5D Rotations

Higher-dimensional geometry (previously "Polyshapes").

5D Rotations

Postby The_Science_Guy » Fri May 12, 2006 12:46 am

I know that in n dimensional space, that rotations happen around an n-2 dimensional figure, so, according to that, a 5D object would rotate around a stationary cube. Are there any animations out there that can show this in action?
WARNING: Reading this signature line may cause extreame confusion and laughter. Please consult a comedian before reading

!sdnah ruoy no emit eerf fo tol lufwa na evah tsum uoy ,woW
User avatar
The_Science_Guy
Dionian
 
Posts: 59
Joined: Sun Apr 23, 2006 8:12 pm
Location: perpendicular to you

Postby bo198214 » Fri May 12, 2006 9:22 am

So have you explored the 4d rotation animations already to such an excess that it became boring and you need more dimensions? :o
For higher dimensions its by the way easier to regard the rotation plane, i.e. not the n-2 dimensions but the other 2 dimensions.
bo198214
Tetronian
 
Posts: 692
Joined: Tue Dec 06, 2005 11:03 pm
Location: Berlin - Germany

Postby Nick » Fri May 12, 2006 10:12 am

You would need an extremely excellent understanding of 4d before you can move on to 5d. 4d would need to be as easy as 3d is to you now. That would take years of studying! :shock:
I am the Nick formerly known as irockyou.
postcount++;
"All evidence of truth comes only from the senses" - Friedrich Nietzsche

Image
Nick
Tetronian
 
Posts: 841
Joined: Sun Feb 19, 2006 8:47 pm
Location: New Jersey, USA

Postby The_Science_Guy » Fri May 12, 2006 4:17 pm

irockyou wrote:You would need an extremely excellent understanding of 4d before you can move on to 5d. 4d would need to be as easy as 3d is to you now. That would take years of studying! :shock:
Oh. :oops:
WARNING: Reading this signature line may cause extreame confusion and laughter. Please consult a comedian before reading

!sdnah ruoy no emit eerf fo tol lufwa na evah tsum uoy ,woW
User avatar
The_Science_Guy
Dionian
 
Posts: 59
Joined: Sun Apr 23, 2006 8:12 pm
Location: perpendicular to you

Postby pat » Fri May 12, 2006 4:46 pm

I think it's easier to think of rotations as happening parallel to two-dimensional planes rather than as happening around (n-2)-dimensional spaces.

For a basic rotation, take two linearly independent vectors in n-dimensional space. These two vectors define a 2-dimensional plane. A rotation of an arbitrary point in n-dimensional space parallel to that plane takes the projection of the point into the plane, rotates that, then "unprojects".

For more complex rotations, concatenate basic rotations.
pat
Tetronian
 
Posts: 563
Joined: Tue Dec 02, 2003 5:30 pm
Location: Minneapolis, MN

Postby wendy » Sat May 13, 2006 8:48 am

I have a fully working phase-space for 3 and 4 dimensions, which live in 3d and 6d respectively. The phase-space for 5d rotations is supposed to live in 10 dimensions or something, from what i read.

Many rotation modes are not around a (n-2) axis. The natural mode assumed by a spherical planet is one of the clifford rotations, where everything goes around the centre of the thing. This is a truely wonderous form, where from the planet's surface, every star rises on the horizon, and sets directly opposite (on the antipode). It is a truly beautiful sight to watch.

In five and six dimensions, the rotations are not easy for me to imagine. If you want to understand a small fragment of those in 7d and 8d, try looking for the conway-smith novel 'quarterions and octonions'. it's pretty heavy going, but these require phase-spaces in 21 and 28 dimensions respectively.

i discussed some aspects of rotation with john conway, but i tend to disagree with what he had to say in part, more by ommission than error.

W
The dream you dream alone is only a dream
the dream we dream together is reality.

\ ( \(\LaTeX\ \) \ ) [no spaces] at https://greasyfork.org/en/users/188714-wendy-krieger
User avatar
wendy
Pentonian
 
Posts: 2014
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia

Postby pat » Mon May 15, 2006 3:48 am

Wendy, you've met John Conway? Cool....
pat
Tetronian
 
Posts: 563
Joined: Tue Dec 02, 2003 5:30 pm
Location: Minneapolis, MN

Postby wendy » Mon May 15, 2006 8:50 am

John Conway and I exchanged a number of emails on hyperbolic tilings. Apparently some new book in the works is partly inspired by the work that Marek and I did on H2 tilings.

He tought me something on the way that simple rotations (PG great arrows) are grouped in 4d, particularly his model of it.

I showed him how to turn the poincare dodecahedron into a poincare decahedron. He was impressed.

We have a severe terminology difference, but nothing that can't be worked out.

W
The dream you dream alone is only a dream
the dream we dream together is reality.

\ ( \(\LaTeX\ \) \ ) [no spaces] at https://greasyfork.org/en/users/188714-wendy-krieger
User avatar
wendy
Pentonian
 
Posts: 2014
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia

Postby thigle » Tue May 16, 2006 10:49 am

wendy, so what is the difference between his model of space of great arrows for 4d rotations ?

if i grasped it right, your model for phase space of simple rotations in 4d is a terix naturally dwelling in 6d - a bi-glomohedric prism.
for rotations and rotations^2, you use a bi-glomohedric pyramid which is a petix in 7d.

question1: is that so or do i misinterpret you?

question2: how does John Conways' model of phase space for 4d-rotations differs from yours ?

question3: elsewhere you state that shape of phase space of rotations for 3d is a sphere. i wonder if that is not just for simple rotations. it seems to me that the phase-space of ALL rotations for 3d could be a 6d sphere-point, modellable perhaps after 6d hypersphere.
thigle
Tetronian
 
Posts: 390
Joined: Fri Jul 29, 2005 5:00 pm

Postby wendy » Wed May 17, 2006 8:43 am

A phase-space represents a point for every mode of inertial rotation. That means, we are looking at rigid spheres rotating around the centre, without being disturbed externally.

The phase-space in 3d represents the north-pole. A sphere spinning in any direction must have a north-pole, and therefore therefore is a one-to-one mapping of sphere-rotations + north poles to points on the sphere.

The phase-space for great arrows in 4d represents a bi-glomohedric prism. This is a 4-fabric (terix) in 6d. However, this presumes that there is no rotation perpendicular to this. This is what John was trying to tell me.

When one accounts for orthogonal rotations, one gets a bi-glomohedric pyramid, nominally, a 5-fabric (petix) in 7d. But you can map this onto the surface of a 6-sphere, by using a tegum-product, and fractional parts.

Wendy
The dream you dream alone is only a dream
the dream we dream together is reality.

\ ( \(\LaTeX\ \) \ ) [no spaces] at https://greasyfork.org/en/users/188714-wendy-krieger
User avatar
wendy
Pentonian
 
Posts: 2014
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia

Postby thigle » Wed May 17, 2006 4:52 pm

The phase-space for great arrows in 4d represents a bi-glomohedric prism. This is a 4-fabric (terix) in 6d. However, this presumes that there is no rotation perpendicular to this. This is what John was trying to tell me.

that is also what i was also pointing out in this thread: glomar(glomic?)rotations...
in my question:
how to generalize this process [of rotation]? ...
r0 - rotate something (= ordinary rotation around axis)
r1(r0) - rotate this rotation (= 'spherical rotation', 'spin', 'double rotation')
r2(r1(r0)) - rotate the spherical rotation (= 'triple rotation')
rn(...r3(r2(r1(r0)))...) - 'n-tuple rotation' ???
r(infinity) - 'infinity-tuple rotation ???

you wrote there:
2: then, in 4d, rotating for exemple around plane(xy), while simultaneously rotating this plane(xy) around another plane(wz), what do we get - a 'glomar rotation' ?

This is too hard for me to figure out. the rotations sound like they would add in some way, but i can't see how.

but elsewhere you talk about clifford rotations (link from quickfur's page). these seem to be the way how these rotations add. is that what you mean by orthogonal rotations - that you rotate an axis of rotation about an axis orthogonal to it ?
When one accounts for orthogonal rotations, one gets a bi-glomohedric pyramid, nominally, a 5-fabric (petix) in 7d. But you can map this onto the surface of a 6-sphere, by using a tegum-product, and fractional parts.

i unfortunately do not understand the tegum product yet, so i quite cannot grasp that mapping.

could you perhaps link JohnConway to that thread so he looks at it ? could he perhaps answer that question on generalisation of nesting of rotations ? does he have any model for these hyperrotations (aka 'hypeRotations') ?
thigle
Tetronian
 
Posts: 390
Joined: Fri Jul 29, 2005 5:00 pm

Postby wendy » Thu May 18, 2006 10:09 am

A "great arrow" rotation is a simple rotation in say, wx plane, holding yz constant. This is the sort of rotation that a wheel would do, since the wx gives "height + forward" while the yz space gives "steering". It is hard to steer your car if the steering axis is rotating!

If one sets the yz plane also rotating, then one gets a "double-rotation". In simple terms, consider a point where y=z=0. This still rotates in a wz plane as it did before. As one goes further out, one starts rotating separately in the wx and yz planes. You can for example, imagine a circle in the wx plane as a cylinder that has been bent end to end (like a torus). If you unfold this, then

rotation in wx plane = going higher on the cylinder
rotation in yz plane = going around the cylinder.

A general double rotation then corresponds to going around the cylinder n times as you go upwards. It is kind of like a screw, really.

Regardless of how big you make the cylinder, or how squat, there is exactly the same number of revolutions in yz for each revolution in xy.

When you make these all equal (ie +1 turn or -1 turn per height), something magic happens. All rotations go around the centre, and you can shift the wx and yz axis to wherever you please.

CLIFFORD ROTATIONS

This magic is the clifford-rotation. Under this sort of rotation, all things appear to revolve around the centre point (rather than some kind of extended axis).

If you were sitting on a planet that was revolving clifford-wise, the heavens would not only go from east to west, but would rotate around the horizon. A bright star that rose over a hill, would reach its zenith at an angle 90 degrees from that hill, and set at an opposite point (antipode) of the horizon (remember, the horizon is now a surface sphere).

Because there is no way of deriving a wx or yz axis system from this, it is more fundemental than a single arrow rotation. This particular rotation corresponds to the multiplication by quaterions, for what this is worth.

Clifford rotations come in two flavours (left and right), which amounts to the horizon rotating during the day, such that the direction 'east' is either the north-pole or south-pole (ie it goes clockwise or anticlockwise around the line from east to west).

THE PHASE SPACES

The clifford-rotations form a lot of arrows that each represent a point on a sphere. If for example, some bright star rises at the east point here, then it will reach its zenith directly overhead, and set to the west point. The places on the planet that have this same bright star form a circle. You can map all of the stars in the heavans onto different or (same) circles.

A second star that rises say, 50 degrees from the first, will rise for all points of the globe, the arc between their rising points will be 50 degrees. If one makes a model of rising distances, one gets a sphere whose diameter is not 180 degrees, but say, 90 degrees. This makes a rising sphere.

This sphere surface forms one of the "glomohedrices" mentioned on the product.

The thing with the left and right rotations, is that for a given great circle, there is only one other great circle, which appears in both the left and right rotations. In terms of our cylinder, it is kind of like going up the cylender clockwise or anticlockwise. These are only the same when the cylinder is flat (no height), or infinitely thin (no width). For any other route, there are two distinct paths.

Therefore, a given great arrow that is left-parallel to arrow X, is not left-parallel to any other arrow that is right-parallel to X. That is, every great arrow has a left- and right- parallelness to a right- and left- parallel of X.

This maps onto phase-space as follows:

a great arrow is a point. Left-parallel-ness is in x1x2x3 space, and right-parallelness is in y1y2y3 space. For a given point X1X2X3Y1Y2Y3, one can go first right-parallel to X1X2X3y1y2y3, and then left-parallel to x1x2x3y1y2y3, or go otherwise via x1x2x3Y1Y2Y3.

Replacing x1x2x3 with x, and y1y2y3 as y, we can then see the four special points:

x,y is the great arrow itself. (ie in wx plane)
-x,-y is the great arrow reversed (ie spinning backwards)
x,-y and -x,y is the completely perpendicular arrow (yz plane) spinning in different directions.

This is john conway's model. It only covers great arrows, and not double rotations, etc.

The phase space for double rotations, has to some how include a mode for dealing with a great arrow at speed x, and a reversal at speed y.

So what we have, is then a% x,y and say b% x,-y, that is, different speeds on different axies. (a+b)/(a-b) is the number of times you go around the cylinder as you go up.

We can then say that one is the faster, and have a tetrahedron, forming four sixes of the square, where

1,1,1 = x,y -1,1,-1 is x,-y 1,-1,-1 is -x,-y and -1,1,-1 is -y,x.

The top and bottom edge are the diameter of the sphere representing clifford rotations. Going from say, 1,1,1 to -1,1,-1 slows down the orthogonal rotation y until it becomes 0 (at 0,1,0) and then speeds up to -1. Any faster speeding would then make y the promary axis, and we go down the next edge of the tetrahedron, ie from x,-y, we make x more -x.

The points at 0,x,y are the great arrows rotations, that is, that which has a static inverse. The values at, say 1,x,0 or -1,0,y are the clifford-rotations, and all other point are "double rotations".

The pyramid product is then in seven dimensions, the points

2,2x1,2x2,2x3,0,0,0 to -2,0,0,0,2y1,2y2,2y3

which passes through the six-dimensional prism (like the mid section of a tetrahedron edge 2 is a square edge 1), gives

0,x1,x2,x3,y1,y2,y3

removing the 0 gives the six dimensional bi-glomohedric prism.

We can replace a range of +2 to -2, for perpendicular axies, by an line from 2 at x1,x2,x3,0,0,0 to -2 at 0,0,0,y1,y2,y3. This gives a tegum, which in turn can be expanded out to a glomopetix (sphere in 6d, literally, globe-shaped 5-fabric).
The dream you dream alone is only a dream
the dream we dream together is reality.

\ ( \(\LaTeX\ \) \ ) [no spaces] at https://greasyfork.org/en/users/188714-wendy-krieger
User avatar
wendy
Pentonian
 
Posts: 2014
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia

Postby thigle » Thu May 18, 2006 1:26 pm

i had to print this, now i'll reread it few times while jotting down madly all that clicks. it seems to be clear and coherent logically, but my imagination is stumbling beyond the intellect this time. thank you.
thigle
Tetronian
 
Posts: 390
Joined: Fri Jul 29, 2005 5:00 pm


Return to Other Geometry

Who is online

Users browsing this forum: No registered users and 11 guests

cron