Is it possible to construct a model of an object performing

Higher-dimensional geometry (previously "Polyshapes").

Is it possible to construct a model of an object performing

Postby The_Science_Guy » Sat May 20, 2006 4:54 pm

a Clifford rotation? I mean actually building one, not showing a reperesentation on a screen.
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Postby thigle » Sat May 20, 2006 5:38 pm

perhaps the circle-framed thiings used for training of cosmonauts, where you have a certain number of circles one within another, connected via joints at antipodal points, now each circle can rotate within a circle, and if you're in it, its quite heavy on stomach for many people. :lol:
or maybe what happens with EM fields in toroidal-knots coils might be considered as clifford rotations ? :oops:
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Postby thigle » Sat May 20, 2006 5:55 pm

so it's not only possible to construct such an object, but also to fit in it and experience it bodily. if you have an access to such a gadget. or if you're sleeping with a cosmonaut :lol:
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Postby moonlord » Sat May 20, 2006 6:37 pm

You ain't got four spacial dimensions or more, so you can't.
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Postby wendy » Sun May 21, 2006 7:58 am

There are some interesting pictures of sections of clifford-rotations on the web, under the name 'swirl-prism'.

Apart from that, they exist in 3d, and so that a trip from side to side of the universe makes but a rotation (or by the poincare-dodecahedron model), 1/10 of a rotation.

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Postby moonlord » Sun May 21, 2006 5:58 pm

Um, don't Cliffords require two independent planes of rotation? That sums up to 4 dimensions...
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Postby wendy » Mon May 22, 2006 8:02 am

Yeah, but a wind that rages at a distance of 4000 miles from the centre rages in a 3d space. So there you go.
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Postby The_Science_Guy » Wed May 24, 2006 4:09 pm

A clifford rotation is also defined as when every point in a 3D object rotates around the center point. So, according to that, it is possible to have a clifford rotation happen in 3D.
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Postby moonlord » Wed May 24, 2006 5:59 pm

"is also deffined"... Aren't things supposed to only have a definition?
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Postby papernuke » Thu Jul 27, 2006 3:14 am

I can't awnser your question, but your topic is on google :D ! I was seeing what a Clifford rotation was and I saw it said tetraspace here http://www.google.com/search?hl=en&q=Clifford+rotation. :D
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Postby bo198214 » Thu Jul 27, 2006 7:41 am

moonlord wrote:"is also deffined"... Aren't things supposed to only have a definition?

No, usually there are more definitions of a term depending on the field of application.

On the other hand I dont understand what science_guy means with rotation around a point in 3d.
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Postby The_Science_Guy » Sat Aug 19, 2006 12:23 am

^ What i mean is, that in a Clifford rotation, every point on and in a 3d object rotates around the point in the center of said object.
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Postby bo198214 » Sat Aug 19, 2006 12:55 am

dont get it. Clifford rotation is a rotation in 4D. So how we can Clifford rotate a 3d object?
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Postby wendy » Sat Aug 19, 2006 7:36 am

Any rotation that happens on the surface of a sphere can be mapped onto the plane. This is a mapping of S2 onto E2.

Likewise, any rotations that happen on the surface of a glome can be mapped onto the surface of horochorix (or E3 space).

For such a rotation, imagine you have a circle in xy, and a straight line in z. The circle rotates in xy, and the straight line is an unravelled (with infinite point) line in z.

The rotation is right-handed, suppose. (for lefthanded, replace z with -z).

The circle rotates clockwise when viewed from under neath. The Line goes from -oo to +oo (ie upwards).

For a general circle (we assume stereographic projection), it passes through two points on the xy plane, and passes as a linked circle (so one point is inside and one is outside). The points of crossing is set by negative inversion (ie, r, -1/r). It is possible, then to set up a set of circles, that cross the plane at such a circle. These circles are set so that the angle made with the plane is the arc-cos of the radius of the inner intercept.

One then has a kind of rotation that as the disk rotates, the entirity of space also rotates through the disk and at 180°, through the outer half of the disk.

This is the stereographic projection of the clifford rotation, represented in 3d.

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Postby pat » Fri Aug 25, 2006 3:48 am

Here's an applet that shows Clifford rotations of a 3-cube, a pair of concentric 3-cubes, a 4-cube, or a 3-octahedron.
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Postby bo198214 » Fri Aug 25, 2006 7:45 am

oh thank you, so it isnt a rotation in 3d, that was unclear to me.
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Postby pat » Fri Aug 25, 2006 4:21 pm

The_Science_Guy wrote:A clifford rotation is also defined as when every point in a 3D object rotates around the center point. So, according to that, it is possible to have a clifford rotation happen in 3D.


That depends on what you mean be a rotation. There is no way that you can throw a ball up into the air with a spin that has only a single fixed point. Sure, you could hold the ball in your hands and spin it erraticly. But, you'd more or less be just doing a series of rotations where each rotation has a fixed axis of rotation, but you rapidly change which axis is fixed.

Unless, of course, you're talking about some non-rigid object, like a liquid mass.... then, I suppose you could have every point rotate around the center....
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Postby bo198214 » Fri Aug 25, 2006 8:51 pm

pat wrote:That depends on what you mean be a rotation.

By a rotation I mean an isometric and orientation preserving transformation.

Reminds a bit of the hairy ball theorem. :)
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Postby pat » Sun Aug 27, 2006 4:58 pm

Yes, I was thinking of the hairy ball theorem after my last post. I was thinking, even with a non-rigid body, it would have to be discontinuous to not have any fixed points (taking the hairs to be the momentum vectors of the surface points). And, it's pretty hard to conceive of a "body" of any sort where particles didn't have to worry about their neighbors.
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Postby thigle » Thu Sep 14, 2006 6:53 am

pat wrote
The_Science_Guy wrote:
A clifford rotation is also defined as when every point in a 3D object rotates around the center point. So, according to that, it is possible to have a clifford rotation happen in 3D.

That depends on what you mean be a rotation. There is no way that you can throw a ball up into the air with a spin that has only a single fixed point. Sure, you could hold the ball in your hands and spin it erraticly. But, you'd more or less be just doing a series of rotations where each rotation has a fixed axis of rotation, but you rapidly change which axis is fixed.

Unless, of course, you're talking about some non-rigid object, like a liquid mass.... then, I suppose you could have every point rotate around the center....


well, what about acrobatic ski-jump ? when they are in the air, and do for ex. 720 with a twist...
...it means that their rotation takes place around horizontal axis orthogonal to front-back direction (720)
AND simultaneously around vertical axis (360twist)
BOTH relative to body position at each moment.

that means only a point stays invariant.

isn't this so ?

also when juggling, one can throw it in air in sucha way that it revolves around 2 axies and so rotates clifford-style - only point invariant.

also, if atoms are only rigid models for spherical standing waves, then all the 'matter' teems or jitters cliffordishly.
*
also, could you add 3-simplex and 4-simplex to your clifford rotation applet ? people ususally skip these simpletons when doing multidimensional visualisation, byt i find that these minimal systems serve best to grasp the idea.

i have done these anims few years ago, first is 3simplex, second just its vertices, performing rotation around 3 planes:
http://s24.photobucket.com/albums/c18/thigle/?action=view&current=8-fold_jointsonly.flv
http://s24.photobucket.com/albums/c18/thigle/?action=view&current=_0.flv
are these clifford rotations of 3simplex ? it revolves around 3 planes, so that's one more.?
could you add to the applet a possibility of rotation about 3, 4 planes ? what that would be ? a hyperClifford ? :roll:
also please consider putting up a numeric input possibility in the interface together with the sliders.

also, what projection method is used for the applet ? is the 4the coordinate used for distance from the centre ? (like here: http://www.lboro.ac.uk/departments/ma/gallery/hyper/mov/hyperCube2.mpeg, in the video where hypercube with one cell color coded is rotating around 3 planes at different rates ?)
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Postby pat » Thu Sep 14, 2006 3:30 pm

thigle wrote:well, what about acrobatic ski-jump ? when they are in the air, and do for ex. 720 with a twist...


A jumper in the air is bends and twists his body and such to effect the rotation. But, if you assume that the jumper gets the motion going at the very beginning of the jump and just rides out the motion after that, then he has a fixed axis of rotation (if we disregard wind-resistance). That axis isn't aligned with what you'd consider his major axises to be if he were just standing there. But, there is a fixed axis.

Consider tossing a ball up into the air. Once it leaves your hand, there is a fixed axis of rotation. You can see this most easily with a basketball, but any textured/patterned ball will do. Once the jumper stops bending and twisting his body, he is effectively oriented some way within the ball. It may be easier to imagine that the skier has a ball inside at his center of gravity. That ball has a fixed axis of rotation. Thus, the skier has a fixed axis of rotation.

also, could you add 3-simplex and 4-simplex to your clifford rotation applet ? people ususally skip these simpletons when doing multidimensional visualisation, byt i find that these minimal systems serve best to grasp the idea.


I had intended to add them, but I didn't because I didn't want to recalculate the coordinates for a regular 4-simplex. I should have realized that I already had code that contained the coordinates. ;) Anyhow, those are added now.

could you add to the applet a possibility of rotation about 3, 4 planes ? what that would be ? a hyperClifford ? :roll:
also please consider putting up a numeric input possibility in the interface together with the sliders.


Yep, I'd have to go to numeric input to add more planes. And, sure, I can do all six planes at some point.

also, what projection method is used for the applet ?


The projection for the XYZ left-eye goes like this:
Code: Select all
      [ 1, 0, 0, 0, 0   ]
P43 = [ 0, 1, 0, 0, 0   ]
      [ 0, 0, 1, 0, 0   ]
      [ 0, 0, 0, 1, 1.5 ]


      [ 10,  0,  0, -0.75 ]
P32 = [  0, 10,  0,  0    ]
      [  0,  0, 10, 15    ]

p' = P32 * V * P43 * R * p


Here, p is the homogenous coordinates for the initial point [ x y z w 1 ]
and p' is the homogenous coordinates for the final point [ x' y' d ]. The matrix R is a 5x5 matrix that specifies the object's orientation (the upper-left 4x4 submatrix is the Clifford rotation). The matrix V is a 4x4 matrix that specifies the orientation of the view (because it was pretty boring looking straight on to the object).

So, we start with the homogenous coordinate p = [ x y z w 1 ]. We rotate it by R. Then, we project it from 4-dimensions to 3-dimensions with P43. Then, we rotate the thing a bit to get a better angle on it. Then, we project it from 3-dimensions to 2-dimensions with P32. To get the final point [ x' y' d ]. This is mapped to screen coordinates as:
Code: Select all
sx =  sw/2 + x' * ss / d
sy =  sh/2 - y' * ss / d

Where sw and sh are, respectively, the screen width and height. And, ss is a scaling factor which I just set equal to sh/2.

For the XYW view, the P43 matrix is:
Code: Select all
      [ 1, 0, 0, 0, 0   ]
P43 = [ 0, 1, 0, 0, 0   ]
      [ 0, 0, 0, 1, 0   ]
      [ 0, 0,-1, 0, 1.5 ]

For the right-eye views, change the sign on the 0.75 in the P32 matrix.
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Postby thigle » Thu Sep 14, 2006 7:13 pm

very cool. the simplexes. thank you.

i don't understand the projection code coz i don't know how matrices work. :cry: but it looks incredibly simple. just few lines of code. what are those values 0,1,0.75,... ? where do you get them from and what do they mean ?

ok. the acroski-jumper might not be rigid body par excellence, but what about spherical standing waves model of atoms ?

or what about this, assuming no friction:
a spherical body is set in rotation by a force F1 that hits it in tangential direction that cuts its radius r orthogonally at 1/4 from centre. now another force, F2, hits the rotating spherical body in direction orthogonal to plane rF.

depending on the magnitudes of F1 & F2, can there be a possible ratio when they set the body in motion that leaves only a point invariant, in other words how can one get a spherical body in spherical rotation with 2 forces ? or is there no way to make an object spin ? how come then that particles spin ? what made them do so ? or are they multidimensional ?
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Postby pat » Thu Sep 14, 2006 10:34 pm

thigle wrote:a spherical body is set in rotation by a force F1 that hits it in tangential direction that cuts its radius r orthogonally at 1/4 from centre. now another force, F2, hits the rotating spherical body in direction orthogonal to plane rF.


Still, no dice. For any F1 and F2, there is an angle and force which you could have used on the non-moving ball to get the same final result... the same spin in the same plane. The force F2 changes where the axis of rotation is, but there is still an axis of rotation.

As for "how come then that particles spin"? You have to be really careful there. The "spin" that particles have is not much like the spin of a rigid body. If somehow it made sense to give a basketball spin 1/2 or spin 3/2, it would also be perfectly possible for a basketball to have spin 5/7. Particles don't have that option.

As for the spherical standing wave model of atoms, several points... if it is a standing wave, it has lots of fixed points.... beyond that, the thing doing the waving isn't a rigid body and we're talking about rotating, not oscillating, right?

In my notation above, I played a little loose with the notation. I should have written the vectors as columns instead of rows.
Code: Select all
    [ x ]
    [ y ]
p = [ z ]
    [ w ]
    [ 1 ]

That is a (column) vector which is a special-case of a matrix. It is a 5x1 matrix. It has five rows and one column. Often, one writes this like p = [ x, y, z, w, 1 ]<sup>T</sup>. The "T" stands for "transpose". It means, flip it... make the rows columns and the columns rows.

If you want to multiply two matrices, they have to be compatible first. That means that the first one has to have the same number of columns as the second one has rows. So, you can multiply a 5x3 matrix by a 3x4 matrix in that order, but not in the opposite order.

Actually, this is going to get long and more complex than I want to do in BBCode. I'll try to get into it more some other time.

The one's in the matrices pretty much select which coordinate you want. If you multiply the matrix:
Code: Select all
[ 1, 0, 0, 0, 0 ]
[ 0, 0, 1, 0, 0 ]
[ 0,-1, 0, 0, 0 ]
[ 0, 0, 0, 0, 1 ]

by the vector [ x, y, z, w, 1 ]<sup>T</sup>, you'd get the vector [ x, z, -y, 1 ]<sup>T</sup>.

The 0.75 thing is an offset in x. If you multiply the matrix:
Code: Select all
[ 1, 0, 0, 0, a ]
[ 0, 1, 0, 0, 0 ]
[ 0, 0, 0, 1, 0 ]
[ 0, 0, 0, 0, 1 ]

by the vector [ x, y, z, w, 1 ]<sup>T</sup>, you'd get the vector [ x + a, y, w, 1 ]<sup>T</sup>.

And, to really show how matrices work, if you multiply these matrices:
Code: Select all
                 [ x 2 6 ]
[ a, b, c, d ] * [ y 3 7 ]
[ e, f, g, h ]   [ z 4 8 ]
                 [ w 5 9 ]

You would get the matrix:
Code: Select all
[ ax+by+cz+dw, 2a+3b+4c+5d, 6a+7b+8c+9d ]
[ ex+fy+gz+hw, 2e+3f+4g+5h, 6e+7f+8g+9h ]
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Postby pat » Thu Sep 21, 2006 9:46 am

Okay, I've updated the applet
http://www.nklein.com/products/crot/
to allow you to set the rotation speed for any of the planes.

Also, I added some more shapes: 3-Ball, 4-Ball, Torus, Klein bottle...
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Postby thigle » Thu Sep 21, 2006 7:12 pm

whoa, that's great. thanxalotalot
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