but they didn't guarantee the correctness of all of the computer calculations.

Hales wasn't happy.

He decided to do a completely rigorous proof using computer logic systems, so that automated proof-checking software could check it.  He worked on it for about 10 years with a large team of people.

He announced that it was done on 10 August 2014.  You can see it here:

To verify the proof, the main thing you need to do is check 23,000 complicated inequalities. Checking all these on the Microsoft Azure cloud took about 5000 processor-hours.

When it was done, Hales said:

"An enormous burden has been lifted from my shoulders.  I suddenly feel ten years younger!"

Personally I prefer shorter proofs.  But this is quite a heroic feat.

I actually wrote about this because I want to talk about packing tetrahedra.  But I figured if you didn't know the more famous story of packing spheres, that would be no good.

For more, check out Hales' free book, which starts with a nice history of the Kepler problem:

• Thomas C. Hales, Dense Sphere Packings: a Blueprint for Formal Proofshttps://flyspeck.googlecode.com/svn/trunk/kepler_tex/DenseSpherePackings.pdf.

For more on computer-aided proof, try this paper:

• Thomas C. Hales, Developments in formal proofs, http://arxiv.org/abs/1408.6474.

The image here was created by Christophe Dang Ngoc Chan and the words translated to English by "Muskid":

https://commons.wikimedia.org/wiki/File:Empilement_compact.svg

#spnetwork arXiv:1408.6474 #formalProofs
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Now this is one way to 'bee' cool and friendly to the Earth.
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This is the {6,3,6} honeycomb, as drawn by .  It's also called the order 6 hexagonal tiling honeycomb. The reason is that has a lot of sheets of regular hexagons, and 6 sheets meet along each edge of the honeycomb.

I've already showed you honeycombs where 3, 4, or 5 sheets of hexagons meet along each edge.  They all live in hyperbolic space, and you can see them all starting here:

http://blogs.ams.org/visualinsight/2014/05/01/636-honeycomb

But now we've reached the end!   Like life itself, some patterns in math are finite - and all the more precious for that reason.  You can do 6, but you can't build a honeycomb in hyperbolic space with 7 sheets of hexagons meeting at each edge.

Puzzle: what do you get if you try?

I don't know, but it's related to this:

• You can build a tetrahedron where 3 triangles meet at each corner.  For this reason, the tetrahedron is also called {3,3}.  You can build a hyperbolic honeycomb where the edges coming out of any vertex go out to the corners of a tetrahedron... and these edges form hexagons.  This is the {6,3,3} honeycomb.

• You can build an octahedron where 4 triangles meet at each corner. The octahedron is called {3,4}.  You can build a hyperbolic honeycomb where the edges coming out of any vertex go out to the corners of an octahedron... and these edges form hexagons.  This is the {6,3,4} honeycomb.

• You can build an icosahedron where 5 triangles meet at each corner.  The icosahedron is also called {3,5}.  You can build a hyperbolic honeycomb where the edges coming out of any vertex go out to the corners of an icosahedron... and these edges form hexagons.  This is the {6,3,5} honeycomb.

• You can build a tiling of a flat plane where 6 triangles meet at each corner.  This triangular tiling is also called {3,6}.  You can build a hyperbolic honeycomb where the edges coming out of any vertex go out to the corners of a triangular tiling... and these edges form hexagons.  This is the {6,3,6} honeycomb.

The last one is a bit weird!   The triangular tiling has infinitely many corners, so in the picture here, there are infinitely many edges coming out of each vertex.   For other weird properties of the {6,3,6} honeycomb, read my blog article.

But what happens when we get to {6,3,7}?

#geometry