Triplex algebra

[kram1032]

wait...

Unfortunately, for most triplex numbers, this log formula and the exp formulas above don't appear to be inverses of one another.

That could be a problem...

Or was that an error?

[David Makin]

wait...

Unfortunately, for most triplex numbers, this log formula and the exp formulas above don't appear to be inverses of one another.

That could be a problem...

Or was that an error?

I think it's probably correct, given that for fully triplex numbers (z*z)*z != z^3

[BradC]

Unfortunately, for most triplex numbers, this log formula and the exp formulas above don't appear to be inverses of one another.

That could be a problem...
Or was that an error?

I think that's just the way it comes out. If you use power series to define exp and ln for triplex numbers, then the resulting functions don't turn out to be inverses of each other. For example, and . I'm not sure what to think about this. It would be nice if they were inverses, but I don't know how important it is that they're not. Or it's possible that I made a mistake somewhere. I'm interested to see what bugman comes up with if he looks at it some more.

[kram1032]

hmmm...

could you do the plane-analogy you did for e^z with ln(z), ln(e^z) and e^(ln(z)) ?

this could visually express the error

[bugman]

I expect the log of the triplex to have an infinite number of valid solutions because that is true of the complex case:

"every nonzero complex number z has infinitely many logarithms"
http://en.wikipedia.org/wiki/Complex_logarithm

"If z = re^iθ with r > 0 (polar form), then w = ln r + iθ is one logarithm of z; adding integer multiples of 2πi gives all the others"

[kram1032]

yeah, an infinite number of solutions but with a fixed distance.
And there is one solution considered as main solution...

[BradC]

I expect the log of the triplex to have an infinite number of valid solutions because that is true of the complex case:

That's true, but exp(ln(z)) in particular would ideally still reduce to the identity function because in the complex case, all the various possible branches of ln all get mapped to the same point by exp, since exp is periodic with period 2 pi i.

I wonder if it would be possible, rather than defining ln as a series, to define it by solving (a,b,c)=exp(x,y,z) for x, y, and z. It looks hard, and Mathematica (32-bit) ran out of memory.

[BradC]

The "z <- exp(z) + c" fractal looks kinda like stretched foam outside of the xy-plane. My ray tracer couldn't deal with this one so I had to resort to animating 2D slices. The slices are parallel to the xy-plane:

   

I let z animate through the interval [0, 2 pi). Two frames stand out, z=0 and z=pi. The z=0 plane is just the complex exp fractal. The z=pi plane is unique because of the sin(z) factor in the 3rd component of the exp formula: with z=pi, this just maps everything into the xy-plane.

[Snakehand]

A picture of the triplex exponential map:

I think this figure might explain how the isopotential surface of the power 8 bulb transforms from genus=0 (no holes) at 3 iteration, to being full of holes (genus > 0) at 4 iterations. The exact mechanism is still unclear, but I am somewhat less puzzled by this transformation now that I have seen this figure.

[kram1032]

BradC could you do a triplex log picture as you did a triplex exponential one?
(and also a gif animation of the two other planes )

Could be interesting

[vancefab]

I am confused why the standard quaternions confined to the three dimensional sphere don't work just fine for what you are doing.  They are defined by:

a+b*i+c*j+d*k

with i*j=k, j*k=i, k*i=j, and reversing the products negates the results

and i*i=j*j=k*k=-1

and b^2 + c^2 +d^2 = 1.

the idea is that the vector (b,c,d) is the axis of the rotation and a is the cosine of the angle.  If you raise a single one to a power, you get the Chebyshev polynomials in the first term.  If you multiply them together, after normalization, it produces the new axis and the new angle.

[BradC]

How do you mean exactly? If you mean that the rotations could be expressed as quaternions instead of rotation matrices R_y and R_z to derive this stuff, then I think you're right. I think Paolo Bonzini did something along these lines here http://www.fractalforums.com/theory/the-real-math-of-the-mandelbulb/

[BradC]

BradC could you do a triplex log picture as you did a triplex exponential one?
(and also a gif animation of the two other planes )

The log formula posted above has some discontinuities in it that are making it hard to plot. They look like they're probably branch cuts. The exp formula had a discontinuity along the z-axis, but the log is more complicated...

[kram1032]

so it basically looks like a mess?

And would a ln(exp(x,y,z)) or an exp(ln(x,y,z)) work? - in theory they should give x,y,z but you said they kinda don't... would be interesting to see the exact difference...

[fractalrebel]

Hi everyone,

In my Ultrafractal 5 library I have a fairly complete set of quaternion and hypercomplex functions. I am starting a project to build functions for triplex algebra. Simple power, multiplication and division functions are already in the library, as these are the ones I use to generate my UF 5 raytraced images. I am adding BradC's expoential and log functions, along with a triplex power function. There will actually be two triplex power functions since multiplication doesn't commute. Does anyone have thoughts of transcendental functions that use triplex variables?