## More Tesseracts Than One?

Ideas about how a world with more than three spatial dimensions would work - what laws of physics would be needed, how things would be built, how people would do things and so on.

### More Tesseracts Than One?

I was puzzling over something today and perhaps someone can help to agree with, or more probably, dispel the notion...
I was looking at a cube directly at one of its square faces by itself with the top-left edge going back away from me.
I imagined that, if the cube were a 3D slice of a tesseract, that the retreating Z edge line was actually a square; even though I could only see a line of it.

I imagined that retreating square rotating around between the z and w axes and realised it would get longer in our view as it rotated to being diagonally in line with us at 45°.
It would return to its original length at 90° with this cycle repeating on and on...
I'm guessing that this would also change the appearance of the front face which would have to elongate and shrink as well.
I'm not sure exactly what else would happen.

So as a side question: Would the front face just keep rotating around itself also or would the opposite face come emerging through?
I should contemplate it but this other question is puzzling me more first...

I realise that the W-axis is going off in an invisible direction (just as the X-axis goes off in an invisible direction for the 2Der).
The W-axis is perpendicular to the front face just as the Z-axis is perpendicular to the front face.
There are actually a full 360° of perpendicular directions to the front face and not just the W and Z axes.

So this lead me to wonder if I could make the tesseract's Z line actually offset some into the W direction while making the tesseract's W line offset an equal amount towards the Z line.
My question is: In 4D are there actually a whole 90° of varieties of tesseracts of same size that don't actually occupy the same identical space as each other?

If that is not correct can you help me understand why please?

The closest 2D-3D analogy I can give is that the x-line can be extruded either left or right.
In both instances the result will be cube's that occupy the exact same space but they'll just have opposite side colour's.
However in 3D-4D I'm wondering if two 90° axes can have a variety of perpendicular orientation's to two other 90° axes for a tesseract build?
And if that is so then does this support the idea of non-identical equally dimensioned tesseracts in 4D?
gonegahgah
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### Re: More Tesseracts Than One?

Thinking to the analogy of the three eyed 4D being who is looking forward while standing.
We see only one eye at a time, if we currently see an eye in their current orientation.
They can rotate in place while continuing to face forward and remaining standing.
While they do this we see one eye come into view, followed and replaced by the second eye, then the third eye, and then repeating.

So their aspect goes through all the forward positions of their face.
This answers the first question which is that the back of their head won't emerge through to the front (thank goodness).
Nor for the tesseract. Simply spinning it around in the W-Z won't bring the back to the front.

What we see are the different front aspects of one cube which basically reveals the volume (as we think it) of the cube one rotated slice per rotation angle.
This grows and ebbs in width sideways (as described in the first post) as we rotate through the width-diagonal seen faces of the face cube's 3D slices.
gonegahgah
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### Re: More Tesseracts Than One?

Within 3D you can keep one direction and spinning the other two by means of an according rotation.
So x-direction, say, would be kept, while x-direction rotates into y, and y-direction rotates into -x.

Similar in 4D: a simple rotation can take place wrt. z- and w-edges of a tesseract, while its x- and y-edges remain unchanged.
Esp. the full square face, spanned by the x- and y-edges, remains unchanged.

In fact, in 4D we even have somthing which is called a Clifford rotation or double rotation, which is nothing but a simple rotation within z and w together with a completely independent further rotation within x and y. You can consider this as an overlay of two simple rotations, the first rotates z and w, thereby keeping x and y unchanged, the other is rotating x and y, thereby keeping the (then new) directions of z and w.

--- rk
Klitzing
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### Re: More Tesseracts Than One?

I can see what you mean Klitzing.

If the X corner rotates around to the Y corner the Z edge remains where it is and just rotates around itself.
There is no reason why the W edge would not do the same.

That also means that the Z corner can rotate around to W corner and leave both the X and Y edges in place which just rotate around themselves.
Conversely that means that the full face rotates around/within itself.
Just like when a line rotates all the dots along its length rotate around themselves.

So there is only one tesseract. Thanks for clarifying that Klitzing.

The leads me to the next thing I seek which is an absolute way to represent rotation of the tesseract.
For a cube for example, if you rotate: down, right, up, left :even though these sum to no rotation you don't end up with the same facing cube.
I've seen that they use rotation matrix math but won't the separate stored figures go out very slowly over time?

If you just have a square in 2D space it is easy:
- You only need to know the rotation of the yellow face upwards.

It gets more complicated in 3D space.
- You can rotate the yellow face upwards and record the yellow up rotation.
- But you can also rotate the yellow face sideways.
- As mentioned 0 total rotations can end up with a differently facing or orientated cube...
So is there a standard derivable mathematical form of representing this?
- I guess you can make one edge (say the Z edge) and keep a measure of how far it is rotated towards where Y was.
- You can then also keep a measure of how far it (same Z edge) is rotated towards where the X was.
- Additionally you need to keep a measure of how much it has been rotated in the process.

So for 2D space you need only to track one value, in 3D space you need to track three values...
So what do you need to track the rotations of an object in 4D space?
gonegahgah
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### Re: More Tesseracts Than One?

When you rotate an object - say point (x,y) - within 2D about an angle phi around the origin, and let s = sin(phi), c = cos(phi), then this point becomes (cx+sy, -sx+cy). Thus this rotation could be described in matrix notation as
Code: Select all
`/        \   /        \   /   \|  cx+sy |   |  c   s |   | x ||        | = |        | * |   || -sx+cy |   | -s   c |   | y |\        /   \        /   \   /`

If you apply rotation within 3D you just add an identity transformation in the remaining coordinate. E.g. when fixing the z axis you'd get
Code: Select all
`/        \   /            \   /   \|  cx+sy |   |  c   s   0 |   | x ||        |   |            |   |   || -sx+cy | = | -s   c   0 | * | y ||        |   |            |   |   ||    z   |   |  0   0   1 |   | z |\        /   \            /   \   /`

Now assume you first apply a rotation around the z axis (i.e. keeping that one fixed, rotation just applying within the x-y-plane) about an angle phi and thereafter a second rotation around the y-axis with an amount phi', and let s' = sin(phi'), c' = cos(phi'), then you'd have
Code: Select all
`/                \   /             \   /        \   /             \   /            \   /   \|  c'(cx+sy)+s'z |   |  c'  0   s' |   |  cx+sy |   |  c'  0   s' |   |  c   s   0 |   | x ||                |   |             |   |        |   |             |   |            |   |   ||    -sx+cy      | = |  0   1   0  | * | -sx+cy | = |  0   1   0  | * | -s   c   0 | * | y ||                |   |             |   |        |   |             |   |            |   |   || -s'(cx+sy)+c'z |   | -s'  0   c' |   |    z   |   | -s'  0   c' |   |  0   0   1 |   | z |\                /   \             /   \        /   \             /   \            /   \   /`

Same idea applies to any further rotation (just use the appropriate coordinates) resp. to any further dimension. E.g. a single rotation within 4D within x-y-plane (keeping z and w fixed) would read
Code: Select all
`/                \|  c   s   0   0 ||                || -s   c   0   0 ||                ||  0   0   1   0 ||                ||  0   0   0   1 |\                /`

and an according double rotation (Clifford rotation) about the angle phi within x-y-plane and about phi' within z-w-plane then just reads
Code: Select all
`/                 \   /                \   /                \|  c   s   0   0  |   |  c   s   0   0 |   | 1   0   0   0  ||                 |   |                |   |                || -s   c   0   0  |   | -s   c   0   0 |   | 0   1   0   0  ||                 | = |                | * |                ||  0   0   c'  s' |   |  0   0   1   0 |   | 0   0   c'  s' ||                 |   |                |   |                ||  0   0  -s'  c' |   |  0   0   0   1 |   | 0   0  -s'  c' |\                 /   \                /   \                /`

For ratations within a slanted 2-plane you would have to solve first for the according rotations which turn that plane into some coordinate plane. Then you will have to transform your original problem into a coordinate plane rotation by means of this turn, then apply the appropriate coordinate plane rotation, and finally the according turn back into the slanted position. Here the last rotation clearly is the inverse to the first. Kind of S^(-1)*R*S.

--- rk
Klitzing
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### Re: More Tesseracts Than One?

Hi Klitzing, I think I've worked out how to orient a cube using three angles.
I'm now trying to work out how to further the orientation of a line...
By that I mean...

Take a line that is at an angle described by 20° from z to y, and 30° from z to x.
How would I add 60° further to the rotation away from the z line in the current direction of rotation?
gonegahgah
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