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Using the work of Bugman (i.e. http://www.fractalforums.com/theory/triplex-algebra/) and Soler (i.e. http://soler7.com/Fractals/Matrices%20to%20Triplex.pdf), I've been able to identify and define the well-behaved triplex algebra with triplex multiplication---see http://vixra.org/abs/1305.0085 (specifically Section 4.1.2) which was published in the Hadronic Journal. If you get a chance to check this out, please let me know if it needs to be fixed.

it reminds me of finishing my bachelor thesis about triplex numbers, lol

can you describe what it is all about?! i read the abstract but have no clue about the content
are you an astronomer ?! is it used for visualisating something ?!  can you put it in a few
words what are you doing ?! hadronic journal sounds like hadron-collider something!

yeah man. i'm engaged in computer, math, and physics research. i spent one year attacking the problem of quark confinement with a colleague and was successful in providing a solution (i.e. http://vixra.org/abs/1208.0219). from this, i realized that complex numbers work great for 2D space, but are insufficient for 3D space---a triplex number is needed in order to encode all types of 3D states, with basically limitless application to science and engineering. Thus, I spent time researching complex numbers and trying to extend them. this is largely a math problem, but it can be applied to physics, computation, etc. eventually, I came across the work of Bugman, and Soler. after an intense analysis of their work, i was able to find out how triplex multiplication operates in a well-behaved triplex algebra, which is in Section 4.1.2 with eqs. (70--71). this is some exciting stuff

the hadronic journal is great for what I'm doing largely because they keep an open mind to scientific theories. no other journal (including the arxiv.org database) that I've delt with will accept a theory on quark confinement or the triplex numbers. the hadronic journal was founded by R. M. Santilli (i.e. http://en.wikipedia.org/wiki/Ruggero_Santilli), who does research in clean energy fuels such as magnegas (i.e. http://magnegas.com/).

Sorry I'm not great at math so I may be mistaken but since you asked if there are errors I will point out that in (66) you define the inclination of y as the phase of x1. I think you meant the phase of x2???  A small typo but it had me confused for a bit. I see that (68) has the correct definition.

I would also like to see (70) written in trigonometric form cause now my brain is swimming and I can't really tell the difference between the alternative and White-Nylander:)

the link he provided analyses 48 variants of the triplex multiplication, i think it comes when combining the order of rotations, we as fractal-fanatics are only interested in those that
keep the komplex number plane intact, hence we have only 2 variants,  on that rotates
around the x axis and the other rotates around the y axis

http://soler7.com/Fractals/Matrices%20to%20Triplex.pdf

to bring back a thought of myself, regarding triplexes, in my eyes a real multiplication
of "-1" should inverse ALL axisses, which is not the case in the nylander-white triplex
algebra, but as far as we know it is the current state of the research to live with em,
and the mandelbulb proves the complexity ;) :D

1) yes your right concerning the typo, thanks for reading and the debug. i'll submit fix.

2) hmm ok, here is my first attempt. a triplex number, such as y_3 in the result of eq. (70), adheres to the trigonometric constraints of eq. (34). thus, let y be a triplex number with Cartesian components (y_R, y_I, y_Z) and Polar components (|y|, <y>, [y]) using the notation of the paper, where the objective is to express y in trigonometric form. first, we use eq. (34) to solve for the y_R = y_I / (arctan^-1(<y>)) , y_I = y_R arctan^-1(<y>), and y_Z = |y| arccos^-1([y]). thus, we have y = y_R + y_I + y_Z = y_I / (arctan^-1(<y>)) + y_R arctan^-1(<y>) + |y| arccos^-1([y]). then, we can divide the outcome's components by the radius |y| to get the trigonometric form y = |y| ( y_I / (|y| arctan^-1(<y>)) + y_R arctan^-1(<y>) / |y| + arccos^-1([y]) ). try that and tell me what you think. still swimming?

latex can be used in the forums as well, check the buttons while editing!

ok, thanks. here's a latex restatement of the last part...

...2) hmm ok, here is my first attempt. a triplex number, such as in the result of eq. (70), adheres to the trigonometric constraints of eq. (34). thus, let be a triplex number with Cartesian components and Polar components using the notation of the paper, where the objective is to express in trigonometric form. first, we use eq. (34) to solve for the , , and . thus, we have . then, we can divide the outcome's components by the radius to get the trigonometric form . try that and tell me what you think. still swimming?

do you see then that lines 2 and 3 of eq. (71) in http://vixra.org/pdf/1305.0085v2.pdf correspond to the two distinct axis rotations that are required for the triplex multiplication? to rotate around the z-axis, all we do is add the two phase angles. and to rotate around the y-axis, all we do is add the two inclination angles.

Actually I was asking for the multiplication of two triplex numbers as trig and really I wanted the exponentiation formula which I believe would just be
y^p=|y|^p(cos(<y>p)sin([y]p),sin(<y>p)sin([y]p),cos([y]p)
...since 34 and 66 only differ in the y_Z definition (acos vs. asin).

Am I missing something or does this look right?

(modified as I forgot to multiply the angles by p)

yes, this looks good.