4D Kaleidoscope...

Higher-dimensional geometry (previously "Polyshapes").

4D Kaleidoscope...

Postby pat » Thu Jan 26, 2006 10:10 pm

Maybe this should be under 'Relativity and Space-Time', but I'm really in it for the geometry.... so I put it here.

Yesterday, I threw together a little C++ program to take a sequence of PNG images, consider them as a 3-D block, and view it all through a 4-D kaleidoscope.

Image

Sadly, my hosting site is really sledding at the moment, so you may have trouble checking it out until they sort out their issues.

But, here's the idea... Take a hollow (but not wireframe) tetrahedron. Extrude it along an axis perpendicular to the three-space in which it sits. Now, look through one end and put an image-cube at the other end.

Here's the link: http://www.nklein.com/products/kchron/

(Things seem to have cleared up a bit with my hosting site now. Hope it works for you.)
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Re: 4D Kaleidoscope...

Postby quickfur » Fri Jan 27, 2006 8:24 am

pat wrote:[...]Yesterday, I threw together a little C++ program to take a sequence of PNG images, consider them as a 3-D block, and view it all through a 4-D kaleidoscope.

Nice!

Image

Nice... I had a look at your animation. Pretty weird, in a good way. :-)

Sadly, my hosting site is really sledding at the moment, so you may have trouble checking it out until they sort out their issues.

It seems to be ok for me.

But, here's the idea... Take a hollow (but not wireframe) tetrahedron. Extrude it along an axis perpendicular to the three-space in which it sits.

That would make it a tetrahedral prism: 4 triangular prisms and 2 tetrahedra, one of the prismatic uniform polychora. :-)

Now, look through one end and put an image-cube at the other end.[...]

Nice! I like it.
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Re: 4D Kaleidoscope...

Postby bo198214 » Fri Jan 27, 2006 11:18 am

pat wrote:But, here's the idea... Take a hollow (but not wireframe) tetrahedron. Extrude it along an axis perpendicular to the three-space in which it sits. Now, look through one end and put an image-cube at the other end.


Had really to think about whats a normal kaleidoscope, yeah a triangle which extends perpendicular into the viewing direction. A 2D kaleidoscope is then only two parallel mirror lines :(
Is this the result if one plays to much with 4d-raytracers? ;)
So and now what about a red/cyan 3d movie?
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Re: 4D Kaleidoscope...

Postby pat » Fri Jan 27, 2006 4:11 pm

bo198214 wrote:So and now what about a red/cyan 3d movie?


Hmmm... and use depth instead of time? or start with a red/cyan 3d movie and kaleidscope it?

Actually, it wouldn't really work so well to start with a red/cyan movie. Each reflection would invert the depth-cueing. I'd instead have to start with a heightmap movie, kaleidoscope it, then red/cyan encode it....
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Re: 4D Kaleidoscope...

Postby bo198214 » Fri Jan 27, 2006 4:25 pm

pat wrote:I'd instead have to start with a heightmap movie, kaleidoscope it, then red/cyan encode it....

Yes, thats what I meant.

Instead of a height-map (=depthmap?) movie, you can also randomly place arbitrary glittering stars and diamonds in the original tetrahedron.
(Was it a personal movie by/of you, btw?)
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Postby pat » Fri Jan 27, 2006 4:33 pm

That movie was from about a year ago. It was just when my son started taking his first steps. I teased him across the kitchen floor with a fig newton.
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Postby Marek14 » Sat Jan 28, 2006 7:06 am

Just a thought: Does it make a difference that norm kaleidoscope uses triangle which tiles plane, but the tetrahedron doesn't tile space?
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Postby bo198214 » Sat Jan 28, 2006 9:20 am

Good point, perhaps pat should use the rhombic dodecahedron instead of the tetrahedron or the truncated octahedron :? Can you do this for us, pat?

Btw. is there any research regarding tessalation of the 4D space? (One never knows whether sometime one has to build a 5D kaleidoscope ;) )
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Postby wendy » Sat Jan 28, 2006 11:37 am

One can make a tetrahedron that does tile space.

For example, the tetrahedron formed by the four points

(0,0,0), (0,0,2), (1,1,1), (1,1,3) does indeed tile space. It has two right angles (opposite edges), and the remainder are 60-degree angles.

It is the symmetry of the group 4q.

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Postby bo198214 » Sat Jan 28, 2006 11:49 am

wendy wrote:One can make a tetrahedron that does tile space.


Yes, but there is no tetrahedron, that tiles the space without rotations of it (i.e. with only translations allowed).
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Postby pat » Sat Jan 28, 2006 10:36 pm

But, mirrors allow for rotation. Triangles don't tile the plane without rotation, either.
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Postby bo198214 » Sat Jan 28, 2006 11:19 pm

wendy wrote:For example, the tetrahedron formed by the four points
(0,0,0), (0,0,2), (1,1,1), (1,1,3)

:o the points are linearly dependent (0,0,2)+(1,1,1)=(1,1,3)

But probably you mean that the space will be tiled by cubes and when you put a point into the middle of a cube and connect it to all vertices of the cube you get 6 equal pyramids (with square base) and each pyramid can be splitted into two equal tetrahedrons. And those then necessarily also tile the space. Did you mean this?

pat wrote:Triangles don't tile the plane without rotation, either.
Oops, yes, my remark was nonsense. What I wanted to say, that the tiling didnt look the same in each vertex, regardless what tetrahedron. But the rhombic dodecahedron doesnt either. I loose a bit the thread in 3d tilings ...
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Postby wendy » Fri Feb 10, 2006 9:04 am

The tetrahedron i describe is made of four isoceles triangles, of edges sqrt(3):sqrt(3):2. All four vertices are in fact identical, and so tile as such (they each form a tetrahedral {3,3} kaleidoscope.

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