Animations

Higher-dimensional geometry (previously "Polyshapes").

Animations

Postby BClaw » Mon Mar 01, 2004 11:22 pm

I've been doing some animation of higher dimensional objects, and I'd love to hear any opinions anyone has about them:

4-Measure 1
4-Measure 2
4-Measure 2 (alt)
4-Measure 2 (DivX)

5-Measure 1
5-Measure 2
5-Measure 2 (alt)

For these last two, I started out in the 0th dimension and went up to the 10th dimensional simplex. You can tell which point was extruded by it's color:

0 - black
1 - dark red
2 - dark blue
3 - dark green
4 - white
5 - bright red
6 - bright blue
7 - bright green
8 - yellow
9 - magenta
10 - cyan

n-simplex
n-simplex (2)
n-simplex 2 (alt)

And here is the 16-Cell (4-Cross)
16-Cell (DivX)

And here is the Glome!
Hypersphere
Hypersphere III (DivX)

The 24-Cell, my favorite!
24-Cell (DivX)

Thanks for looking!
Last edited by BClaw on Fri Mar 26, 2004 6:26 am, edited 7 times in total.
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Postby pat » Mon Mar 01, 2004 11:59 pm

Unfortunately, Quicktime will only play the '(alt)' versions you have above. I really wanted to see the simplexes thing.

And, as mentioned before in another thread, they look wonderful (the ones that I can see). I'd love to know the sequences of rotations.
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Rotational Specifics

Postby BClaw » Tue Mar 02, 2004 3:54 am

I'll do another version of the simplexes one. Pat? I thought I emailed you a copy of the rotations...

Hi, this is Burt, also known as BClaw on the Tetraspace Forum. I don't know to what level of detail you want to go to, so I'll give you some details, and if you need more, I'll send the code snippets. First of all, the way I did the translations was just as you described, but they are always applied in order on the base object. That way I never have to worry about how complex rotations are going to come back to their origin. I apply the rotations in this order: xy, xz, yz, xw, yw, zw, (xv, yv, zv, wv). OK here we go!

4-Cube 1: total animation, frame 0 to 600
xy: 0-90 deg, frame 30 to 215
xz: 0-90 deg, frame 105 to 290
yz: 0-90 deg, frame 180 to 365
xw: 0-90 deg, frame 255 to 440
yw: 0-90 deg, frame 330 to 515
zw: 0-90 deg, frame 405 to 590

4-Cube 2: total animation, frame 0 to 720
xy: 0-360 deg, frame 210 to 420
xz: 0-360 deg, frame 120 to 330
yz: 0-360 deg, frame 390 to 600
xw: 0-360 deg, frame 30 to 240
yw: 0-360 deg, frame 300 to 510
zw: 0-360 deg, frame 480 to 690

5-Cube 1: total animation, frame 0 to 1080
xy: 0-90 deg, frame 30 to 105: 90-0 deg, frame 150 to 240
xz: 0-90 deg, frame 120 to 195: 90-0 deg, frame 240 to 330
yz: 0-90 deg, frame 210 to 285: 90-0 deg, frame 330 to 420
xw: 0-90 deg, frame 300 to 375: 90-0 deg, frame 420 to 510
yw: 0-90 deg, frame 390 to 465: 90-0 deg, frame 510 to 600
zw: 0-90 deg, frame 480 to 555: 90-0 deg, frame 600 to 690
xv: 0-90 deg, frame 570 to 645: 90-0 deg, frame 690 to 780
yv: 0-90 deg, frame 660 to 735: 90-0 deg, frame 780 to 870
zv: 0-90 deg, frame 750 to 825: 90-0 deg, frame 870 to 960
wv: 0-90 deg, frame 840 to 915: 90-0 deg, frame 960 to 1050

5-Cube 2: total animation, frame 0 to 630
xy: 0-90 deg, frame 30 to 120: 90-0 deg, frame 150 to 210
xz: 0-90 deg, frame 210 to 300: 90-0 deg, frame 330 to 390
yz: 0-90 deg, frame 75 to 165: 90-0 deg, frame 195 to 255
xw: 0-90 deg, frame 300 to 390: 90-0 deg, frame 420 to 480
yw: 0-90 deg, frame 390 to 480: 90-0 deg, frame 510 to 570
zw: 0-90 deg, frame 120 to 210: 90-0 deg, frame 240 to 300
xv: 0-90 deg, frame 345 to 435: 90-0 deg, frame 465 to 525
yv: 0-90 deg, frame 435 to 525: 90-0 deg, frame 555 to 615
zv: 0-90 deg, frame 255 to 345: 90-0 deg, frame 375 to 435
wv: 0-90 deg, frame 165 to 255: 90-0 deg, frame 285 to 345

The last thing I need to tell you is that in all of these rotations, I've tried to smooth them out by having them start rotating very slowly, then speed up to a maximum rotation speed at the mid-point of the rotation, then slow back down to barely moving. If you need that level of accuracy, I'll give you the formulas I used to calculate this.

Good luck, and I can't wait to see what you come up with! Also, I was very impressed when I took a look at your website. At some future point I'd like to talk to you about how to generate the 24, 120, and 600-cells.
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Re: Rotational Specifics

Postby pat » Tue Mar 02, 2004 5:58 am

BClaw wrote:Pat? I thought I emailed you a copy of the rotations...


It's possible that they somehow hid themselves very well in the several hundred spams my Junk Folder accumulated. Sorry... Thanks for these, though. That's wonderful... I'll try to get together my raytracer's version of these this week. Thanks.
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Re: Rotational Specifics

Postby pat » Tue Mar 02, 2004 1:23 pm

BClaw wrote:Good luck, and I can't wait to see what you come up with! Also, I was very impressed when I took a look at your website. At some future point I'd like to talk to you about how to generate the 24, 120, and 600-cells.


Thank you for the compliments. And, here's the first animations I've come up with given your rotations.

4-cube 1
4-cube 2

I haven't put much time into making the cube look too pretty. Right now, it is just colored based upon location on the unrotated cube. The red channel varies with the y-coordinate. The green channel varies with the z-coordinate. The blue channel varies with the w-coordinate.

I may try to explode your animation into frames a make a composite animation of one of these. I think seeing the wireframe flattened version could really help in understanding my version.
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Re: Rotational Specifics

Postby BClaw » Tue Mar 02, 2004 5:35 pm

pat wrote:
Thank you for the compliments. And, here's the first animations I've come up with given your rotations.

4-cube 1
4-cube 2

I haven't put much time into making the cube look too pretty. Right now, it is just colored based upon location on the unrotated cube. The red channel varies with the y-coordinate. The green channel varies with the z-coordinate. The blue channel varies with the w-coordinate.

I may try to explode your animation into frames a make a composite animation of one of these. I think seeing the wireframe flattened version could really help in understanding my version.


I watched the two animations together, yours in quicktime and mine in windows media player. Very cool! Since we're trying to visualize 4D objects, naturally we can't *see* exactly what is going on, but putting the two methods together is fascinating, and I think gives a much clearer view of what is really going on up there in the 4th dimension. :D
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Postby pat » Tue Mar 02, 2004 5:56 pm

There is now also an animation corresponding to your '5 cube 2' rotations:

5-cube 2

Unfortunately, it's a very sizeable file and is already overcompressed. Argle.

I updated my raytracer page a slight bit, too, to include these animations and a couple of other sneak-peeks at the upcoming version.
Last edited by pat on Mon Mar 15, 2004 5:24 am, edited 1 time in total.
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Postby BClaw » Tue Mar 02, 2004 8:24 pm

pat wrote:There is now also an animation corresponding to your '5 cube 2' rotations:

5-cube 2

Unfortunately, it's a very sizeable file and is already overcompressed. Argle.

I updated my raytracer page a slight bit, too, to include these animations and a couple of other sneak-peeks at the upcoming version.


Cool! I haven't downloaded it to look at yet, but I'll go do that presently. Meanwhile, here's the new version of the n-simplex thingee:

n-simplex 2 (alt)
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Postby pat » Tue Mar 02, 2004 8:38 pm

BClaw wrote:Meanwhile, here's the new version of the n-simplex thingee:


That's awesome. Thanks!
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Postby pat » Wed Mar 03, 2004 5:05 am

And, here are three duo-circles (each with a different initial orientation) rotating the same way that the hypercube did in "4-cube 2" above.

Rotating Duo-circles
Last edited by pat on Thu Mar 04, 2004 3:42 pm, edited 1 time in total.
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Ooooh!

Postby BClaw » Thu Mar 04, 2004 6:16 am

:D Hey, nice animation! I *still* don't really get what a duocircle is, but I get a bit of a sense for it from your animation. Oh, and the 5D cube is *very cool*!!! :shock:
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Re: Ooooh!

Postby pat » Thu Mar 04, 2004 3:59 pm

BClaw wrote:Oh, and the 5D cube is *very cool*!!!


I agree... it is really cool how you can see various square-shapes
in the w-v plane and such.
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DivX

Postby BClaw » Fri Mar 12, 2004 11:16 pm

OK, I downloaded the DivX codec, and tried an animation with it. The nice thing, is that the file size wasn't uch different if it was 200X200 or 800X800, so I rendered it full size. If most people can view this without too much trouble, I'll adopt DivX as my standard for the web. :)

4-Measure 2 (DivX)

Let me know if it works for you!

P.S. I'm working on an animation of the 16-Cell (4-cross) :P
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Re: DivX

Postby pat » Sun Mar 14, 2004 5:14 am

BClaw wrote:Let me know if it works for you!


I'm working on a Mac. I had to download a DivX Codec for Quicktime, but after that... it all worked great.

Awesome.

alter,
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16-Cell arrives amid minor fanfare!

Postby BClaw » Mon Mar 15, 2004 1:45 am

And here is the 16-Cell (4-Cross)
16-Cell (DivX)
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Hypersphere!

Postby BClaw » Mon Mar 15, 2004 4:40 am

And here is the Glome!
Hypersphere

A little explanation...

I put a vertex in every 20 deg in the x/y (phi), every 18 deg in the z (theta), and every 30 in the w (?). I hoped that this would give some differentiation. Also the coloring goes from dark red to bright red in the x, dark green to bright green in the y, dark blue to bright blue in the z, and black to white in the w direction.l

Hope you like!
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Postby pat » Mon Mar 15, 2004 5:29 am

pat wrote:5-cube 2


I updated this animation to encode it with DivX. It let me do a much higher resolution (temporal and spatial) for about the same distortion.
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Glomar coordinates animation

Postby pat » Tue Mar 16, 2004 7:49 am

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Re: Glomar coordinates animation

Postby BClaw » Tue Mar 16, 2004 3:44 pm



:D Woah! Cool animation! It's hard to believe that your animation and mine represent the same object!

I'm working on another one of the hypersphere where it shows just a single circle in each of the six hyperspatial planes. I had an interesting thought about this when I was working it out: One way to represent a sphere in 3-space with 3 circles, one at the equator, and two longitudinally (like wire-frame models of spheres in AutoCAD). This is one circle each on the xy-plane, then xz-plane, and the yz-plane. You could do the same thing with 4 spheres on a hypersphere! One each for xyz, xyw, xzw, and yzw! I might do one of those also, but with a different than traditional representation for the spheres. Perhaps Icosidodecaheda. I'll have to think about that one.
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Re: Glomar coordinates animation

Postby pat » Tue Mar 16, 2004 5:36 pm

BClaw wrote:You could do the same thing with 4 spheres on a hypersphere! One each for xyz, xyw, xzw, and yzw!


That is a great idea... hope you don't mind: Axial Spherinders
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Re: Glomar coordinates animation

Postby BClaw » Tue Mar 16, 2004 7:57 pm

pat wrote:
BClaw wrote:You could do the same thing with 4 spheres on a hypersphere! One each for xyz, xyw, xzw, and yzw!


That is a great idea... hope you don't mind: Axial Spherinders


No, of course I don't mind! And your animation is beautiful!!! I really like how you've made each of the four axial spherinders a little different in size, so you can *really* see what is going on!
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Re: Glomar coordinates animation

Postby pat » Tue Mar 16, 2004 10:03 pm

BClaw wrote:I really like how you've made each of the four axial spherinders a little different in size, so you can *really* see what is going on!


Well, actually, I made them each a little different in size because when they were all the same size, there were some strange artifacts (due to rounding errors) where two surfaces were coincindent. :roll: But, I'm glad you like it.
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Glome

Postby BClaw » Thu Mar 18, 2004 11:37 pm

Here's the one I was talking about for the glome where I'm just showing six circles, one each for the six planes of four-space:
Hypersphere III (DivX)

This animation compressed very nicely, unlike the 1st glome animation. Probably because it's not as complex..

Hey, Pat, here's another thought! This animation has six planer discs to represent the glome, you did one with four spherinders, and I *think* another representation could be three duo-circles. I don't really understand how a duo-circle works, but in looking at the definition: r[sup]2[/sup]=x[sup]2[/sup]+y[sup]2[/sup]=z[sup]2[/sup]+w[sup]2[/sup], I had a thought... Aren't there just three ways to orient a duo circle exactly along axes in tetraspace? Consider 1:xy-zw, 2:xz-yw, and 3:yz-xw. What do you think?
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Icositetrachoron!

Postby BClaw » Fri Mar 26, 2004 7:08 am

24-Cell (DivX)

I did it! Thanks to some help on the coordinates from the folks at Polytopia on Yahoo. I'm so excited! :D

Here's a sneak peek at the 120-Cell animation I'm working on:
120-cell (Frame 0).jpg
120-cell (Frame 512).jpg

And some new variations on the Hypersphere:
Hypersphere II (Frame 426).jpg
Hypersphere II (Frame 527).jpg
Hypersphere IV (Frame 437).jpg
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Re: Icositetrachoron!

Postby pat » Fri Mar 26, 2004 2:37 pm

BClaw wrote:I did it! Thanks to some help on the coordinates from the folks at Polytopia on Yahoo. I'm so excited! :D


Awesome... I can't wait to see the other animations. The 120-cell looks cool.
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New page!

Postby BClaw » Wed Mar 31, 2004 5:19 am

New Hyper Page!

Well, I finally put together a little page with my animations. Dunno how long it will last, but there it is!

I added an animation of a truncated 16-Cell, and another version of a Hypersphere (IV).

Let me know what ya think!
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Postby pat » Wed Mar 31, 2004 2:34 pm

Last night, I added a rotating 24-cell animation. This animation is saved as MPEG-4. Hopefully, other folk will have an easier time playing it than I do. I have to download it first to get it to play in Quicktime. I'll try updating the MIME-types with a .htaccess file.
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