Answer to Paul (about unanswered questions)

Hi Paul,

(1)

As you may know, my definition of the sonance (or log complexity)

C = C(X) = |x|*log 2 + |y|*log p + |z|*log q

corresponds to the Tenney's Harmonic Distance. What I wanted to show is the
effect of the representation of this sonance by the distance

MC = MC(X) = |x| + |y| + |z|

in <2,p,q> as Z-module and the distance

RMC = RMC(X) = |y| + |z|

in <p,q> where octave modularity is used.

I underline that it is not question of complexity metric but topological
invariance using MC and RMC to represent sonance (or log complexity). The
first reduction corresponds to a variation in axis elongation ratios and
the second to a counterclockwise rotation. The invariance concerns the
spatial location of the intervals between them and the convexity property
is not changed by these two transformations.

(2)

I hope you don't doubt that for me, the parallelopiped defined by N unison
vectors in dimension N+1, is the fundamental unit of periodicity. This
shape insure that there is no possibility to have inside it two intervals
of the same class and the periodicity insure that all classes have a
representant in this block. I shall define precisely later a prime block,
what a true fundamental unit has to be, and a such fundamental unit cannot
have a center ...

... but I shall come with that in next parts ...

Pierre

Pierre, thanks for your replies. Honestly, I think I have very little
understanding of what you posted, and hance my remarks were on the most
superficial level. This one is no exception:

>I hope you don't doubt that for me, the parallelopiped defined by N unison
>vectors in dimension N+1, is the fundamental unit of periodicity.

First of all, N unison vectors define a periodicity block in N dimensions,
and secondly, I feel that the parallelopiped, the hexagonal prism, and the
rhombic dodecahedron are all equally "fundamental" shapes that the
periodicity block can take on in 3 dimensions. For a particular set of
unison vectors, I'd choose whichever one of the three shapes that best
approximates a spherical volume.

--- In tuning@y..., "Paul H. Erlich" <PERLICH@A...> wrote:
>I feel that the parallelopiped, the hexagonal prism,
and the
> rhombic dodecahedron are all equally "fundamental" shapes that the
> periodicity block can take on in 3 dimensions. For a particular set
of
> unison vectors, I'd choose whichever one of the three shapes that
best
> approximates a spherical volume.

There are exactly 5 shapes that fill 3D space by translation only (no
rotation).

Cube
Hexagonal prism
Rhombic dodecahedron (12 rhombic faces)
Elongated rhombic dodecahedron (8 rhombic faces and 4 hexagonal)
Truncated octahedron (6 square faces and 8 hexagonal)

Of course any linear transformation of these (e.g scale, skew) also
fills space, hence Paul's use of "parallelipiped" rather than "cube".

Regards,
-- Dave Keenan

Dave Keenan wrote,

>There are exactly 5 shapes that fill 3D space by translation only (no
>rotation).

>Cube
>Hexagonal prism
>Rhombic dodecahedron (12 rhombic faces)
>Elongated rhombic dodecahedron (8 rhombic faces and 4 hexagonal)
>Truncated octahedron (6 square faces and 8 hexagonal)

Right, but the truncated octahedron is not suitable for periodicity blocks,
as Paul Hahn explained. The elongated rhombic dodecahedron -- what does it
look like?

--- In tuning@y..., "Paul H. Erlich" <PERLICH@A...> wrote:
> Dave Keenan wrote,
>
> >There are exactly 5 shapes that fill 3D space by translation only
(no
> >rotation).
>
> >Cube
> >Hexagonal prism
> >Rhombic dodecahedron (12 rhombic faces)
> >Elongated rhombic dodecahedron (8 rhombic faces and 4 hexagonal)
> >Truncated octahedron (6 square faces and 8 hexagonal)
>
> Right, but the truncated octahedron is not suitable for periodicity
blocks,
> as Paul Hahn explained.

Can you please point me at that explanation. A search on "truncated
octahedron" did not reveal.

>The elongated rhombic dodecahedron -- what does it
> look like?

This is from George Hart and Henri Piciotto's new book 'Zome
Geometry'.

A rhombic dodecahedrom can be built in zometool by adding a yellow
pyramid to each face of a blue cube, then removing the blue struts.

"To build an elongated rhombic dodecahedron ... take a rhombic
dodecacahedron and locate a square equator that divides it in half.
Separate the two halves and add zomeballs to the bare ends. Now
reconect the two halves in their original orientation, but separated
from each other by adding four blue struts. You are expanding four
rhombi into four [non-regular] hexagons."

For those not already into Zometool, I thoroughly recommend this
package of book and kit
http://www.zometool.com/store/math_geometry_kit.html

Those who are already into Zometool may not know that green struts are
available now. Don't bother with the "Green line starter kit", it has
too many useless lengths.

Even though it doesn't say so on the web site, you can order greens in
various other packs. I recommend getting 48 of each of the short,
medium and long (whole) greens (G0, G1 and G2) and 48 short
blue-greens (Gb1), and of course the Zome Geometry book.

Regards,
-- Dave Keenan

>> Right, but the truncated octahedron is not suitable for periodicity
blocks,
>> as Paul Hahn explained.

>Can you please point me at that explanation. A search on "truncated
>octahedron" did not reveal.

It may have been a private e-mail. Paul? But the idea is that the
parallelogram-to-hexagon shift than I illustrate in the "excursion", in 3-d,
becomes parallelopiped-to-hexagonal prism, or parallelopiped-to-rhombic
dodecahedron, but there's no way to get it to truncated octahedron, since
the lattice of those guys has a different symmetry group or something . . .
I'm sure, if you try it, you'll understand better.

--- In tuning@y..., "Paul H. Erlich" <PERLICH@A...> wrote:
> >> Right, but the truncated octahedron is not suitable for
periodicity
> blocks,
> >> as Paul Hahn explained.
>
> >Can you please point me at that explanation. A search on "truncated
> >octahedron" did not reveal.
>
> It may have been a private e-mail. Paul? But the idea is that the
> parallelogram-to-hexagon shift than I illustrate in the "excursion",
in 3-d,
> becomes parallelopiped-to-hexagonal prism, or
parallelopiped-to-rhombic
> dodecahedron, but there's no way to get it to truncated octahedron,
since
> the lattice of those guys has a different symmetry group or
something . . .
> I'm sure, if you try it, you'll understand better.

Nope. I thought you just wanted the most spherical thing you could
fill space with. Having the largest number of faces (14) of any of the
5 "bricks" might well qualify the truncated octahedron as "most
spherical".

What "excursion"? URL? What do you mean by "get it to"?

-- Dave Keenan

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

> What "excursion"? URL? What do you mean by "get it to"?

Go to the Gentle Introduction to Fokker Periodicity Blocks; click the
link on the bottom to get to part 2; then click on the bottom of that
to get to the excursion.

Basically, my understanding is that truncated octahedra fill space in
such a way that it is not possible to translate from one to all the
others using three and only three vectors -- you need four, I think --
which means that, in general, the contents of each truncated
octahedron won't be equivalent to the contents of all the other ones -
- only half of the other ones, I think -- and that the contents of
each won't, in general, be any kind of well-distributed scale.
However, I think that each _pair_ of truncated octahedra, connected
by the fourth vector that is not one of the three unison vectors,
_will_ be a periodicity block -- though a rather oddly-shaped one.

Paul Hahn, if you're out there, could you confirm this?

--- In tuning@y..., PERLICH@A... wrote:
> --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
>
> > What "excursion"? URL? What do you mean by "get it to"?
>
> Go to the Gentle Introduction to Fokker Periodicity Blocks; click
the
> link on the bottom to get to part 2; then click on the bottom of
that
> to get to the excursion.

Aargh! Searching on "gentle periodicity" or the like doesn't work. How
do we get Yahoo to index the whole archive including the old Onelist
stuff?

Paul, can you give me a URL or at least an approximate date these were
posted?

> Basically, my understanding is that truncated octahedra fill space
in
> such a way
...

Ok. That gives me a direction to think about. Thanks.

-- Dave Keenan

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> --- In tuning@y..., PERLICH@A... wrote:
> > --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> >
> > > What "excursion"? URL? What do you mean by "get it to"?
> >
> > Go to the Gentle Introduction to Fokker Periodicity Blocks; click
> the
> > link on the bottom to get to part 2; then click on the bottom of
> that
> > to get to the excursion.
>
> Aargh! Searching on "gentle periodicity" or the like doesn't work. How
> do we get Yahoo to index the whole archive including the old Onelist
> stuff?

It should already do that, but no need -- a Google search on "gentle periodicity" takes you
immediately to the URL: http://www.onthenet.com/interval/td/erlich/intropblock1.htm.
I was surprised to see "onthenet" instead of "ixpres" -- did Monz move his entire site?