I'm trying to generate a 3D-Mandelbrot-image.
My idea is: when you multiply 2 complex numbers, you
can do this by
1. component-wise (r1,i1)*(r2,i2)=(r1*r2-i1*i2,r1*i2+r2*i1) or
2. with spherical coordinates l=length1*length2, arg=arg1+arg2
Now I add a second imaginary part ii.
I name my new thing triplex. If 2. is right for complex-numbers,
2. should be right for triplex-numbers.
My problem is to find a component-wise multiplication from
l=length1*length2; length1,2 = sqrt(r^2+i^2+z^2)
arg.alpha = arg1.alpha+arg2.alpha
arg.beta = arg1.beta +arg2.beta
Programs like maple or derive reduces the terms for triplex_number*triplex_the_same_number to
r = l^2*COS(2*a)*COS(2*b)
i = l^2*SIN(2*a)
z = l^2*COS(2*a)*SIN(2*b)
these formulas work, but there's an ugly division.
So I've constructed a component-wise-multiplication
with the idea in mind, that, if you fix either the i or z - part
and vary the r-part, you will get an normal mandelbrot-set
in i or z - plane.
(r,i,z)^2=(r*r - i*i -z*z, 2*r*i, 2*r*z)
But I really would like an mathematical correct transform between
spherical and kartesian-koordinates.
Can anyone do the mathematics ?
> : So your mathematics will be interesting, but they will not represent
> : actual numbers, i.e. math will not be consistent.
> Alas, your conclusion is incorrect. Quaternions do not satisfy all the
> field properties (they have non-commutative multiplication) but it hardly
> follows that they are not "actual numbers" or that quaternion math is
> "inconsistent". The algebra of quaternions is quite consistent, it is
> just a different algebra than that of the complex numbers, which form a
No, I was trying to say that three-component numbers are impossible.
I did not mean to imply that four-component numbers were inconsistent. I'm
sorry if I was unclear. According to Hamilton it's not possible to create
a system of numbers with 3 components per number, you have to use 4. If
you use four components your mathematics will be consistent. Quaternions
are actual numbers, quaternion math is consistent. Hypercomplex are actual
numbers, hypercomplex math is consistent. Trinions (or whatever they're
called) are not actual numbers; their math is inconsistent.
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: Quoting from "Chaos and Fractals: New Frontiers of Science" by Peitgen,
: Vergens, and Saupe (Springer-Verlag, 1992)
: "Complex numbers are a two-dimensional extension of real numbers. It is
: possible to extend the space of complex numbers further.
(Interesting text deleted)
: So your mathematics will be interesting, but they will not represent
: actual numbers, i.e. math will not be consistent.
Alas, your conclusion is incorrect. Quaternions do not satisfy all the
field properties (they have non-commutative multiplication) but it hardly
follows that they are not "actual numbers" or that quaternion math is
"inconsistent". The algebra of quaternions is quite consistent, it is
just a different algebra than that of the complex numbers, which form a
As I have described in another message here, quaternions were not the only
four dimensional extension of the complex numbers possible. The
hypercomplex numbers (also investigated by Hamilton, who for fairly
arbitrary reasons abandoned them in favor of quaternions for his purposes)
are another example. This algebra has commutative multiplication but you
cannot always take the multiplicative inverse of a non-zero element, so
this algebra is not a field either. In terms of abstract algebra, it is a
"commutative ring with unity". And I assure you hypercomplex numbers are
no more or less "actual numbers" than any other. But because familiar
functions such as sin, cos, log, exp etc. generalize to hypercomplex
numbers with their main features intact, hypercomplex numbers are great
for generating fractals using the same formulas we know and love based on
complex numbers. (I should add that I do worry I am overstating the case -
there may be a related way to generalize these same functions to
quaternions also and I am just unaware of it.)
What's to stop anyone from defining a 3-part number: T = x + i*y + j*z,
where T1 + T2 = (x1+x2) + i*(y1 + y2) + j*(z1 + z2), and T1 * T2 =
(x1*x2 - y1*z2 - y2 * z1) + i*(x1*y2 + x2 * y1 - z1*z2) +
j*(x1*z2 + x2*z1 - y1*y2) ?
Once addition and multiplication are defined, it seems to me that the other
functions and relationships can be built up as needed. They may or may not
resemble what we know about real, complex, and quaternion math, but does that
make it wrong?