glomar (glomic ?) rotation & spins of spins in general

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glomar (glomic ?) rotation & spins of spins in general

Postby thigle » Sat Aug 06, 2005 7:36 pm

actually, this turned out to have more questions than I supposed. so I'm gonna number these for clarity:

1: although it is generally assumed that rotation in 3d happens around an axis, leaving it invariant, there also exists what is called 'spherical rotation' in 3d : rotating around an axis A, while simultaneously rotating this rotation axis A around another axis B, gives spinning motion - so called spherical rotation. then ( speed of rotation on A / speed of rotation on B ) & (angle between A and B) determines the kind of loop (if any) one gets. does this leave a point of intersection of A and B invariant? or not even that (because it would get somehow 'twisted' i imagine) ?

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' ?
3: then, what (if any) influence does it have in 4d, whether 1 of 2 coordinates of the first rotation plane is the same as one of second rotation plane's 2 coordinates ?
I mean: is there a difference between 'glomar rotations' for these 3 couples of rotation planes: [(xy)(zw), (xz)(yw),(xw)(yz)] and these 12: [(xy)(xw), (xy)(xz), (xy)(wy), (xy)(zy), (xz)(xw), (xz)(zw), (xz)(zy), (zw)(wx), (zw)(wy), (zw)(zy), (yw)(wx), (yw)(yz) ] ?

{btw, can anyone give a lesson in notation ? i'm not mathematician, but surely there must be a concise way to note the above questions... }

4: finally, how to generalize this process ? firstly its iterative character:
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 ???
5: then to understand it across increasing dimensionality...

6: or, because of some (to me yet not understood) reasons, this sequence of 'rotational iterations' breaks down and stops ? (like for exemple the 4 divisor algebras end up at octonions), or does something like the Gimbal lock in 3d, occuring recursively, forces the whole thing into stasis ?

.done
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Postby wendy » Sun Aug 07, 2005 11:39 pm

From what i know,

the space of great arrows (ie where a point represents a simple rotation in a 2d subspace), forms a curved figure in N(N-1)/2, this means for 3d, it is 3d, for 4d it is 6d, for 5d, it is 10d &c.

I know the 3d shape (a sphere, where the points represent the north-pole), and the 4d shape (a bi-glomohedric prism, a 4d manifold (terix) in 6d.

When one accounts for all modes of rotation, the 4d figure becomes a bi-glomohedric pyramid: a 5d manifold (petix) in 7d.

That's as far as i got.

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Postby thigle » Sat Oct 01, 2005 12:48 am

still studying your polygloss, but:

is the following correct ?: space of great arrows for 3d is SO4 ? or that's double as much viewpoints as needed? i interpret a point in space of great arrows as viewpoint + orientation, i.e. you can have many front-wise views, but all with different arounds, according to the orientation of horizon, which is rp1 in rp2 witch closes e3 by infinity in this case. the quats with the same vector part will have the same axis of view, but their around-positions differ as their scalar part specify. 2 same values with opposite sign are identified.

why for your 3d shape, the points represent the north-pole? does it have anything to do with stereographic projection?

for 4d, you say that all possible sur_viewpoints form together a bi-glomohedric prism, a terix in 6d. why in 6d?

does idea of counterspace anything to do with this at all?

also when observer gets to infinity, what does happen to his around potential ? is the observer the dweller in the centre of projection ? the projection centre can be before(ex.perspective), at(stereographic) & under(???) the projection space. when it's out and zooms out even more, into infinity, what happens ? if he zooms through, does the vision reverse through paralle projection to inverse perspective (as small children before perspectival enculturaltion often draw - small things far, bigger things the closer they are)?

how come you go got this 8d limit to your explorations? is it enough?
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Postby wendy » Sat Oct 01, 2005 8:54 am

I am not familiar with exactly what SO4 means. I have seen it, and John Conway mentioned it to me in some conversation we had, but i derive things from first principles.

A great arrow is simply a great circle with an arrow on it. A great arrow rotation rotates only in the great arrow, ie in 2 dimensions, with a static axis of N-2 dimensions.

In four and higher dimensions, one can have compound rotation, eg rotation in wx, yz planes at different rotations.

One can create a "phase-space" (to use the physicist's term), where each possible rotation is represented by a point.

In three dimensions, only great-arrow rotations are possible, and one can set the rotation such that the phase-point corresponds to the north pole. You can use any point, but usually the north or south pole does. It has no particular role. One could set the north pole at london or lhasa.

In four dimensions, the phase-space is much more complex, but it can be easily derived from watching the night sky in four dimensions (which is how i figured it out).

The phase-space of great arrows is a bi-glomohedric prism: the prism-product of two sphere-surfaces. Suppose we let the point x, y represent a point on the two sphere-bases.

The point x represents the north pole (it can be any general point). It creates a great arrow rotation on the equator.

The point -x also creates a great-arrow rotation on the equator, but opposite to x, so -x would be the antipode of x

The point x,y corresponds to a particular great arrow in 4D.

The point x,w is a different great arrow, equidistant from xy, by half the angle between w and y on their sphere. Likewise, g,y is equidistant from x,y. The points equidistant from the great arrow x,y form a torus-shape, which unfolds to a rectangle. Lines like x,w and g,y form the crossing diagonals of this rectangle.

The points x,-y and -x, y are the circle completely orthogonal to x,y. One sees that changing the sign keeps the great circle but reverses the direction.

The point -x, -y is the same as x,y. but completely reversed.

The general phase space for modes of rotation correspond to a biglomohedric pyrimad. This is a petic in 7D. When one selects either of the bases, one gets a "clifford" rotation, right and left modes. Anything in between can be made by the sum of two clifford-rotations.

This is how the great-rotations are effected in quaterion multiplication, with left and right multipliers.

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Postby thigle » Wed Oct 05, 2005 1:37 pm

now this is as close to an answer to my clumsy nebula of questioning (the 'clouds' condensing on the sky of my diffused interest in things of nature multidimensional) that I was ever given. ec.static

i'll happily ponder this for a while and then ask.

meanwhile:

SO4 is group of orthogonal rotations in E4. if i get it right. some simple noninteractive applet is at: http://gregegan.customer.netspace.net.au/APPLETS/Applets.html

so each element of the group SO4 corresponds to a single point of the phase-space of biglomohedric prism ? or pyramid ? it seems to me they overlap much, these 2: biglomohedric pyramid (or prism ?) & SO4.

and two SU(2) (unit spheres in quaternions) correspond to base-spheres of biglomo...prism or pyramid ? now i would say pyramid.

i'll give much thought to this. i cannot see it clearly yet.
Last edited by thigle on Wed Oct 05, 2005 1:51 pm, edited 1 time in total.
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Postby thigle » Wed Oct 05, 2005 1:37 pm

deleted by thigle. this was double post. (some moderators, please, can you erase this all?)
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Postby wendy » Wed Oct 05, 2005 11:04 pm

Answers.

1: although it is generally assumed that rotation in 3d happens around an axis, leaving it invariant, there also exists what is called 'spherical rotation' in 3d : rotating around an axis A, while simultaneously rotating this rotation axis A around another axis B, gives spinning motion - so called spherical rotation. then ( speed of rotation on A / speed of rotation on B ) & (angle between A and B) determines the kind of loop (if any) one gets. does this leave a point of intersection of A and B invariant? or not even that (because it would get somehow 'twisted' i imagine) ?


This is procession. It requires an external force to make the axis wobble.


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.

The phase-space i give for the 4d glome is purely for inertial rotations: there is no outside force, no processions, tumblings or whatever. The phase space for the higher dimensions go as the triangular numbers, ie (n)(n-1)/2, to which you add additional dimensions for the phased space.

The simple space of great arrows in five dimensions is a manifold in 10 dimensions. I don't know how many dimensions the manifold is, but it is spread out over 10D. The addition of double rotations may add an extra two or three dimesniosn to the whole kit.

The phase-space for a simple force-derived object (ie the equal of procession), is still too complex for me to grasp.
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Postby thigle » Thu Oct 06, 2005 10:51 am

with this distinction of 'inertial' / 'processional' rotations:

...procession. ...requires an external force to make the axis wobble."


what is this external force for spin of 'particles' ? if a particle has a 1/2 spin, it means that it rotates 360deg 'inertially' while 720deg rotation takes place 'processionally', about axis cutting the first one at right angle ?

this external force for each particle results from what ? some self-reflexivity or 'microfolding' of subquantum-vacuum creating tension & different densities, resulting in overall 'swarming' motion at the micro-scale ?

just speculating in this case.

also is the rot(360while720) same as rot(720while360) ?
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Postby wendy » Thu Oct 06, 2005 11:06 pm

i am not even sure if electrons actually "spin" per se. They're not little spheres that spin like the earth. much of what electrons are is pure speculation.
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Postby wendy » Thu Oct 06, 2005 11:27 pm

regards procession in four dimensions. [thinking aloud]

i have no real means or desire to do the calculations as they stand, and so instead i watch hammers &c spin in four dimensions, and follow the trace on a glomic coordinate system. But even this is complex for cases other than the clifford-rotations.

Consider a planet that rotates at equal speeds on perpendiculars [clifford-rotation]. Every point goes around the centre of the planet, and in theory it is less likely to be oblate (or flattened, like the earth is).

The planet orbits its sun, so the sun appears to go around the sky on its own great circle. The track of the sun is not likely to correspond to the rotation-circles of the earth, and so would be anti-clifford to exactly one great circle (the tropic ring).

Some distance from this (eg 23.5 deg), lies the "tropic torus", eg the tropic of capricorn, cancer, leo, aquarius, &c. The zenith sun moves across the great arrows that make this torus, covering every one once a year. [Somewhere it is summer, always]. The effect of this is that while the planet still rotates relative to the stars in clifford-motion, it rotates relative to the sun in some kind of rotation mode.

For something like procession to occur, we might suppose that the ray through the earth and the sun and a fixed star in aries, makes the tropic of aries at a fixed great arrow, and that over the course of 25920 years, that the tropic of aries makes a complete cycle through all the allowed values.

For this to happen, we might suppose there exists some kind of counter rotation, a procession of axies, that is very slow, and travels in a counter clifford-rotation, that rotates centred on the tropic (direct) and artic (reversed) ring.

However, while i can easily visualise this, i still struggle to see what the effect on the day is for random localities.
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Postby thigle » Fri Oct 07, 2005 1:07 am

as for the actual spin of electrons: i don't believe they are 'spheres', nor that they "spin" per se. i am more inclined to believe that they, as well as all that we consider matter, are undulations in the fabric of space.
however, dynamics of these spherical standing spining waves (of whatever nature might they ultimately be), i do believe are of this 'processional' geometrodynamics. my reasons for this are of purely non-scientific reasons: i've seen them (as well as many other 'building blocks' of physus)

as for your visualisation abilities, i am amazed and inspired.

closely bound with these thread's questions are the following (speaking explicitly of 'spherical rotations') websites:
http://www.quantummatter.com/ on quantumatter
http://www.quantummatter.com/body_spin.html on physical origin of electron spin
http://www.quantummatter.com/documents/Einstein-WebPage.pdf einstein's last question: what is electron ?
http://www.quantummatter.com/point.html Beyond the Point Particle - A Wave Structure for the Electron

it seems this guy is sayin that electrons are spherical standing waves of space.
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Postby thigle » Thu Feb 09, 2006 3:18 pm

isn't it that what is at quickfur's page under name clifford rotations is what I ask for in questions 2-3 in the original post that started this thread ?

still, the questions 1,4-6 stay unanswered :cry:
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