Effects of bi-rotation (direction, gravity)

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Effects of bi-rotation (direction, gravity)

Postby gonegahgah » Thu Apr 16, 2015 2:40 pm

First off I just wanted to ask a question about something that just occurred to me as possibly being the case.
I was just wondering if a 4th dimensional glome were rotating in two axial directions around the other two axi at equal speeds if it would be rotating around all possible axis at equal speed?

The best way I have to picture this is that if you look at a sphere in our 3D space that belongs to a glome that is rotating around the middle sideways and you rotate the 3D space through the 4th dimension that you will see the rotation moving from that sideways rotation to slanting and turning from the horizontal until we see it rotating through the top and bottom.
I'm thinking that it will appear to be rotating from any angle that you look at it with the same speed but in slightly different inclined directions; from this visualisation? Is that so?
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Re: Effects of bi-rotation (direction, gravity)

Postby gonegahgah » Thu Apr 16, 2015 11:37 pm

It occurred to me last night that this might also reflect in what we see when the glome were rotating along both axis.
Tell me if the following is correct please?
1. Along the equator of rotation we would see what we would expect where the left moves around to the right (or the opposite direction).
2. Around the rest of the sphere face part that we see we would see a constantly changing face without visual coherence to what was to the left or right.
3. Every spot on the face of the sphere part we see would change at the same speed; even at the poles without apparent rotation.
Does that sound correct?

I imagine it would be a little like what a 2Der would see if they saw our globe rotating through the circle surface (to us edge) that they see.
The greater the angle our pole is to their space alignment the greater the de-coherence would be that they would see along that edge.
If our equator aligned with their space they would see bottom moving to top or vice-versa.
If our equator were perpendicular to their space they would see just a constantly changing face without the upward or downward movement.
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Re: Effects of bi-rotation (direction, gravity)

Postby gonegahgah » Fri Apr 17, 2015 6:31 am

So effectively what we would see is a combination of what the 2Der would see for our globe rotating with the axis vertical in their world and as well as perpendicular to their world both at once.

What it occurs to me as well is that that spinning may help to create a balance that makes orbits possible.
It is my belief that it is spin at the sub-atomic level that keeps things in balance. At that level orbits just aren't practical.
The interesting thing about spin - which science, to my mind, chooses to ignore - is that it displaces effect.
I think at the subatom level that electrons aren't attracted to the nucleus but to a point 90° perpendicular to their direct line to the nucleus.
I think there is evidence for this in that shooting an electron over a magnetic pole deflects it neither towards the pole or away from the pole but towards a point 90° sideways perpendicular to the pole.

The same thing may occur for objects in a 4D universe.
If you spin them fast enough, around light speed, they may divert enough of the gravity to mean that they are falling at a similar rate to things falling in our universe.
So if you were to spin a glome along its two perpendicular axis at a very fast speed it may orbit a larger glome with a stable orbit possibly?
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Re: Effects of bi-rotation (direction, gravity)

Postby gonegahgah » Sat Apr 25, 2015 2:01 pm

I was pondering what effect this would have on a tesseract.

I found the following animation which depicts the 'Clifford rotation' of a cube at http://eusebeia.dyndns.org/4d/vis/10-rot-1.
It turns out that Clifford rotation is the established term for what I called bi-rotation.
Image

Because it is a cube it is flat on both sides - the front side and back side - in a 4D world. This is why in the animation it goes from looking like a cube to a square and back continuously.
I realised it probably wouldn't be hard to depict a tesseract 3D-slice rotating with Clifford rotation in a simple, when aligned, fashion.

I think it would be similar to the above but you would see the cube remain a cube; without the flattening; which was rotating like above.
The other rotation would appear as the cube spinning end-over-end (top-to-side-to-bottom-to-side-backto-top).

In our 3D world that movement is difficult because the the end-over-end spin changes the orientation of the sideways rotation causing it to tilt resulting in an exotic looking tumble (like astronauts and pilots have to try to avoid).
But, I don't think that is the case in 4D. Instead the apparent slice can look like a cube happily tumbling end-over-end while maintaining a stable steady sideways spin. (That is depicting the simplest bi-rotations and view angles).
Am I correct?
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Re: Effects of bi-rotation (direction, gravity)

Postby ICN5D » Sat Apr 25, 2015 5:00 pm

I believe you are correct, since 4D has two orthogonal 2D planes, which can also be the planes of rotation. Like the way a 3D sphere rotates, the equator sits on a 2D plane, while the poles sit stationary on a 1D axis. A 4D sphere will have two such equatorial 2D planes, and no stationary poles, allowing the ability to double rotate, without the exotic tumble.
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Re: Effects of bi-rotation (direction, gravity)

Postby gonegahgah » Tue Apr 28, 2015 1:49 pm

Thanks ICN5D. According to the same link it mentions that the only point not moving (though it is spinning) is the very middle of the object under Clifford Rotation. So I guess that answers my question about whether a glome would be moving at the same speed on all points of its surface if its bi-rotation were at equal speeds. Cool.
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Re: Effects of bi-rotation (direction, gravity)

Postby quickfur » Wed May 06, 2015 12:08 am

What's even more cool, is that when the two rates of rotation are equal, all points on the glome travel in great circles instead of spirals. When this happens, the number of stationary planes (2D planes that do not change under the rotation) become infinite, instead of just two. Each circle traced out by each point on the glome interlocks with adjacent circles in an interlocking, spiralling way, forming what we call in maths a "fibre bundle". This is, in fact, the well-known Hopf fibration of the glome.

In this state, the rotating glome loses one degree of orientation, and yet it still retains a sense of the rotation! What I mean by "losing one degree of orientation" is this: in 3D, when a sphere (e.g. the earth) rotates, it has a single stationary plane, which also unambiguously defines a unique north-south axis. When you look at any rotating sphere in 3D, you can always identify this axis and the stationary plane perpendicular to it. So you may say that the axis defines a fixed orientation for the rotating sphere. Now, in 4D, we know that an object, say a glome, rotating only around a single plane does not have a unique axis. However, it does have a unique stationary plane, that lies perpendicular to the plane of rotation. So even though we can't assign a single directional axis to the rotating glome, we can still identify its stationary plane and the perpendicular rotational plane, both of which have a fixed orientation in 4D space. Even when you introduce a second rotation around the stationary plane of the first, as long as the two rates of rotation are unequal, you can still identify a unique rotational plane for the first rotation, and a unique rotational plane for the second rotation lying perpendicular (in fact, orthogonal) to the first. So you can still "orient" the double-rotating glome with these two planes. These two planes are distinguished from other planes because points of the glome that lie on them trace out circles, whereas points outside of them trace out spirals.

However, something strange happens when both rates of rotation are equal: while the original two orthogonal planes still remain as stationary planes, now a whole bunch of other pairs of planes also become stationary planes -- an infinite number of them, actually. This makes it impossible to distinguish the original two orthogonal planes from the infinite number of other orthogonal pairs that also behave like stationary planes! So now you can no longer use the two planes to fix an orientation for the rotating glome. So it has "lost one degree of orientation". All of these stationary planes are completely congruent to each other, so none of them stands out as being a special landmark that you could use to orient the glome. So in a sense you could say this rotating glome is "non-orientable" (not in the mathematical sense, though).

The bizarre thing, though, is that even this "non-orientable" rotating glome still has a distinct "sense" of rotation. In 2D, there's a distinction between clockwise and anticlockwise, and in 3D, while technically this distinction is absent, in its place we have the direction of the rotational axis which we can use to discriminate between different orientations of rotation. In 4D, however, the double-rotating glome does not have this distinction, because the infinite number of stationary planes means that no matter how you roll the glome around, it makes no difference at all to its rotation! However, at the same time, this crazy orientation-less double rotation actually comes in two breeds, which are distinct from each other! Their distinction is not in clockwise/anticlockwise, nor in orientation ('cos all orientations are equivalent), but in the way the great circles spiral around each other. One breed has the great circles spiral around like a left-handed helix, while the other has the circles spiral around like a right-handed helix. And no matter what you do, it's impossible to rotate one breed into the other -- they are distinct double rotations, yet they have no orientation either!

Furthermore, this strange effect only exists in 4D: in 5D and beyond, you can always rotate the glome through the additional dimension(s) such that a left-handed double rotation becomes a right-handed double rotation. It's only in 4D where this distinction exists.

Weird, huh?
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Re: Effects of bi-rotation (direction, gravity)

Postby wendy » Wed May 06, 2015 7:14 am

What quickfur says is true, but it's even stranger than that.

Suppose you were on a clifford-world. If you imagine you were standing in a spherocylinder facing east-west, then the stars would cross the mid-line at a particular point on the sphere. The star that reaches zenith would of cause be at the top.

But the same spherinder would serve at every location, rotated at a different orientation.

For a given clifford rotation, there is an opposite clifford rotation that shares exactly one great arrow (directed great circle) with the rotation. It forms a grid.
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Re: Effects of bi-rotation (direction, gravity)

Postby granpa » Thu May 07, 2015 1:15 am

yes I can see that you will have an infinite number of planes of rotation but it seems to me that only two of those planes will be at right angles to each other
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Re: Effects of bi-rotation (direction, gravity)

Postby quickfur » Thu May 07, 2015 1:45 am

There are an infinite number of pairs of rotational planes that are perpendicular (indeed, orthogonal) to each other. It's not very easy to visualize, but it's true!
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Re: Effects of bi-rotation (direction, gravity)

Postby granpa » Thu May 07, 2015 2:19 am

I'm not convinced
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Re: Effects of bi-rotation (direction, gravity)

Postby quickfur » Thu May 07, 2015 2:28 am

There's nothing to be convinced about. It's evident in the math.
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Re: Effects of bi-rotation (direction, gravity)

Postby Secret » Fri May 08, 2015 3:14 pm

Not sure if I understood the isoclinic rotation properly
Can gonegahgah, quickfur or ICN5D made an animation of this?

backup 1.PNG
(271.48 KiB) Not downloaded yet


Key: blue and dark green lines are supposed to connect the same position on each plane of a set of rotational planes in the isoclinic rotation so as to give some idea what shape the whole infinite set of rotating planes look like

The eyes corresponds to the 4 different projections to 3D in 4 space. The black duocylinder basically helps me to draw the isoclines by considering each of the toroid shaped partitions of the 3-sphere
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Re: Effects of bi-rotation (direction, gravity)

Postby quickfur » Fri May 08, 2015 4:39 pm

Don't have time to do animations right now, but one way to visualize the multi-stationary plane symmetry of isoclinic rotations is to look at spidrox in this post. Ignoring the individual cells in this polytope, each of the tube-like stacks of cells represents a torus that wraps around the polytope. The center of each torus runs along a great circle wrapping around the corresponding glome that the polytope approximates. There are 12 toruses made of prisms and antiprisms, in 6 pairs of mutually-orthogonal planes, each pair wrapped around the other in a spiralling fashion. There are also 20 toruses made of square pyramids; these also correspond to great circles wrapped around each other in a spiralling fashion, and they also come in 10 sets of orthogonal pairs. Each orthogonal pair corresponds with a pair of stationary planes in the isoclinic rotation that preserves the symmetry of the whole polytope. So we have 32 pairs of planes here, wrapped around each other in a spiralling fasion, that are stationary under this isoclinic rotation. From here, it should not be hard to generalize to the case of the glome itself, where you interpolate between these plane pairs to get an infinite number of pairs of stationary planes, all of which follow a spiralling layout wrapping around each other across the glome.

It's important to note that not every possible plane is a stationary plane; only those planes that are part of the spiralling set is stationary. So even though there are an infinite number of stationary planes, there are also an infinite number of non-stationary planes.
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Re: Effects of bi-rotation (direction, gravity)

Postby granpa » Fri May 08, 2015 9:34 pm

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Re: Effects of bi-rotation (direction, gravity)

Postby gonegahgah » Sun May 10, 2015 2:15 pm

I guess the best way for me to try to depict it in my way would be to try to actually formalise the rotational projection formulas.
I could also do it slowly and not perfectly using Sketchup - which might be nice for a mock up - but perhaps it is better to go for the former.

In my original depictions I've relied upon the objects standing on 'flat' ground so that the upright faces are perfectly vertical.
It is interesting to note the ways we can turn a 4D object and that it will remain upright...

A 2Der would be surprised by the notion that we can turn a square 'sideway's somehow (around a centre axis) and it will remain upright.
Their only conception of turning a square requires that it not remain upright and turn end-over-end.

The same goes for turning a cube in 4D space. It can be turned any 360° of sideways without affecting the vertical orientation of the cube.
This is unlike us where we can only turn it left or right around an axis.
To turn it anything other than the 180° antipodal directions, flat to the surface, means that we must escape the surface and topple the cube through the vertical like tossing a dice.

So it is possible to rotate the base of a tesseract towards any of its 360° of sideways directions without toppling that tesseract and causing it to escape being flat to the surface.
It will happily remain upright turning it this way. Pretty cool!

I suppose for any projection we need to consider where the camera eye is in relation to the object. This consideration involves both height and distance.
We then need to consider what angle the defining lines of the tesseract make off into the 4th dimension at any time as well as those that are presently in our space at any time.
Rather than displacing the projection - which is what gives us the https://en.wikipedia.org/wiki/Rotations_in_4-dimensional_Euclidean_space#Double_rotations undulations - we can hopefully consider rotating the 4th dimension up so that it is just like our space but displaced fractionally by its quantity of rotation.
This will hopefully give us a clearer concept of this whereas that animation only serves to confuse.

It might take me awhile; if I'm able to do it.
One of the things I hope to eventually have are resultant levels of transparency with less transparent depicting the more bulky parts of 4D objects.
Also the different 4-faces will hopefully come through with some distinctness as shades and colours superimposed. I hope so.
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