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Feb 24, 1995, 1:16:10 PM2/24/95

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Hi !

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

l:=SQRT(r^2+i^2+z^2)

a:=ATAN(i/SQRT(r^2+z^2))

b:=ATAN(z/r)

r = l^2*COS(2*a)*COS(2*b)

i = l^2*SIN(2*a)

z = l^2*COS(2*a)*SIN(2*b)

reduces to:

r= (r^2-z^2)*(z^2-i^2+r^2)/(z^2+r^2)

i= 2*i*SQRT(z^2+r^2)

z=2*r*z*(z^2-i^2+r^2)/(z^2+r^2)

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 ?

my e-mail is:

akr...@techfak.uni-bielefeld.de

and

andre....@stud.uni-rostock.de

Thanks !

Mar 8, 1995, 5:11:12 PM3/8/95

to

In article <3jgprn$c...@gryphon.phoenix.net>, twe...@phoenix.phoenix.net

(Tim Wegner) wrote:

(Tim Wegner) wrote:

(Quoting Me)

> : 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

> field.

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.

--

This .sigfile is a thought virus. Its mission in life is to ensure that it does not get read, thought about, or duplicated. So DO NOT read this, do not think about it, and DEFINITELY do not duplicate it in your own .sig.

Mar 7, 1995, 12:10:15 AM3/7/95

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Jonathan Rynd (jr...@cornell.edu) wrote:

: In article <3il7pa$8...@taiwan.informatik.uni-rostock.de>,

: andre....@stud.uni-rostock.de wrote:

: In article <3il7pa$8...@taiwan.informatik.uni-rostock.de>,

: andre....@stud.uni-rostock.de wrote:

: 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

field.

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.)

Tim

Mar 10, 1995, 3:50:49 AM3/10/95

to

In article <jrr7-08039...@j302603012.resnet.cornell.edu>,

jr...@cornell.edu (Jonathan Rynd) writes:

> 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.

> 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.

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?

Kerry Mitchell

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