Re: vum

Hi Gene,

> A convex polyhedron can be arbitarily close to a sphere, so these must
> exist. What's dicier is trying to get a good Euclidean approximation
> to something like the Tenney norm; there's only so far you can take
> it. However, the Hahn metric and the symmetrical metric on 7-limit
> note classes can look a lot alike in practice.

Not so easy as you think, because they have to tile space.
Put spheres together and the best you can achieve is a
sphere packing, which won't generate a euclidean taxicab
metric approximation at all. At the time that I did the
paper I didn't know of any published results on the
subject at all, though I suppose I'll have to do another search
to see if there are any now, if I try to publish it again.

Robert

Hi Gene,

> >it seems to me what you are really doing is imagining a Euclidean
> >space with a Euclidean metric, and then using it to measure taxicab
> >distances between lattice points.

> Yes, that sounds right.

Indeed that's how it is done in geometrical investigations of
taxicab geometry. Define the notion of a tile geometrically,
Mathematically, a tiling of a space is a collection
of open sets (sets not including their boundaries) such
that none of them intersect and their closure (including
the boundaries) covers the space.

Normally you have other conditions as well as normaly
you want to exclude tiles with fractal boundaries and such
like, but I can't remember the usual way of doing that
now - it will be given in in Grunbaum and Shephard.
Sometimes one might also study fractal tilings
too.

Then anyway, you define a path joining two tiles as
a sequence of tiles such that the closure of each
tile intersects with the next at more than one point
(at a common edge).

Then you define the taxicab metric as one less than
the number of tiles along the shortest path betweem the two tiles.

It's all done within Euclidean geometry all the way.

Robert

>> A convex polyhedron can be arbitarily close to a sphere, so these must
>> exist. What's dicier is trying to get a good Euclidean approximation
>> to something like the Tenney norm; there's only so far you can take
>> it. However, the Hahn metric and the symmetrical metric on 7-limit
>> note classes can look a lot alike in practice.
>
>Not so easy as you think, because they have to tile space.
>Put spheres together and the best you can achieve is a
>sphere packing, which won't generate a euclidean taxicab
>metric approximation at all. At the time that I did the
>paper I didn't know of any published results on the
>subject at all, though I suppose I'll have to do another search
>to see if there are any now, if I try to publish it again.

Hiya Robert,

Is there a pdf?

-Carl

Hi Carl,

> Is there a pdf?

No sorry. Actually don't know where
I put it now. The first drafts I think
I did at the maths dept & perhaps may have
used LaTex, but I did a much
improved one since then.

I don't even know where I put the
hard copy right now. But I'll take
a look for it. Maybe this is a time
to do that - though it could have
got stored away somewhere hard to find
amongst all my papers.

I can remember the results and some of the
details of the proofs. This is an incentive
to maybe look it out and write it up again.

Here is a kind of abstract of it as a
taster from what i remember:

I first defined a measure of how
close a taxicab metric approximates
to Euclidean geometry, using
the idea of the shape of
the region of all the tiles
at a fixed distance from a given
cell - and the limiting form
of that shape as the distance tends
to infinity. Obviously if the limiting
shape is a disk (sphere) then it
is Euclidean in the limit.
Usually it is some polygon with a
number of sides.

Then I proved that if the
limiting shape is a disk
(or sphere) then the tiling
has to be non periodic.

I showed that you can have periodic
tilings that generate taxicab
metrics which are arbitrarily close
to Euclidean if there are enough
tiles in the repeating unit.

Also then the question arises,
what is the best approximation
you can obtain for a given
number of tiles in a periodic
tiling - there I found some
tilings with small numbers of tiles
that were good approximations and
it seemed like the sort of puzzle
where you get better and better
versions as time goes on but can
only probably prove particular tilings
to be optimal for small numbers of
cells.

E.g. hexagons are optimal
for tiles with a repeating unit
consisting of a single tile - but
I found a tiling better than hexagons
if you have a single tile shape
and more than one tile to the
repeat unit. I found that a nice
puzzle to try. It wasn't large,
maybe eight tiles or so.

I showed that if you allow two tile shapes
then you can get as close as you like to euclidean
by using them to make a large enough repeating
unit, but it remained open whether
you can do the same with a single
tile shape I think, if I remember
rightly now.

Then I went on and proved
that you can do the same thing
in 3D. That was rather more complicated.
Again I managed to show you can do it
with two tile shapes, while the single
tile shape case remained open.

Also constructed non periodic tilings euclidean
in the limit for 2D and 3D.

Then I started work on a way of
studying geometry using
particular arrangements of cells
as your points and lines.

That then is as far as it went.

The actual tilings in 2D really weren't
so hard to find once you got a couple
of key ideas, and I remember
well how they were constructed,
and it is a nice puzzle to try
and find such tilings.

Only one of three techniques that
I found also worked in 3D, but it was
one of the two non techy ones.

The proof that a euclidean taxicab tiling
must be non periodic was a bit techy
and I can't remember it now
so will just have to look up the
paper to find out - well sort of
remember the general line of the proof
and could recover it probably with some
thought too. It ran to two or three pages,
though not long by modern proof standards.

Robert

>Here is a kind of abstract of it as a
>taster from what i remember:
>
>I first defined a measure of how
>close a taxicab metric approximates
>to Euclidean geometry, using
>the idea of the shape of
>the region of all the tiles
>at a fixed distance from a given
>cell - and the limiting form
>of that shape as the distance tends
>to infinity. Obviously if the limiting
>shape is a disk (sphere) then it
>is Euclidean in the limit.
>Usually it is some polygon with a
>number of sides.
>
>Then I proved that if the
>limiting shape is a disk
>(or sphere) then the tiling
>has to be non periodic.
>
>I showed that you can have periodic
>tilings that generate taxicab
>metrics which are arbitrarily close
>to Euclidean if there are enough
>tiles in the repeating unit.
>
>Also then the question arises,
>what is the best approximation
>you can obtain for a given
>number of tiles in a periodic
>tiling - there I found some
>tilings with small numbers of tiles
>that were good approximations and
>it seemed like the sort of puzzle
>where you get better and better
>versions as time goes on but can
>only probably prove particular tilings
>to be optimal for small numbers of
>cells.
>
>E.g. hexagons are optimal
>for tiles with a repeating unit
>consisting of a single tile - but
>I found a tiling better than hexagons
>if you have a single tile shape
>and more than one tile to the
>repeat unit. I found that a nice
>puzzle to try. It wasn't large,
>maybe eight tiles or so.
>
>I showed that if you allow two tile shapes
>then you can get as close as you like to euclidean
>by using them to make a large enough repeating
>unit, but it remained open whether
>you can do the same with a single
>tile shape I think, if I remember
>rightly now.
>
>Then I went on and proved
>that you can do the same thing
>in 3D. That was rather more complicated.
>Again I managed to show you can do it
>with two tile shapes, while the single
>tile shape case remained open.
>
>Also constructed non periodic tilings euclidean
>in the limit for 2D and 3D.
>
>Then I started work on a way of
>studying geometry using
>particular arrangements of cells
>as your points and lines.

This sounds fascinating. I'd love to read it if
you can make a pdf or djvu file.

-Carl

> This sounds fascinating. I'd love to read it if
> you can make a pdf or djvu file.

Since it could be useful in music theory it seems it could be uploaded
to the files area here.

Hi Carl,

Rightio, thanks for the encouragement.
I'll look forward to your comments when you
read it.

It is a finished paper in the sense that everything
is worked out and presented in the form of a paper
- though now that I think it over, I think it is simply handwritten
in the latest and most complete version (I'm a fast writer
and can touch type fairly quickly, but can write faster
- and my fast hand writing is nearly totally illegible
to anyone else, almost more like shorthand in
appearance than ordinary writing). (I also need
to find the thing!).

Everything I think is there except
that the references need to be
got up to date but can show it around
for comments etc before doing that.
There were rather few of them
- not much directly on it. But
there will be more material
available now.

Also when I get back to it I'm sure
to find ways to improve the exposition
and I've just had a nice idea today
about a new way to define points and lines
in the last section on finite discrete
geometries which I'd like to explore
to see how it pans out.

So, anyway hopefully I'll find the
leisure to explore it all, and finish it.
Maybe only a week or two of work when
I have the time.

Nice to have someone who wants to read it :-).

More later I hope.

Robert