How do I triangulate an N-dim lattice?

I'm working on an Excel spreadsheet to animate the 2-D projection of an
eikosany rotating in 5 dimensions, in answer to Joe Pehrson's questions.
But I decided I had bitten off more than I could chew and backed up to
doing a hexany first (only 3 dimensions).

The part I'm stuck on is going from N axes at right angles, to N axes that
are 60 degrees from each other. i.e. triangulating the lattice. I can't
figure out what the linear transformation is for 3 dimensions (so tetrads
are regular terahedra and hexanies are regular octahedra), let alone 5
dimensions.

2 dimensions is easy.

x' = 1*x + sin(30)*y
y' = 0*x + cos(30)*y

or it could be written

x' = 1*x + cos(60)*y
y' = 0*x + sin(60)*y

But how does this generalise to 3 dimensions and beyond?

-- Dave Keenan
http://dkeenan.com

Is 5 dimensions enough so every hexany will look like an octahedron, or at least have the same
shape?

David C Keenan wrote:

> I'm working on an Excel spreadsheet to animate the 2-D projection of an
> eikosany rotating in 5 dimensions, in answer to Joe Pehrson's questions.
> But I decided I had bitten off more than I could chew and backed up to
> doing a hexany first (only 3 dimensions).
>
> The part I'm stuck on is going from N axes at right angles, to N axes that
> are 60 degrees from each other. i.e. triangulating the lattice. I can't
> figure out what the linear transformation is for 3 dimensions (so tetrads
> are regular terahedra and hexanies are regular octahedra), let alone 5
> dimensions.
>
> 2 dimensions is easy.
>
> x' = 1*x + sin(30)*y
> y' = 0*x + cos(30)*y
>
> or it could be written
>
> x' = 1*x + cos(60)*y
> y' = 0*x + sin(60)*y
>
> But how does this generalise to 3 dimensions and beyond?
>
> -- Dave Keenan
> http://dkeenan.com
>
>

-- Kraig Grady
North American Embassy of Anaphoria island
www.anaphoria.com

Ok. I figured out 3 dimensions but still can't see how to get to 5.

You can see the tumbling hexany at:
http://dkeenan.com/Music/HexanyRotation.xls

1 dimension.

x' = 1*x

2 dimensions.

x' = 1*x + sin(30)*y
y' = 0*x + cos(30)*y

3 dimensions.

x' = 1*x + sin(30)*y + sin(30)*z
y' = 0*x + cos(30)*y + tan(30)*sin(30)*z
z' = 0*x + 0*y + cos(30)*z

4 dimensions?

x' = 1*x + sin(30)*y + sin(30)*z + sin(30)*w
y' = 0*x + cos(30)*y + tan(30)*sin(30)*z + ? *w
z' = 0*x + 0*y + cos(30)*z + ? *w
w' = 0*x + 0*y + 0*z + cos(30)*w

5 dimensions?

x' = 1*x + sin(30)*y + sin(30)*z + sin(30)*w + sin(30)*v
y' = 0*x + cos(30)*y + tan(30)*sin(30)*z + ? *w + ? *v
z' = 0*x + 0*y + cos(30)*z + ? *w + ? *v
w' = 0*x + 0*y + 0*z + cos(30)*w + ? *v
v' = 0*x + 0*y + 0*z + 0*w + cos(30)*v

Yes Kraig, as Paul Erlich confirmed, 5 dimensions is enough. Maybe you are
confusing dimensions with axes of symmetry. The hexanies within the
eikosany will never look like octahedra because I am leaving out exactly
the same lines that Wilson does. As you probably know, if all consonances
were shown (as they are in the single hexany above) the eikosany would just
look like a tangled mess. I expect there will be some angles from which the
hexanies all look the same, since I expect the eikosany to look like
Wilson's standard diagram from some angles.

Come on guys, it's nearly ready to roll, I just need that 5D triangulation
matrix. :-)

Regards,
-- Dave Keenan
http://dkeenan.com

David C Keenan wrote:

>
> Yes Kraig, as Paul Erlich confirmed, 5 dimensions is enough. Maybe you are
> confusing dimensions with axes of symmetry.

Well maybe to see what you want to see. This is not my analysis, but Erv's. The eikosany is
symmetrical from the 12 points 1,3,5,7,9,11 all squared and there recipicales. From these
points the whole structure can be generated. thus the structure will look the same from each
one of these points. Yes it is a question of symmetry , but you need so many dimensions to
have this property. maybe 5 is enough. Ask paul I refuse to comment any more about it!

> The hexanies within the
> eikosany will never look like octahedra because I am leaving out exactly
> the same lines that Wilson does. As you probably know, if all consonances
> were shown (as they are in the single hexany above) the eikosany would just
> look like a tangled mess. I expect there will be some angles from which the
> hexanies all look the same, since I expect the eikosany to look like
> Wilson's standard diagram from some angles.
>
> Come on guys, it's nearly ready to roll, I just need that 5D triangulation
> matrix. :-)
>
> Regards,
> -- Dave Keenan
> http://dkeenan.com
>
>

-- Kraig Grady
North American Embassy of Anaphoria island
www.anaphoria.com

--- In tuning@egroups.com, David C Keenan <D.KEENAN@U...> wrote:

http://www.egroups.com/message/tuning/14749

This is absolutely the most incredible thing I have ever seen.

I had *NO* idea Excel could do this! Everybody here at "work" is
going crazy over it too... but this is not an "engineering" shop...
so nobody knows anything about it.

very cool

___________ ____ __ __ _
Joseph Pehrson

>Come on guys, it's nearly ready to roll, I just need that 5D triangulation
>matrix. :-)

I can't believe you did that in Excel. You the _man_, Dave! Next up,
the ExcelOS, from KeenanSoft.

-Carl

David Keenan wrote,

>You can see the tumbling hexany at:
>http://dkeenan.com/Music/HexanyRotation.xls

ROCK ON!

>The hexanies within the
>eikosany will never look like octahedra because I am leaving out exactly
>the same lines that Wilson does.

Excuse me, David, but Wilson leaves out different sets of lines in different
depictions. I think you should put all the lines in, perhaps using dotted
lines for some (like Wilson sometimes did) or different colors!

>Come on guys, it's nearly ready to roll, I just need that 5D triangulation
>matrix. :-)

I dunno -- search the web, join a math list, or do some heavy thinking. Now,
you'll probably want to rotate around more than just one axis when you're
done, to bring out the full 5-dimensional symmetry . . .

Don't s'pose the tumbling eikosany is viewable without Excel? (For once
I may regret making my Mac a Microsoft free zone)

- Adam

Hi Adam-

>Don't s'pose the tumbling eikosany is viewable without Excel? (For once
>I may regret making my Mac a Microsoft free zone)

What if I make a multi-level GIF? Would that be better for you?

Hi Dave,

Instead of jumping right to 5 dimensions, how about trying the dekany in 4?

By the way, Adam and Joseph, I made an .mpg of the rotating hexany, but
mine's opaque unlike Dave's: go to
http://www.egroups.com/files/tuning/perlich/, and save die.mpg to your local
machine, then view it.

Enjoy!

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/14782

> Hi Dave,
>
> Instead of jumping right to 5 dimensions, how about trying the
dekany in 4?
>
> By the way, Adam and Joseph, I made an .mpg of the rotating hexany,
but mine's opaque unlike Dave's: go to
> http://www.egroups.com/files/tuning/perlich/, and save die.mpg to
your local machine, then view it.
>
> Enjoy!

Very nice. "Scary" file name, though. Sounds like it could be a
virus...

JP

>Very nice. "Scary" file name, though. Sounds like it could be a
>virus...

Sorry, I was thinking "dice". Did you ever play Dungeons and Dragons? There
were many different-shaped dice, and this would be the 8-sided die. There
was a 30-sided die too, and I think that can be understood as a
five-dimensional Euler genus . . . the movie is all set . . .

Dave, maybe examining this program will halp you in some way, besides being
of interest in its own right:

http://freeabel.geom.umn.edu/apps/quasitiler/about.html#dimensions

Paul Erlich wrote:

>Excuse me, David, but Wilson leaves out different sets of lines in different
>depictions. I think you should put all the lines in, perhaps using dotted
>lines for some (like Wilson sometimes did) or different colors!

Ah. So if I don't put _all_ the lines in, will it only look like the donut
(10-star-in-10-gon) in one position, not 12?

>Now,
>you'll probably want to rotate around more than just one axis when you're
>done, to bring out the full 5-dimensional symmetry . . .

I'm on top of that. Notice how the hexany rotates about two axes at once.
It rotates about the second one more slowly than the first. For each full
circle about the first, it rotates about the second by the same amount that
it turns about the first in one frame.

Trouble is, the eikosany (doing 4 rotations at ever-decreasing rates) will
take several hours! to pass thru every position.

I only realised by working on the 5D eikosany, that the idea of rotating
"about an axis" only works in 3 dimensions. Notice that a 2D rotation is
not "about an axis". There are no others to rotate about. In general a
rotation is a rotation of one axis toward (or away from) another. So what
we call rotation about the z axis in 3D, is more generally called rotating
x to y.

So what can you do with whatever tool you used to make that mpg. Can you do
pseudo-3D projections of higher-D objects doing multiple rotations?

Adam Bushell wrote:

>Don't s'pose the tumbling eikosany is viewable without Excel? (For once
>I may regret making my Mac a Microsoft free zone)

Hi Adam,

So far it's only a tumbling hexany (octahedron). I applaud your stance.
Have you tried importing the Excel spreadsheet into your Mac spreadsheet
application? I'm happy to work with you by email to try to get it to work.
Maybe you need me to save it in an earlier version of the Excel file format
or something. I have a Mac as well as a PC, but its spreadsheets are a
little out of date. I have Claris Resolve and Claris Works 4.0.

Regards,
-- Dave Keenan
http://dkeenan.com

"Come on guys, it's nearly ready to roll, I just need that 5D triangulation
matrix. :-)"

If I understand that what you're trying to do is map the coordinate vectors
of a higher dimension onto equally spaced unit vectors in a plane, you need
3 unit vectors (60� apart) for 3 dimensions, 4 unit vectors (45� apart) for
4 dimensions, and 5 unit vectors (36� apart) for 5 dimensions.
So without further ado, the coveted 4D and 5D perspective matrices:

x' = cos(0�)x + cos(45�)y + cos(90�)z + cos(135�)w
y' = sin(0�)x + sin(45�)y + sin(90�)z + cos(135�)w
(there is no z' or w' because this is a 2D perspective)

x' = cos(0�)x + cos(36�)y + cos(72�)z + cos(108�)w + cos(144�)v
y' = sin(0�)x + sin(36�)y + sin(72�)z + sin(108�)w + sin(144�)v

I include terms like "cos(0�)" which are zero only for consistency and
clarity.

Have fun!
-Keenan P., who is still 13 years old (14 next week!)

Hi Keenan Pepper.

>If I understand that what you're trying to do is map the coordinate vectors
>of a higher dimension onto equally spaced unit vectors in a plane

No, Dave Keenan is essentially trying to find the coordinates of the
vertices of a regular triange, tetrahedron, etc. etc., in 2-, 3-, 4-, and
5-dimensions.

I wrote,

>No, Dave Keenan is essentially trying to find the coordinates of the
vertices of a regular >triange, tetrahedron, etc. etc., in 2-, 3-, 4-, and
5-dimensions.

The first "etc." is actually called a pentatope:

http://mathworld.wolfram.com/Pentatope.html

Unfortunately, the big figure on the upper left doesn't show up on my
browser :( Help! I want to see it!

Dave,

from this page

http://www.swin.edu.au/astronomy/pbourke/geometry/4dobjects/5cell.ascii

I would assume that the coordinates of the 5-cell (pentatope) in 4
dimensions are:

0 0 0 2
-1.11803 1.11803 1.11803 -0.5
1.11803 -1.11803 1.11803 -0.5
1.11803 1.11803 -1.11803 -0.5
-1.11803 -1.11803 -1.11803 -0.5

I'd suggest you try to see if this has some 3-d projections that give you a
tetrahedron and any that give you the pentagram inside a pentagon. If so,
perhaps you can use this to make some of the 4-d (5-factor) CPS structures
like the dekany and double dekany.

"No, Dave Keenan is essentially trying to find the coordinates of the
vertices of a regular triange, tetrahedron, etc. etc., in 2-, 3-, 4-, and
5-dimensions."

Sorry, totally misunderstood that. I knew it couldn't have been that easy!
:) The triangulation problem is very similar, though. Here are the REAL
answers:

4-D:

x' = x + (1/2)y + (1/2)z + (1/2)w
y' = (sqrt(3)/2)y + (sqrt(3)/6)z + (sqrt(3)/6)w
z' = (sqrt(6)/3)z + (sqrt(6)/12)w
w' = (sqrt(10)/4)w

5-D:

x' = x + (1/2)y + (1/2)z + (1/2)w + (1/2)v
y' = (sqrt(3)/2)y + (sqrt(3)/6)z + (sqrt(3)/6)w + (sqrt(3)/6)v
z' = (sqrt(6)/3)z + (sqrt(6)/12)w + (sqrt(6)/12)v
w' = (sqrt(10)/4)w + (sqrt(10)/20)v
v' = (sqrt(15)/5)v

It may suprize you that all I used was simple algebra, used creatively.

-Keenan P.

Keenan Pepper wrote:

>
> It may suprize you that all I used was simple algebra, used creatively.

Show off!;-)

-- Kraig Grady
North American Embassy of Anaphoria island
www.anaphoria.com

>Sorry, I was thinking "dice". Did you ever play Dungeons and Dragons? There
>were many different-shaped dice, and this would be the 8-sided die. There
>was a 30-sided die too, and I think that can be understood as a
>five-dimensional Euler genus . . . the movie is all set . . .

The rhombic triacontahedron?

http://www.georgehart.com/virtual-polyhedra/vrml/rhombic_triacontahedron.wrl

-Carl

Carl wrote,

>The rhombic triacontahedron?

Yup -- you can read more about it at
http://www.innerx.net/personal/tsmith/pwtile.html.

>http://www.georgehart.com/virtual-polyhedra/vrml/rhombic_triacontahedron.wr
l

The VRML viewer download failed. This sucks -- no Java, no VRML!

>>www.georgehart.com/virtual-polyhedra/vrml/rhombic_triacontahedron.wrl
>
>The VRML viewer download failed. This sucks -- no Java, no VRML!

My fav. is CosmoPlayer, http://www.cosmosoftware.com/. See my previous
post about Java, let me know off-list if you can't figure it out.

-Carl

--- In tuning@egroups.com, Carl Lumma <CLUMMA@N...> wrote:

http://www.egroups.com/message/tuning/14818
>
>>www.georgehart.com/virtual-polyhedra/vrml/rhombic_triacontahedron.wr
l
> >
> >The VRML viewer download failed. This sucks -- no Java, no VRML!
>
> My fav. is CosmoPlayer, http://www.cosmosoftware.com/. See my
previous post about Java, let me know off-list if you can't figure it
out.
>
> -Carl

Hmmm. I tried to install this... In fact, I used to work with Cosmo
before. However, now I can't even find an "executable" file to work
with Netscape. Where did they put it??

________ ___ __ _
Joseph Pehrson

"Paul H. Erlich" wrote:
>
> Hi Dave,
>
> Instead of jumping right to 5 dimensions, how about trying the dekany in 4?
>
> By the way, Adam and Joseph, I made an .mpg of the rotating hexany, but
> mine's opaque unlike Dave's: go to
> http://www.egroups.com/files/tuning/perlich/, and save die.mpg to your local
> machine, then view it.

I like the seamless loop. Smooth.

--
* D a v i d B e a r d s l e y
* 49/32 R a d i o "all microtonal, all the time"
* http://www.virtulink.com/immp/lookhere.htm