the Lattice Theory Homepage

Wow! ... check out the diagram and text at the top
of this page :

http://www.math.hawaii.edu/LatThy/

Note what it says about cylindrical wrapping!

-monz

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--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> Wow! ... check out the diagram and text at the top
> of this page :
>
> http://www.math.hawaii.edu/LatThy/

I'm afraid that's the wrong kind of lattice--as came up before, there are two different things called "lattice" in English-language mathematics. This kind is a kind of partial ordering, which is important in universal algebra among other things, which is why the univeral algebraists in Hawaii care about it.

> From: genewardsmith <genewardsmith@juno.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Monday, January 21, 2002 12:39 PM
> Subject: [tuning-math] Re: the Lattice Theory Homepage
>

> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> > Wow! ... check out the diagram and text at the top
> > of this page :
> >
> > http://www.math.hawaii.edu/LatThy/
>
> I'm afraid that's the wrong kind of lattice--as came up
> before, there are two different things called "lattice"
> in English-language mathematics. This kind is a kind of
> partial ordering, which is important in universal algebra
> among other things, which is why the univeral algebraists
> in Hawaii care about it.

Are there any other types of lattices or just these two?
(not counting the kind which hold up rose-bushes, etc., of course!)

While I hardly understand it, I'm surprised to see that
Minkowski reduction applies to "our" lattices as well as
the regular mathematical kind, since I knew they are different.

-monz

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--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> > I'm afraid that's the wrong kind of lattice--as came up
> > before, there are two different things called "lattice"
> > in English-language mathematics. This kind is a kind of
> > partial ordering, which is important in universal algebra
> > among other things, which is why the univeral algebraists
> > in Hawaii care about it.
>
>
> Are there any other types of lattices or just these two?
> (not counting the kind which hold up rose-bushes, etc., of course!)
>
> While I hardly understand it, I'm surprised to see that
> Minkowski reduction applies to "our" lattices as well as
> the regular mathematical kind, since I knew they are different.

This is not true, Monz. Both kinds of lattice are "regular
mathematical" kinds. Minkowski reduction only applies to "our"
definition.

Monz,

If you're looking at MathWorld, the definition of lattice that we
care about is found under "Point Lattice".

> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Tuesday, January 22, 2002 9:34 AM
> Subject: [tuning-math] Re: the Lattice Theory Homepage
>
>
> Monz,
>
> If you're looking at MathWorld, the definition of lattice that we
> care about is found under "Point Lattice".

Thanks, Paul!

What about all the stuff listed under "see also"? :

"Barnes-Wall Lattice, Blichfeldt's Theorem, Browkin's Theorem, Circle
Lattice Points, Coxeter-Todd Lattice, Ehrhart Polynomial, Elliptic Curve,
Gauss's Circle Problem, Golygon, Integration Lattice, Jarnick's Inequality,
Lattice Path, Lattice Sum, Leech Lattice, Minkowski Convex Body Theorem,
Modular Lattice, N-Cluster, Nosarzewska's Inequality, Pick's Theorem, Random
Walk, Schinzel's Theorem, Schr�der Number, Torus, Unit Lattice, Visible
Point, Voronoi Polygon"

Any relevance of those to what we do here?

-monz

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--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> > From: paulerlich <paul@s...>
> > To: <tuning-math@y...>
> > Sent: Tuesday, January 22, 2002 9:34 AM
> > Subject: [tuning-math] Re: the Lattice Theory Homepage
> >
> >
> > Monz,
> >
> > If you're looking at MathWorld, the definition of lattice that we
> > care about is found under "Point Lattice".
>
>
> Thanks, Paul!
>
> What about all the stuff listed under "see also"? :

At least "Torus" is relevant.
>
> Any relevance of those to what we do here?

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> "Barnes-Wall Lattice, Blichfeldt's Theorem, Browkin's Theorem, Circle
> Lattice Points, Coxeter-Todd Lattice, Ehrhart Polynomial, Elliptic Curve,=

> Gauss's Circle Problem, Golygon, Integration Lattice, Jarnick's Inequalit=
y,
> Lattice Path, Lattice Sum, Leech Lattice, Minkowski Convex Body Theorem,
> Modular Lattice, N-Cluster, Nosarzewska's Inequality, Pick's Theorem, Ran=
dom
> Walk, Schinzel's Theorem, Schröder Number, Torus, Unit Lattice, Visible
> Point, Voronoi Polygon"

> Any relevance of those to what we do here?

Blichfeldt's theorem, Minkowski's theorem, and the Jarnick-Nosarzewska ineq=
uality are quite relevant, and the Voronoi cell is a bit of terminology I've=
been considering introducing. I keep thinking I'll apply Pick's theorem, bu=
t never have, however I have had occasion to mention random lattice walks.

If you want specific lattices to look at, the root lattice An and its dual =
An* are the key ones, I think.

> From: genewardsmith <genewardsmith@juno.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Tuesday, January 22, 2002 4:17 PM
> Subject: [tuning-math] Re: the Lattice Theory Homepage
>
>
> Blichfeldt's theorem, Minkowski's theorem, and the
> Jarnick-Nosarzewska inequality are quite relevant, and
> the Voronoi cell is a bit of terminology I've been
> considering introducing.

Paul used Voronoi cells in several diagrams he made a couple
of years ago, illustrating harmonic entropy.

> I keep thinking I'll apply Pick's theorem, but never have,
> however I have had occasion to mention random lattice walks.

Thanks for all of those, Gene! Lattice School is
now in session!

Looks to me like "Ehrhart Polynomial" is worth
looking into as well.

> If you want specific lattices to look at, the root lattice
> An and its dual An* are the key ones, I think.

I don't understand that. Where can these be found?

-monz

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--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> > If you want specific lattices to look at, the root lattice
> > An and its dual An* are the key ones, I think.
>
>
> I don't understand that. Where can these be found?

If you have a university library available, "Sphere Packings, Lattices and Groups" by Conway and Sloane is a good place to read up on it; "A Course in Arithmetic" by Serre might be relevant but I can't find my copy to check. Don't get sucked into Lie algebras; the Mathworld definition and things of that ilk will just lead you into all kinds of irrelevant complexities.

The lattices people often like to draw--with triangles, or in three dimensions, hexany/octahedrons and tetrahedrons, are A2 and A3.