## 5-D rotation phase-space

Discussion of shapes with curves and holes in various dimensions.

### 5-D rotation phase-space

I know ICN5D is into round things with holes, so i've been trying to figure out the 5D rotation space.

A phase space is a space where every point represents a different state. For rotations, we have a space where every point represents a different rotation mode and rotation speed.

For 2D, you have a number line. It spins one way when positive and the other when negative. Any point gives a speed.

For 3D, you have 3d space itself. This means that you pick a point, and the ray through it defines the north pole, and the distance is the speed as before.

For 4D, you have a 6D thing, this is a bi-glomo-hedric prism (or cartesian product of two 3-sphere-surfaces). In essence, three directions x1 x2 x3 define a left clifford rotation, and the axies y1 y2 y3 define a right clifford rotation. The radius is the speed. If you have equal left and right cliffords they give a single rotation in 2 axies and still in the other two. This is the bi-glomo-hedric prism thing. This space divides the 5-sphere (glomo-terix) into two halves, so it makes sense in 4D, that you can have 'left rotations' and 'right rotations' that have to pass through a 'wheel rotation'.

For 5D, it's a 10D thing. I don't have a name for it, but it's something like a bi-glomohedric-prism glomoterix comb. It seems to be a rototope, of the kind

( (iiiii) ((iii)(iii)))

In essence, we replace each diameter of the bi-glomohedric prism with a 5-sphere, which means that we add four dimensions to each of the six extant dimensions of the 4d rototope.

Just have to figure out how the odds and ends tie up.
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wendy
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### Re: 5-D rotation phase-space

Hey Wendy! I'm still trying to wrap my mind around exactly what a phase space is. How does it compare to a euclidean plane? Is there anything I can borrow from the toratopes, in the way of intuition, or patterns, that will translate over nicely? Does it entirely have to do with rotations?
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ICN5D
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### Re: 5-D rotation phase-space

ICN5D wrote:Hey Wendy! I'm still trying to wrap my mind around exactly what a phase space is. How does it compare to a euclidean plane? Is there anything I can borrow from the toratopes, in the way of intuition, or patterns, that will translate over nicely? Does it entirely have to do with rotations?

A phase space is quite general concept. It can have any number of dimensions, but the dimensions are in the mathematical sense, that is, they can represent just about anything. It is used to describe all the possible states of a system. "Phase" is just a name. Maybe it came from the phase diagrams of materials, which describe whether the material is solid, liquid, or gas at various combinations of pressure and temperature.
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### Re: 5-D rotation phase-space

PatrickPowers wrote:"Phase" is just a name. Maybe it came from the phase diagrams of materials, which describe whether the material is solid, liquid, or gas at various combinations of pressure and temperature.

No. "Phase" in this context derives from crystallographers usages.

In fact, the arguments of trigonometric functions (esp. wrt. sine and cosine) originally are called their phases. The same holds true when shifted into complex embedding, i.e. wrt. the complex exponential function (cf. the Euler theorem: exp(i phi) = cos(phi) + i sin(phi)).

X-ray diffraction pictures, say, of real crystals then are described by the Fourier transform of the scattering object (e.g. atomic positions). There the sine functions or complex exponential functions come in. The arguments of these functions will be scalar products of real space coordinates with coordinates of some space, which thus ought have inverse length units. This "reciprocal space" then is also called the "phase space". Usually bearing units in such a way, that the (reciprocal) lattice geometry becomes described by integral positions. And those in turn then are well-known as the Miller indices.

--- rk
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### Re: 5-D rotation phase-space

I'm still trying to wrap my head around this. Still though, those neurons aren't sparking anything I can see yet. Where do inverse length units come into play? I understand how the exponential function relates to complex numbers, the roots of unity, rotations, and the polar coordinate system. How do any of these relate in a way I might be more familiar with?
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### Re: 5-D rotation phase-space

I should not worry too much about it Philip. I thought ye were into round things with holes

I have difficulty understanding what it means, although the shape was shown to me by someone much brighter than I am.
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### Re: 5-D rotation phase-space

Just consider exp(i(k*r)), and remember that the argument of which ought be a mere imaginary number in order to match the Euler equation cited.
But when * is the scalar product and r is some vector of real euclidean space, i.e. having usual length units, then k ought be a vector of reciprocal space, i.e. having reciprocal length units, just in order to cancel them out after multiplication: (k_1 / m, k_2 / m, k_3 /m) * (r_1 m, r_2 m, r_3 m).

--- rk
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### Re: 5-D rotation phase-space

Ha, oh, wendy . So, is it some sort of specific coordinate system? The way you and richard are describing it sounds like it is. Whatever it is, it's more abstract than my geometrical mind can follow, at the moment. If you can define it with a rotatope, which types match up for the lower dimensional phase spaces? What's the sequence that lead to (((III)(III))(IIIII)) for 5D ?
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### Re: 5-D rotation phase-space

Philip, the coordinates r are considered the "real", the physical space coordinates. The coordinates k are coordinates of some abstract, non-physical space. That's why they are called coordinates of phase space. Here they can be used to index the Bragg peaks of the diffraction pattern of crystals, a.k.a Miller indices.

Most probably you already have read in some math books, that some theorems are stated quite generally for some vector spaces. This then is not only some tick in order to sound more special and eloquent. The theorem then will nowhere refer to which sort of vector space it would apply. You might think of the usual vector space of euclidean coordinates. But you well could apply the results to any other, way more abstrct vector spaces too. So, for example, ex, ey, ez are the base vectors of our well-known vector space R3. And any point of this space then can be represented by prefixing some scalar values to these base vectors. This is what coordinates are meant for.

But likely too you might have heard of Taylor series too. E.g. the Taylor series of exp(x) = (1/0!) x0 + (1/1!) x1 + (1/2!) x2 + (1/3!) x3 + etc., where n! = 1 . 2 . 3 ... n and 0! by convention is set to be 1. Again you have the same principle: You have some scalar coordinates ( (1/0!), (1/1!), (1/2!), ...) which prefix basic polynomic functions. And indeed those basic polynomials xn, n some integer, serve as a base of the function space, which then can be considered a vector space.

With respect to the subset of periodic functions one could provide a different base as well. E.g. the set of functions of type sin(nx) or cos(mx) - for integral values of n or m - forms the base for 2pi-periodic functions. In fact the Fourier analysis is nothing but the quest to derive the coefficients to that base when given some specific periodic function. Again we have a vector space.

In the last 2 examples the phase space is a functional space, a space of functions. And just as we are used to sequence x - y - z as a right handed coordinate system (say), the infinite dimensional space of functions is sequenced by the exponents n, resp. by the phase space coordinates n and m. Here phase is 2 dimensional and discrete. But when allowing not only for 2pi-periodic functions but rather for any periodic ones, it becomes clear that the arguments of the trigonometric functions have to be scaled accordingly. And, esp. when you finally aim to describe any type function by that such a trigonometric functional base, then your Intervall of "periodicness" just becomes infinite. Conversely, by corresponding scaling, the discreteness of these n and m becomes infinitely small, i.e. those now become continuous variables of phase space.

--- rk
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### Re: 5-D rotation phase-space

Ah, okay. That makes a bit more sense. Thanks for the clarification. I'll have to mull this over a bit, and perhaps begin to check out some of these math texts for once.
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### Re: 5-D rotation phase-space

The phase space represents every state of something by a single point.

If you have a 2d circle, rotation can be controlled by a slider. You slide it left it goes clockwise, you slide it right, anticlockwise.

In 3d, you have to imagine the thing is like a rotating top. You can slide the north-pole rod in or out, and this makes it go faster or slower. You can rotate the thing around, and it will rotate in a different direction. You can reverse something by pulling the control in a circle, so it is always rotating at the same speed. It gives (III)

In 4d, you have two 3d controls. One handles the left-clifford rotation and the other the right-clifford rotation. When the speeds are equal radius, the rotation is in 2d only. True clifford rotation has something set to a speed of zero. The space of great circles is (III)(III) , a bi-spheric prism. [or rather the edge thereof]

In 5d, the diameters of the 4d space are replaced by a 5-sphere. So I guess something like (IIII[(III)(III)]) is the proper notation. I really have to think hard on this one.
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### Re: 5-D rotation phase-space

Ah Good luck with the maths texts. I could not find any.

Why do you think I make all my own maths. The sources are so obscure and oblique. Part of the early stuff I used was Gauss's "Discussion Arithmetic" in German (when at the time i could not read it), so it amounted to looking at the pictures. It's a bit better now.

Klitzing's thesis was a little easier, but I must fess that I'd point out odd words to my brother for a hint. The numbers were easy to translate. rk use the values from the cos, I use the full chord, which is a doubling. The chords are algebraic integers, the cosines are not. The L. M. S. thing in the second half threw me a bit, too. (The acronym is only too well known in a family following railways.)

John Conway and Derek Smith (Quarterions and Octonions) give some hints about 7d and 8d, but it's pretty heavy stuff if you're relying on the piccies.
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### Re: 5-D rotation phase-space

I came to the conclusion that in 5D rotation, the five transverse dimensions have no effect on the five orthogonal ones. I think, anyway.

The other 5D mistery might be easier to tackle in this heat (it gets to 37C or 104F in the day). It's quite a fancy notion, though.
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