Re:imagination versus the inner workings of the ear.....

Hi there,

Members may like my new wikipedia article about the hexany:

http://en.wikipedia.org/wiki/Hexany

Will see if they keep it. I've made a copy on my web site in case they delete it so it's sort of an experiment.

BTW has anyone found a geometric name for the eikosany since I was last posting regularly here? (five dimensional figure with 20 vertices)? Or for that matter, is the pentadekany (edge dual of the 5-simplex or five dimensional tetrahedron) called anything else, anyone found out?

Robert

the drawing is completely unreadable compared to the originals.

Robert walker wrote:
>
> Hi there,
> > Members may like my new wikipedia article about the hexany:
> > http://en.wikipedia.org/wiki/Hexany <http://en.wikipedia.org/wiki/Hexany>
> > Will see if they keep it. I've made a copy on my web site in case they > delete it so it's sort of an experiment.
> > BTW has anyone found a geometric name for the eikosany since I was > last posting regularly here? (five dimensional figure with 20 > vertices)? Or for that matter, is the pentadekany (edge dual of the > 5-simplex or five dimensional tetrahedron) called anything else, > anyone found out?
> > Robert
> > > -- Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/index.html>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main/index.asp> 88.9 FM Wed 8-9 pm Los Angeles

Robert --

You article looks great. It would perhaps be useful to two simpler representations: (1) a single hexany as a star of David on a triangulated lattice, and (2) mapped onto an ordinary cube, with the extra vertices at the fundamental and guiding tones, as well as a stellated hexany. Also, perhaps this article should be part of a larger article on combination-product sets?

Daniel Wolf

--
Using Opera's revolutionary e-mail client: http://www.opera.com/mail/

> Members may like my new wikipedia article about the hexany:
>
> http://en.wikipedia.org/wiki/Hexany
>
> Will see if they keep it. I've made a copy on my web site in case
> they delete it so it's sort of an experiment.
>
> BTW has anyone found a geometric name for the eikosany since I
> was last posting regularly here? (five dimensional figure with
> 20 vertices)? Or for that matter, is the pentadekany (edge dual
> of the 5-simplex or five dimensional tetrahedron) called anything
> else, anyone found out?

Hi Robert,

Erv's most famous visualizations of the eikosany employ a
"uniform dodecatetron". Dave Keenan has speculated that the
pentadekany in Erv's drawings might be called a "rectified
5-D simplex".

-Carl

Here's an updated version of something originally made by
Dave Keenan. If you're viewing this on the web, you'll need
to choose Use Fixed Width Font under Message Options on the
right to see it correctly.

First, a rendition of the left half of Erv's "A Pascal Triangle
of Combination Set Lattices"...

3-limit dyad
5-limit triad
7-limit tetrad hexany
9-limit pentad dekany
11-limit hexad pentadekany eikosany

Now, with geometric names...

3-limit line segment
5-limit triangle
7-limit tetrahedron octahedron
9-limit pentachoron rectified pentachoron
11-limit 5D simplex ?? uniform dodecatetron

-Carl

These structures are not limit based.
the first hexany was not the 1-3-5-7
the structures are in the letter to Chalmers in the archives.
Erv has quite a few excellent diagrams of the hexany showing all the facets.

Carl Lumma wrote:
>
> Here's an updated version of something originally made by
> Dave Keenan. If you're viewing this on the web, you'll need
> to choose Use Fixed Width Font under Message Options on the
> right to see it correctly.
>
> First, a rendition of the left half of Erv's "A Pascal Triangle
> of Combination Set Lattices"...
>
> 3-limit dyad
> 5-limit triad
> 7-limit tetrad hexany
> 9-limit pentad dekany
> 11-limit hexad pentadekany eikosany
>
> Now, with geometric names...
>
> 3-limit line segment
> 5-limit triangle
> 7-limit tetrahedron octahedron
> 9-limit pentachoron rectified pentachoron
> 11-limit 5D simplex ?? uniform dodecatetron
>
> -Carl
>
> -- Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/index.html>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main/index.asp> 88.9 FM Wed 8-9 pm Los Angeles

Hi Carl,

Yes the pentadekany is indeed a rectified 5-simplex - rectified is synonymous with edge dual.

That's not hard to see. You can see how the construction works with the hexany in the hypercube - in the diagram on the new wkipedia page:
http://en.wikipedia.org/wiki/Hexany#Tuning
scale the whole thing down by a scale factor of 1/2 about the origin (blue vertex) and the octahedron vertices will all lie on the midpoints of the edges of the tetrahedron.

If that's not immediately obvious - note that each vertex is originally at the far vertex of a square away from the origin (with one blue, one yellow and two red vertices) , so scaling the whole diagram by a factor of 1/2 takes the yellow vertex to the midpoint of the original square and that also takes it to the midpoints of the edges of the tetrahedron with the red vertices.

Similarly in 5-dimensions, after a 1/2 saling, the dekany vertices will lie on the midpoints of the edges of the 4-simplex and in six dimensions, after the same scaling, the pentadekany vertices will lie on the mid points of the edges of the 5-simplex, and so on in all the higher dimensions.

Perhaps I'll add a note to the article to that effect.

I've done a web search for uniform dodecatetron, and nothing has turned up. Do you know any more about it?

I think names of five and higher dimensional solids belong to a fairly recondite area of geometry, apart from the hypercube, simplex (generalised tetrahedron) and cross set (generalised octahedron). So it's perhaps not too surprising that nothing turned up.

So, in Dave Keenan's figure you can replace the ?? by rectified 5-simplex, (or rectified hexateron).

I've usually seen it written as 5-simplex rather than 5D simplex. But a search for "5D simplex" turns up a fair number of results so I expect they are alternate spellings. 5-simplex seems to be the one most commonly used at least on the web.

Thanks,

Robert

Hi Daniel,

Yes, sounds great. As it's wikipedia you can add those yourself if you like.

I'm not going to do too much on it myself, until I see what happens to it. (Anyway I've still got rather a lot on right now, I mean more than usual for me - lots of immediate post release things to do to do with the web site).

I gather that often articles on Wikipedia may get deleted, or cut right down to a single paragraph, so it doesn't make much sense to work too much on it until we find out if that happens or not.

I think the reference to the article in 1/1 may "tick the right boxes" for someone who looks at it to see if it is a candidate for deletion, since it is a reference to a printed article in a journal subject to editorial review. At least I hope so. But it mightn't prevent someone from deciding it is too long and should be just a single paragraph.

Yes I agree a separate article on CPS makes sense, but again, idea was to start small. If it is left untouched for a few months maybe it can become a seed for more articles on the subject.

The link to Combination Product Sets from Erv Wilson's page was broken, so I redirected it to the hexany for now. Perhaps there used to be an article on CPS that has already been deleted? Though I can't find anything in the wayback machine.

There must be lots of good material that got deleted from wikipedia. It would be interesting if they had a "wikipedia rejects" site though one can understand why they don't as it would be an invitation for contributors to add even more material that doesn't belong.

Anyway don't want to discourage anyone else from editing it - and improving it. Pariticularly I expect the more references one can add to it the better, especially if they are printed articles in a journal with an editor. I don't know if I'm being over cautious there - I've been somewhat warned off doing too much work on it for the time being by some of the posts I've seen here about deleted material on wikipedia. But the current banner I think isn't too serious - looked it up and it means that whoever added it feels it is probably suitable for inclusion, but just needs more work on it. That was before I added the 1/1 ref.

Robert

Hi Krag,

I've just expanded the section that explains why you can easily see from the diagram that the octahedron is regular. That was the main reason for including a diagram there, which perhaps wasn't clear, the reason for putting it in.

The idea was to show its four dimensional context, particularly, and make it easy to see that the polyhedra are regular - well as easy as could be expected - which I think it does reasonably well??

One expects to find the projection baffling, if one tries to make sense of it, tries to fit it all together into a single picture - unless one happens to be one of the very very few people who are able to think four dimensionally, such as Alicia Stott.

Also one can see that by just rotating the diagram around, then there are four ways of finding the octahedron corresponding to the four major diagonals of the hypercube. Though I don't know if that has any musical significance particularly. There may be other things that it may be useful for, that particular projection, it's a very commonly used projection of the hypercube, very symmetrical. well expect you know that, anyway that may perhaps make it easier to see some things.

As it's wikipedia anyone who wants to add more material or better versions, or edit anything that's confusing can just go ahead and do it.

BTW the quality I'm sure could be improved too. I just took an existing Wikipedia diagram and added colour and lines to it

Robert

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
> These structures are not limit based.

Yes, I know. As I said, it was a rendition of a particular
diagram of Erv's, which was 3-11 "limit".

-Carl

> Carl Lumma wrote:
> >
> > Here's an updated version of something originally made by
> > Dave Keenan. If you're viewing this on the web, you'll need
> > to choose Use Fixed Width Font under Message Options on the
> > right to see it correctly.
> >
> > First, a rendition of the left half of Erv's "A Pascal Triangle
> > of Combination Set Lattices"...
> >
> > 3-limit dyad
> > 5-limit triad
> > 7-limit tetrad hexany
> > 9-limit pentad dekany
> > 11-limit hexad pentadekany eikosany
> >
> > Now, with geometric names...
> >
> > 3-limit line segment
> > 5-limit triangle
> > 7-limit tetrahedron octahedron
> > 9-limit pentachoron rectified pentachoron
> > 11-limit 5D simplex ?? uniform dodecatetron
> >
> > -Carl