Hexagon or Square Lattice?

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matthew

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Aug 18, 2002, 12:56:56 AM8/18/02
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Why do CA experiments use a square lattice instead of hexagonal one?

If nature is indeed based upon CA then the lattice it appears to be
using is hexaganol because of the concept of circle packing. Bees
create hexagaonl honeycombs and not square ones because of the greater
efficiency in space.

Is there any good literature that documents using hexagons instead of
squares for cellular automata? Is it possible to use hexagons with
the CAStep function in Mathmatica?

thanks
matt

David Eppstein

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Aug 18, 2002, 2:11:24 AM8/18/02
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In article <8ce604c2.02081...@posting.google.com>,
ma...@bt1.com (matthew) wrote:

My impression is that the physicists use the square lattice because they
think the specific choice of lattice shouldn't be important and it's
easier to write programs that way.

Cellular automaton enthusiasts use the square lattice because Life runs
on a square lattice while similarly interesting rules on other lattices
seem harder to come by. And it's easier to write programs that way.
And all the programs already written by other cellular automaton
enthusiasts use the square lattice...

But there has been some interesting hexagonal-lattice work, see e.g.
http://www.cse.sc.edu/~bays/trilife3/home.html
http://www.bayarea.net/~maydwell/htdoc/ca/hexgallery.html
http://www.unm.edu/~keithw/strangeUniverse.html
http://www.hexatron.com/hexca/index.html

--
David Eppstein UC Irvine Dept. of Information & Computer Science
epps...@ics.uci.edu http://www.ics.uci.edu/~eppstein/

George Maydwell

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Aug 18, 2002, 2:46:08 AM8/18/02
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On 17 Aug 2002 21:56:56 -0700, ma...@bt1.com (matthew) wrote:

>Why do CA experiments use a square lattice instead of hexagonal one?
>

Interesting hexagonal rules have traditionally been difficult to
discover.

>If nature is indeed based upon CA then the lattice it appears to be
>using is hexaganol because of the concept of circle packing. Bees
>create hexagaonl honeycombs and not square ones because of the greater
>efficiency in space.
>
>Is there any good literature that documents using hexagons instead of
>squares for cellular automata? Is it possible to use hexagons with
>the CAStep function in Mathmatica?

You might be interested in reading my paper on hexagonal cellular
automata techniques, available at:
www.collidoscope.com/ca/hcat.html

A few screen captures from the corresponding hexagonal ca software are
at:
www.collidoscope.com/gallery/welcome.html


George Maydwell
--
Modern Cellular Automata: www.collidoscope.com/modernca
Collidoscope Hexagonal Screensaver: www.collidoscope.com

Tim Tyler

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Aug 18, 2002, 3:34:52 AM8/18/02
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matthew <ma...@bt1.com> wrote:

: Why do CA experiments use a square lattice instead of hexagonal one?

Not all of them do. Some probably do so for the sake of ease of programming.

One of the issues is that most LCD display devices - and most graphics
formats - have already adopted rectangular configurations.

: If nature is indeed based upon CA then the lattice it appears to be


: using is hexaganol because of the concept of circle packing. Bees
: create hexagaonl honeycombs and not square ones because of the greater
: efficiency in space.

Nature is not two-dimensional, though ;-)

Interestingly there is no exact three-dimensional equivalent to hexagonal
packing.

You can densely stack spheres in three dimensions - but there are an
infinite number of ways to do it.

The "best" way is the FCC structure, but (unlike hexagonal packing) that
doesn't have enough symmetries to support properly asymptotically-isotropic
lattice gasses.

: Is there any good literature that documents using hexagons instead of


: squares for cellular automata? Is it possible to use hexagons with
: the CAStep function in Mathmatica?

I link to a number of hexagonal CA from: http://hex.org.uk/links/

You can run hexagonal rules on any general CA simulator that supports the
Moore neighbourhood - using the isomorphism described on:
http://hex.org.uk/map/
--
__________
|im |yler http://timtyler.org/ t...@tt1.org

Siamak

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Aug 18, 2002, 11:06:27 AM8/18/02
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Hi,

Tim Tyler has a nice web page for Hexagonal Engineering:
http://hex.org.uk/

Also Kenichi Morita (http://kepi.ke.sys.hiroshima-u.ac.jp/~morita/)
has worked on hexagonal CAs. (embedding reversibility, computation,
life in hex CAs...)

I completely agree with David Eppstein: The reason people usually
prefer square lattices is that writing programs for them is easier.
Actually it is not difficult to see that Hex CAs can be simulated in
square ones (and vice versa) with a little cost.

But still, I think Hex CAs have worth of more attention. Hex cells
have six neighbors, i.e. more powerful than von Neumann's
4-neighborhood, and still they are very symetrical in appearence,
contrary to Moore's 8-neighborhood.

Siamak Taati

Siamak

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Aug 18, 2002, 3:58:58 PM8/18/02
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Tim Tyler <t...@tt1.org> wrote:
> You can densely stack spheres in three dimensions - but there are an
> infinite number of ways to do it.
>
> The "best" way is the FCC structure, but (unlike hexagonal packing) that
> doesn't have enough symmetries to support properly asymptotically-isotropic
> lattice gasses.

Why FCC? And why not CCP? Doesn't it look more like hexagonal? And
isn't it more symmetric than FCC? Supporting isotropic behavior needs
more symmetries or more density? Which one is denser?

Thanks, Siamak

Tim Tyler

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Aug 18, 2002, 5:31:46 PM8/18/02
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Siamak <ta...@dpir.com> wrote:
: Tim Tyler <t...@tt1.org> wrote:

: Thanks, Siamak

AFAICS FCC and CCP are synonyms - e.g.:

``There are two possibilities for closed packed structures: Cubic Close
Packed (ccp) and Hexagonal Close Packed (hcp)

Cubic Close Packed (ccp) also comes with a different name, Face Centered
Cubic (fcc).''

http://www.chemistry.nmsu.edu/studntres/chem116/notes/crystals.html

If you mean HCP, that has less symmetries, and the same density.

Here's some material (from Bruce B.) relating to the symmetry issue:

``Finally, note that this is the reason that a 3D version of the CA fluid
is so difficult. In 3D, icosahedral symmetry is required to guarantee
the isotropy of a fourth rank tensor. Alas, there is no regular lattice
with this symmetry in 3D. In 4D, however, there IS a regular lattice
with the required symmetry, and the most promising 3D work to date has
done the CA on this 4D lattice and then projected the flow back down to
3D.''

- http://www.fractalus.com/steve/ca/archives.88.html

Siamak

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Aug 19, 2002, 7:26:38 AM8/19/02
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Tim Tyler <t...@tt1.org> wrote in message news:<H126G...@bath.ac.uk>...
> [...]

> AFAICS FCC and CCP are synonyms - e.g.:

:p I don't know exactly the names.:">

By CCP I meant something like this:
1 1 1 1 1 1 1
2 2 2 2 2 2 2
3 3 3 3 3 3 3
1 1 1 1 1 1 1
2 2 2 2 2 2 2
3 3 3 3 3 3 3
1 1 1 1 1 1 1
2 2 2 2 2 2 2
3 3 3 3 3 3 3
1 1 1 1 1 1 1
2 2 2 2 2 2 2
3 3 3 3 3 3 3
1 1 1 1 1 1 1
( it is more reasonable to be named hexagonal:p )

And didn't you mean by FCC something like this?:
1 2 1 2 1 2 1 2 1 2 1 2

2 1 2 1 2 1 2 1 2 1 2 1

1 2 1 2 1 2 1 2 1 2 1 2

2 1 2 1 2 1 2 1 2 1 2 1

1 2 1 2 1 2 1 2 1 2 1 2

Thanks,
Siamak

Tim Tyler

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Aug 19, 2002, 7:53:16 AM8/19/02
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Siamak <ta...@dpir.com> wrote:
: Tim Tyler <t...@tt1.org> wrote in message news:<H126G...@bath.ac.uk>...

:> AFAICS FCC and CCP are synonyms - e.g.:

: :p I don't know exactly the names.:">

: By CCP I meant something like this:
: 1 1 1 1 1 1 1
: 2 2 2 2 2 2 2
: 3 3 3 3 3 3 3
: 1 1 1 1 1 1 1
: 2 2 2 2 2 2 2
: 3 3 3 3 3 3 3
: 1 1 1 1 1 1 1
: 2 2 2 2 2 2 2
: 3 3 3 3 3 3 3
: 1 1 1 1 1 1 1
: 2 2 2 2 2 2 2
: 3 3 3 3 3 3 3
: 1 1 1 1 1 1 1
: ( it is more reasonable to be named hexagonal:p )

: And didn't you mean by FCC something like this?:
: 1 2 1 2 1 2 1 2 1 2 1 2

: 2 1 2 1 2 1 2 1 2 1 2 1

: 1 2 1 2 1 2 1 2 1 2 1 2

: 2 1 2 1 2 1 2 1 2 1 2 1

: 1 2 1 2 1 2 1 2 1 2 1 2

It's not easy to convey three dimensional structures using two dimensional
diagrams.

Here are diagrams showing what I mean by FCC and HCP:

http://www.kings.edu/~chemlab/vrml/clospack.html

At the top it says:

``Face Centered Cubic (fcc) or Cubic Close Packed (ccp)

These are two different names for the same lattice.''

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