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Author Topic: Fractal foundation of Fluid Mechanics  (Read 40054 times)
Description: Discussion.insights, notions and paradigm shifts
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youhn
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« Reply #45 on: February 09, 2014, 05:13:04 PM »

A solid has 6 degrees of freedom but a limitless set of axes of rotation.

I think this is incorrect. You only need 3 axis of rotation in a 3D space to archieve all possible rotations of a solid, in the same manner as translation based on 3 axis is enough for translation in every possible direction.

Sources
<a href="https://www.youtube.com/v/vOFM8eG8kVc&rel=1&fs=1&hd=1" target="_blank">https://www.youtube.com/v/vOFM8eG8kVc&rel=1&fs=1&hd=1</a>
https://en.wikipedia.org/wiki/Degrees_of_freedom_%28mechanics%29
https://en.wikipedia.org/wiki/Euler_angles
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kram1032
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« Reply #46 on: February 09, 2014, 05:27:31 PM »

Really, rotations are not along axes but rather across planes. I mean obviously you can represent a rotation of a plane with a rotation around the normal of the plane, but if you go beyond 3D, thinking of rotations with planes is more intuitive than thinking of it along axes.

As such, in 3D, you have 3 planes that are normal to each other and can be freely rotated. - however, you can choose the precise orientation of your planes arbitrarily, as long as they all are normal to each other.

So it's a matter of view. Or rather a matter of what you are really saying.
If you are talking about how you represent rotations, you only ever need 3 planes.
If you are talking about what you can choose, you have infinitely many options with only one requirement.
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youhn
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« Reply #47 on: February 09, 2014, 05:33:53 PM »

Aha, but if we were talking about the options that you can chose, the solid has infinite freedom both in translation and rotation. My point was that there is no fundamental difference between translation and rotation if we are talking about the degrees of freedom.
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jehovajah
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« Reply #48 on: February 09, 2014, 06:15:21 PM »

Fabulous image of a weir, Hermann. Illustrates how contiguous regions of different densities just morph visually into one another!
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kram1032
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« Reply #49 on: February 09, 2014, 06:28:41 PM »

Rotation and Translation are, in fact, so similar that you can introduce a point at infinity in order to make translations be treated as rotations around that point, in which case you end up with six degrees of rotation, three of them in the usual sense, and three around the point at infinity, all six such that that point does not change.
That's what happens in projective and also conformal models of space.
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jehovajah
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« Reply #50 on: February 09, 2014, 06:59:55 PM »

Thanks youhn, and Kram 1032.

I happen to agree with both of you!

Yes degrees of freedom is too loose a term to describe translations alone , and yes axes of rotation is an idea from my kindergarten days!  However , using only 3 axes of rotation as a basis set can describe any rotation providing you remember to use half angles!

I find the bivector notion of rotation interesting. It is natural to observe complex rotation without being ble to identify an axis or an equivalent axis of rotation. One of the tools we use to specify rotation requires an axes but another tool does not. Often I will revert back unwittingly to sn old way of thinking without realising.

The rigidity of a solid is locally well captured by the 6 degrees of freedom in a 3 orthogonal axis system. This can be reduced to a quadrant of the 6 degrees system, if preferred.

Rotation using this axial system is more complex to describe, whereas using the bivectors of a parallelapioed that contains the object nicely describes local rotations.

The more complex rotations require nested sequences. It is only since my research into The Barycentric calculus that I have realised a more flexible reference frame has always existed. This bears directly onto the Grassmann method of analysis and through that onto the Clifford and Geometric Algebras.

Yes I have still got lts to clear out and relearn, but it is great fun doing it.

I tend to use your last point a lot Kram1032. But I have had to take instruction from Nran on projective geometry to make sure I am not making any naive assumptions.

For many motions I use a local reference frame, but for a general discussion I use a projective reference frame with homogenous coordinates.

I can recommend Normans universal hyperbolic Geometry course for clarity on these points at infinity.
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May a trochoid of ¥h¶h iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
kram1032
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« Reply #51 on: February 09, 2014, 07:42:53 PM »

You do not need any kind of nested sequence for "more complex rotations" - all rotations lie in the same space.
You do need up to three rotations to describe arbitrary rotations in coordinate system.
However, this is only a constraint if you actually fix coordinates.

If you take a coordinate-free approach to rotation (which Geometric Algebra and especially Conformal Geometric Algebra lets you do without a problem), all rotations you can possibly imagine require only a single step.
Quaternion rotation also does this, and in fact, what you do in Geometric Algebra to rotate planes is the exact same thing as what you do with quaternions. It's just that the framework of Geometric Algebra is so much clearer and, in a sense, more polished than what quaternions do, that it seems like an a lot easier operation.

Quaternions seem to come out of the blue and they are messy and confusing, them being 4D and all that.
Inside Geometric Algebra, this makes a lot more sense. You have (for a 3D space) one scalar dimension, three locational dimensions (corresponding to translation and vectors), three rotational dimensions (corresponding to three planes, the bi-vectors and rotations) and one "volume dimension" which is, in a sense, degenerate and acts almost like a scalar. (It is, in fact, a pseudo-scalar.) That last dimension, besides closing the algebra so it all becomes workable, really only is there to change operations into their dual forms. (For instance, if you multiply a bivector that describes a rotation by this trivector/pseudo-scalar, you'll end up with an axial vector that describes the exact same rotation, and vice versa).

Quaternions turn out to be precisely the scalar plus the three rotational dimensions. That's why quaternions are four-dimensional. And that's also why all the action of quaternions typically lies in their three imaginary parts. - Those are precisely the three bivectors.
The reason why they are confusing really only is because they are, in a sense, incomplete. They only focus on rotation and completely leave out translation. Upgrading them to dual quaternions - which is possible and gives that missing power to the algebra - still is confusing, but if you approach this problem from a perspective of Geometric Algebra, it suddenly becomes a lot more obvious.
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youhn
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« Reply #52 on: February 09, 2014, 09:43:50 PM »

Yes degrees of freedom is too loose a term to describe translations alone , and yes axes of rotation is an idea from my kindergarten days!  However , using only 3 axes of rotation as a basis set can describe any rotation providing you remember to use half angles!

I find the bivector notion of rotation interesting. It is natural to observe complex rotation without being ble to identify an axis or an equivalent axis of rotation. One of the tools we use to specify rotation requires an axes but another tool does not. Often I will revert back unwittingly to sn old way of thinking without realising.

Well, excuse me for still being in kindergarten :-)

Since I'm an engineer, my perspective is mostly the physical 3D world (opposed to the much wider and more general mathematical world). Fractals are to me a subject that attracts because of it's (visual) beauty. Add a little curiosity and I sometimes dive into the math, but mostly not very deep. So I've heard about rotation around planes from the documentation of gnofract4D.

But in practice ... what would higher dimensional rotation mean for (an engineer like) me?
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kram1032
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« Reply #53 on: February 09, 2014, 11:39:36 PM »

Well, the way to turn rotations and translations into one and the same concept is to embed your 3D space into a higher dimensional space that adds one extra dimension.

The reasons for this are that certain very practical, not at all unusual transforms become much easier that way. They obtain a much more natural description that is easier to work with.
In particular, you'll get screw-transforms.

Another reason is to stop the center point from being a "special point". In nature, in the real world, there is no inherent difference between any two given points in space. All difference simply comes from what particles are in its direct neighborhood.
In a naive mathematical model of space, you get some kind of singularity in the origin of your coordinate system. This singularity is entirely artificial and you can get rid of it in a particular point in space, simply by shifting your coordinate system around. However, naively you can't get rid of it altogether.
To do that, you need to add another dimension. If you do that, you get homogeneous coordinates.

If you do both those things, you end up with a conformal model of space which is as well-behaved as it could be.
It has lots of advantages:

  • The Origin is no longer a special point of your space
  • You can do all screw transforms in a very direct, simple way, since translation and rotation ends up being the same thing
  • Additionally, points (not vectors), spheres and planes become one and the same object for, respectively, 0, finite and infinite radius, which is very natural and easy to deal with.
  • Conversely, circles and lines also become one and the same, and a circle of radius 0 becomes something like an infinitesimal vector - a point with direction, which is a great notion for a normal vector.
  • Things that actually should be different quantities, like points and vectors, are usually mingled into a single concept which can cause a lot of confusion and head-scratching. In a conformal model of space you get notions that truly separate the two.
  • And on top of all that, in a conformal model you actually get rid of the need of a coordinate system. - All your calculations can be done algebraically without ever referring to coordinates, which is a really natural notion.
    Coordinates are just a mess to deal with, they add a lot of complexity and they are entirely a mathematical construct with no physical relevance what so ever.
    In fact, many misconceptions behind the theories of relativity, and even behind some Newtonian physics stem from failing to see the difference between coordinates and the actual space. So if you can do all your transformations on a space, without ever referring to coordinates, you avoid all these problems. The structure of problems just becomes a lot easier that way.

Other than that, a 4D Minovski space obviously is good for special and general relativistic concepts and you can also go for a conformal description of that space which gives you a 6D working space of which only 4 Dimensions have direct physical relevance.

It might be surprising, but adding in those two extra dimensions actually makes things easier, combining notions that shouldn't be separate at all, separating notions that shouldn't be confused and making transformations easier that are quite relevant in the physical world.

If you, however, are talking about higher dimensional spaces in which each of the dimensions is supposed to have physical relevance, then there might not be much to that in practice.

Unless, that is, you count statistical data as physically relevant: Such data can have a virtually infinite number of dimensions. And often it's necessary to transform data sets to be "shaped better", such that algorithms which, for instance, search for optima or structures in your data behave better.

Beyond that, even for naive 3D Space, without a homogeneous, projective or conformal extension, thinking of rotations to act in planes has certain advantages.

Note, however, that the conformal model is actually closer to the physical reality than the naive form. - At the very least you need a homogeneous model.
The projective part that deals with a point at infinity you might argue against by saying that in reality there is no such thing as a point that is infinitely distant. (I personally wouldn't argue that) However, even then, that addition is one of convenience. It just makes things so much simpler to treat translation and rotation; circles and lines; planes and spheres as one.
And by combining the two you end up with a system you can manipulate without ever referring to coordinates, which definitely is very natural and much closer to reality.
« Last Edit: February 10, 2014, 01:53:43 PM by kram1032 » Logged
jehovajah
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« Reply #54 on: February 10, 2014, 01:33:39 AM »

Quote
If you take a coordinate-free approach to rotation (which Geometric Algebra and especially Conformal Geometric Algebra lets you do without a problem), all rotations you can possibly imagine require only a single step.

Reading this and your reply to Youhn certainly whets the appetite. But I can't say I understand how you would model an n body rotational problem off what you have just written. By this I mean a solar system rotational description with planets having attendant moons.

For this thread I am particularly interested in vortex shedding and compaction.
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May a trochoid of ¥h¶h iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
hermann
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« Reply #55 on: February 10, 2014, 09:35:52 AM »

Quaternions seem to come out of the blue and they are messy and confusing, them being 4D and all that.
Inside Geometric Algebra, this makes a lot more sense. You have (for a 3D space) one scalar dimension, three locational dimensions (corresponding to translation and vectors), three rotational dimensions (corresponding to three planes, the bi-vectors and rotations) and one "volume dimension" which is, in a sense, degenerate and acts almost like a scalar. (It is, in fact, a pseudo-scalar.) That last dimension, besides closing the algebra so it all becomes workable, really only is there to change operations into their dual forms. (For instance, if you multiply a bivector that describes a rotation by this trivector/pseudo-scalar, you'll end up with an axial vector that describes the exact same rotation, and vice versa).

Quaternions turn out to be precisely the scalar plus the three rotational dimensions. That's why quaternions are four-dimensional. And that's also why all the action of quaternions typically lies in their three imaginary parts. - Those are precisely the three bivectors.
The reason why they are confusing really only is because they are, in a sense, incomplete. They only focus on rotation and completely leave out translation. Upgrading them to dual quaternions - which is possible and gives that missing power to the algebra - still is confusing, but if you approach this problem from a perspective of Geometric Algebra, it suddenly becomes a lot more obvious.

For this isssue their exists an excellent book: "Quaternions and Rotaion Sequences" from Jack B. Kuipers very easy to read and understand. Even if you have only highschool maths.
That is the way I all maths books should be written. An example of very good didactic.

By the way with quaternions it is possible to make fantastic fractals:



I came to quaternions through the art work not through the mathematics as first step.
If one looks at the picture it looks like mixing dough. Has something to do with rotation and stretchy dough in a mixer.

Hermann
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kram1032
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« Reply #56 on: February 10, 2014, 02:02:06 PM »

@hermann
I also first heard of quaternions through their use in generating fractals. I'm pretty sure that's the case for most of us here.

And I didn't mean that it's impossible to present quaternions in an easy way, but rather that they are inherently more out-of-the-blue than how they result as a subset of geometric algebra.

@jehovajah
Vortices aren't just rotations. They are indeed much more complicated. Rotation alone doesn't generate a vortex. You'll need some drag forces and friction for that. Friction complicates all matters.
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hermann
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« Reply #57 on: February 10, 2014, 05:44:11 PM »

Conformal geometry in Wikipedia.
http://en.wikipedia.org/wiki/Conformal_geometry
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kram1032
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« Reply #58 on: February 10, 2014, 08:44:01 PM »

This isn't the conformal model of euclidean space, though it is related.
That's more like it:
http://en.wikipedia.org/wiki/Conformal_geometric_algebra
though it's not exactly one of the best wiki-articles out there.
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youhn
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« Reply #59 on: February 10, 2014, 11:04:11 PM »

Wiki sucks for understanding, though it is great for reminders, facts and the links in the categorized web-of-knowledge.

For understanding I need to play around with the concepts and read it in a more story-fashioned way. The cold and bare wiki language doesn't really enlighten me.

Thanks for the candles everyone ... i'm out to find more fire the next days/weeks.
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