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Author Topic: Do quaternions need revisiting?  (Read 1262 times)
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Roquen
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« Reply #15 on: August 01, 2014, 07:28:55 PM »

Perhaps so.  Even probably so.  But as far as I can tell nobody's really explored them in any meaningful way.
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kram1032
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« Reply #16 on: August 04, 2014, 11:07:21 AM »

Quaternions were explored plentifully in all kinds of research. However, thus far it was not game-changing research that has gotten a lot of fame, other than the first invention of Quaternions and the initial flood of followup papers.

It doesn't help that they seem to sort of come "out of nowhere". - Until you construct them as a part of R³-Geometric Algebra, representing the three bivectors, which, to me, feels way more natural.
And there has been plenty of reformalization work translating typical Tensor Algebra into Geometric Algebra.
Just, again, not a whole lot of inventive work.
It's a bit of a race: You kind of need to reach the state of the art in both formulations to get to the parts where inventions can actually be done. But meanwhile, the Tensor approach keeps expanding.
Translational work is much faster than inventive work, of course. However, there also are a lot fewer people focusing on that kind of work.
Still, I think, overally, Geometric Algebra is catching up and I expect it to, eventually, become the standard for the geometric part of the description of the physical system.
And with it, as sub-algebra, the Quaternions, however in a way less opaque manner than straight up using Quaternions.

Meanwhile, for the other half of the topology of physics and computer science and even the entirety of all maths ever, we'll probably see a new era as everything is reformulated from the messy, structure-less set-theory into the way more tractable and even computationally automatable homotopy-type-theory. At least if we can find our way past the two or so major problems that are the computational equivalent of the univalence axiom (isomorphisms are isomorphic to equality - currently, we do not know of a way how to automate reasoning with this) and recursive infinities (which would bring us a long way towards formulating HoTT as its own Metalanguage, which is kind of important, if you want to use it as the foundation of all of Mathematics. You want to reason about the foundations within the foundations. Right now, with HoTT, we have to use a different meta language to reason about HoTT itself.)
Once those difficulties are solved, we'll have a full (comparatively) comprehensive formulation of the very foundations of reality, or at least the proper tools to get there.

(Unless, of course, there are some unexpected stumbling blocks in the way, which wouldn't be a first. But we'll see about that)
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Roquen
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« Reply #17 on: August 04, 2014, 11:47:03 AM »

My context isn't mathematical exploration but rather application to fractals.  Mathematically, like complex, their treatment in historic context doesn't help understanding. And they're actually frequently discussed in an impossible to understand way as most author punt to using linear algebra.  Taking a Clifford algebra perspective both are certainly much easier to understand.
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David Makin
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« Reply #18 on: August 06, 2014, 12:04:10 AM »

The problem with respect to quaternionic fractals is that they lack the main attribute of a true 3D or 4D fractal i.e. the surface of the "Set" is not fractal in all directions, even on the Julias.
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Roquen
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« Reply #19 on: August 06, 2014, 10:56:13 AM »

This is exactly what I mean.  All analytic functions are quaternions must be boring as they are all surfaces of revolution since they are really 2D systems embedded in a 4D space.  No projections of an analytic function can be interesting.
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Roquen
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« Reply #20 on: August 07, 2014, 01:07:52 PM »

Mandelbulb is an ad-hoc system built on manipulating angles in spherical coordinates.  To construct a spherical coordinate system in 3D you arbitrary choose a reference plane and a direction within that plane.  Say the XY plane through the origin and the X axis.  Then the first angle is the point in question projected into the plane and the angle formed with the X axis (azimuth).  The second angle is the remaining part out of the plane which can be measured in a number of ways, such as with respect to the positive Z axis (zenith angle) or the angle formed with the plane (inclination).  The last part of information is of course the magnitude of the vector to the point.

The two major flavors of Mandelbulb appear to be these:

Code:
vec3 mbulbAI(vec3 p, float m, const float n)
{
  float a = atan(p.y, p.x); // on [-pi, pi]
  float i = asin(p.z/m);    // on [-pi/2, pi/2]
 
  // scale the angles and magnitude by power
  i *= n;
  a *= n;
  m = pow(m, n);
 
  // convert back to Cartesian.
  return m*vec3(cos(i)*cos(a), cos(i)*sin(a), sin(i));
}

Code:
vec3 mbulbAZ(vec3 p, float m, const float n)
{
  float a = atan(p.y, p.x);  // on [-pi, pi]
  float z = acos(p.z/m);     // on [0, pi]
 
  // scale the angles and magnitude by power
  a *= n;
  z *= n;
  m = pow(m, n);
 
  // convert back to Cartesian
  return m*vec3(sin(z)*cos(a), sin(z)*sin(a), cos(z));
}

All spherical coordinates can be also thought as a system which takes a predefined reference position on the unit sphere followed by a pair of rotations in a predefined manner and a uniform scaling.

AI: ref-point {1,0,0}, rotate -Y by i, rotate Z by a, scale
AZ: ref-point {0,0,1}, rotate +Y by z, rotate Z by a, scale

Now it's obvious that these two systems are trivially equivalent, say via trig identities or by observing rotation -Y is same as by Y and negating the angle and moving the reference from in Z to in X by adding pi/2.  Of course these are the same since asin(x)+acos(x)=pi/2.

The above reference point, rotation and scaling view-point hopefully sounds familiar because that's one view point I gave above for quaternion formulations.

Taking AI, the first rotation is ry = sin(i){0,-1,0}+cos(i), rz=sin(a){0,0,1}+cos(a).  These are for rotations of 2i and 2a.  The combined is r = rzry = {sin(a)sin(i), -cos(a)sin(i), cos(i)sin(a)} + cos(a)cos(i)

Okay..enough for now.
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