>>> Question: is the intersection of the 3D projections of the 8 
>>> hyperplanes equal to the 3D projection of the intersection of the 8 
>>> hyperplanes?
>>
>>
>> No. and that is not a feature unique to 4D->3D. An example in 2D-1D:
>> Consider the normal vectors (1,1) and (1,-1) and work it out. Another
>> example: Take the sets {(0,0),(1,1)} and {(0,1),(1,0)}. Projection, then
>> intersection yields {0,1}, while the other way around you obtain the 
>> empty
>> set. The first collects all 1D-points that can be extended to some point
>> in each of the original sets. The second collects all 1D-point that have
>> one extension that pertains to both sets simultaneously.
> 
> 
> Right...
> 
> So figuring out how to project the 8 hyperplanes into 3D and then asking 
> POV-Ray for the intersection will get me nowhere fast. (I was already 
> beginning to suspect this conclusion, which is why I asked! ;-)

Here's another question... what about slicing? If I choose a particular 
value for the 4th ordinate and fix that, can I *then* do the 
intersection on 3D? (Taking a slice is after all a set intersection 
operation - and I *believe* those things are associative...)

Andrew @ home.

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Andrew C on Mozilla wrote:
> Out of interest... I'm told there's an amazing 4D object that has 1200 
> sides or something stupid... anyone know how I compute the corners and 
> join them up to make a wireframe?
> 
> Thanks.
> Andrew @ home.

You might want to check out Paul Bourke's discussion of higher dimensions:

http://astronomy.swin.edu.au/~pbourke/polyhedra/platonic4d/

Here he displays, among other things, a 2d projection of a 3d projection 
of a 4d object with "600 tetrahedral cells, 1200 triangular faces, 720 
edges, 120 vertices" -- probably the one you're thinking of.

Dave Matthews

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> Here's another question... what about slicing? If I choose a particular
> value for the 4th ordinate and fix that, can I *then* do the
> intersection on 3D? (Taking a slice is after all a set intersection
> operation - and I *believe* those things are associative...)

Yes, that would work, but only if you post the images (Can't imagine what
the slices would look like).


-- 
merge{#local i=-11;#while(i<11)#local
i=i+.1;sphere{<i*(i*i*(.05-i*i*(4e-7*i*i+3e-4))-3)10*sin(i)30>.5}#end
pigment{rgbt 1}interior{media{emission x}}hollow}//  Mark Weyer

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In article <4096893b$1@news.povray.org>,
 Andrew C on Mozilla <voi### [at] devnull> wrote:

> > I never had problems with that, but the red/green glasses might have
> > helped a lot on that one. The problems I had were with too many lines
> > when having many 4D objects in the same "scene". I'm talking about 4D
> > tetris here.
> 
> ??!?! :-| ?!/"\@_~??!
> 
> Four *&$^ dimensional *@$^%# tetris *@*@#?!?! :-|
> 
> OK, my head hurts... lol!

Tetris is arrangements of axis-aligned blocks, so 4D tetris wouldn't be 
that hard. Viewing it on a 2D screen would be, though. Probably have to 
view it as transparent 3D objects, colored to indicate offset along the 
fourth axis. Even then, objects falling along the fourth axis would 
appear to overlap in the 3-space slice...you would need multiple views 
as well.

Hmm...rotating the pieces might be a bit mind-bending, even if it is 
restricted to 90 degree turns. I'm pretty sure a 180 degree rotation in 
4-space can mirror the 3-space slice, though I'm still having trouble 
figuring out just what a 4-space rotation is.

-- 
Christopher James Huff <cja### [at] earthlinknet>
http://home.earthlink.net/~cjameshuff/
POV-Ray TAG: <chr### [at] tagpovrayorg>
http://tag.povray.org/

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>> Here's another question... what about slicing? If I choose a particular
>> value for the 4th ordinate and fix that, can I *then* do the
>> intersection on 3D? (Taking a slice is after all a set intersection
>> operation - and I *believe* those things are associative...)
> 
> 
> Yes, that would work, but only if you post the images (Can't imagine what
> the slices would look like).

Interestingly, if you slice a normal cube in the right direction, you 
get a hexagon.

Not many people know that...

Presumably if I slice a hypercube right I'll get a pyramid of some 
kind... if I can rotate it right! :-S

Andrew @ home.

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An image is now in povray.binaries.images. (Simple wireframe - shame I 
can't show you the animation!)

Andrew @ home

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Andrew C on Mozilla wrote:
> An image is now in povray.binaries.images. (Simple wireframe - shame I 
> can't show you the animation!)
> 
> Andrew @ home

simple, post the source and in files

we'll render and movie it :-)

stephen

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stephen parkinson wrote:
> Andrew C on Mozilla wrote:
> 
>> An image is now in povray.binaries.images. (Simple wireframe - shame I 
>> can't show you the animation!)
>>
>> Andrew @ home
> 
> 
> simple, post the source and in files
> 
> we'll render and movie it :-)
> 
> stephen

oops - source + ini files for movie frames

i'll write a .par for it

stephen

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> Four *&$^ dimensional *@$^%# tetris *@*@#?!?! :-|

I'll post the program to p.o-t on monday (forgot to bring it today.
And yesterday.)


-- 
merge{#local i=-11;#while(i<11)#local
i=i+.1;sphere{<i*(i*i*(.05-i*i*(4e-7*i*i+3e-4))-3)10*sin(i)30>.5}#end
pigment{rgbt 1}interior{media{emission x}}hollow}//  Mark Weyer

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>> Yes, that would work, but only if you post the images (Can't imagine what
>> the slices would look like).
> 
> 
> Interestingly, if you slice a normal cube in the right direction, you 
> get a hexagon.
> 
> Not many people know that...
> 
> Presumably if I slice a hypercube right I'll get a pyramid of some 
> kind... if I can rotate it right! :-S

See povray.binaries.animations ;-)

Andrew @ home.

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