Complex numbers
Pat
I recently saw some really interesting pictures of the "Mandelbulb." I
am shamelessly addicted to fractals of any kind, so this kept me up all
night experimenting.
The originator of the images gives some arguments as to whether or not
this Mandelbulb constitutes a "real 3D Mandelbrot" set. One of the
points he made is that there really isn't any such thing as a third
complex axis. To produce 3D fractals, apparently, the general approach
is to map something like "number of iterations" to the Z axis, or to
simply take a 2D fractal and rotate it in some way to create a volume.
The Mandelbulb involves hypercomplex numbers, which seem to be just
complex numbers with a second imaginary axis added.
Anyway, the discussion planted an odd notion in my head, which is
probably utter nonsense, but I thought it might be worthy to ask of the
lords of the sci.math continuum...
If we begin with only the natural numbers (0, 1, 2, ...), then our
number system is "closed" under addition and multiplication. If we want
to allow for subtraction, then we must introduce negative numbers. If
we want to allow for division, we must introduce rational numbers. If
we want to allow for exponentiation (and consequently the taking of
roots), then we must introduce imaginary numbers.
At some point in history, it was decided that negative numbers would
look exactly like positive numbers, but prefixed by a "-" symbol. And
graphically, we chose to represent the positive numbers as a ray
extending infinitely in one direction, with the negative numbers
extending in the oppoisite direction.
For imaginary numbers, instead of a new prefix symbol, we invented the
quantity "i" to represent the sqare root of -1. And graphically, we use
an axis orthogonal to our positive/negreal number axis to represent the
positive and negative imaginary numbers.
But weren't those choices kind of arbitrary? Couldn't we, instead, have
invented a quantity, say "h," to represent the unit negative number?
The idea of a negative number is just as "imaginary" as the square root
of a negative number -- you can't hold negative two apples in your hand
any easier than you can hold 2i apples in your hand. So by definition,
let 0 - 1 be equal to "h" and let the square root of "h" be "i."
We would have three axes, one for real, one for negative, and one for
imaginary. An important difference would be that these axes would only
extend in one direction and zero would form the absolute "corner" of the
graph.
Can arithmetic operations be defined in such a coordinate space? And
would this give rise to 3D fractal geometry?
Pat
Tim Little
> We would have three axes, one for real, one for negative, and one for
> imaginary. An important difference would be that these axes would only
> extend in one direction and zero would form the absolute "corner" of the
> graph.
>
> Can arithmetic operations be defined in such a coordinate space? And
> would this give rise to 3D fractal geometry?
One problem is that for (real,neg) pairs, what do you get when you add
(1,0) and (0,1)? If "negative" actually means anything, you should
get (0,0). But then what makes (1,1) different from (0,0)?
In the usual construction of negative numbers, the answer is "nothing
but the label", and so we have sets of labels that all denote the same
number. It turns out that any of these numbers is either equivalent
to (x,0) or (0,x) for some x, so all we need is just some mark to
distinguish the latter from from former.
- Tim
Robert Israel
There is a big difference. 1 + h = 0, but 1 and i are linearly independent
over the reals. Linear combinations of 1 and i give you a two-dimensional
surface, but linear combinations of 1 and h don't give you anything new.
> We would have three axes, one for real, one for negative, and one for
> imaginary. An important difference would be that these axes would only
> extend in one direction and zero would form the absolute "corner" of the
> graph.
>
> Can arithmetic operations be defined in such a coordinate space? And
> would this give rise to 3D fractal geometry?
>
> Pat
--
Robert Israel isr...@math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
Ken Pledger
Pat <nos...@noemail.com> wrote:
> .... Couldn't we, instead, have
> invented a quantity, say "h," to represent the unit negative number?
> The idea of a negative number is just as "imaginary" as the square root
> of a negative number -- you can't hold negative two apples in your hand
> any easier than you can hold 2i apples in your hand. So by definition,
> let 0 - 1 be equal to "h" and let the square root of "h" be "i."
>
> We would have three axes, one for real, one for negative, and one for
> imaginary. An important difference would be that these axes would only
> extend in one direction and zero would form the absolute "corner" of the
> graph.
>
> Can arithmetic operations be defined in such a coordinate space? ....
Well, for every positive real number t, you would have
t + ht = t - t = 0,
so the single real number 0 would be represented by infinitely many
different points (t, t, 0). Is that what you want?
Ken Pledger.
Pat
> Well, for every positive real number t, you would have
> t + ht = t - t = 0, so the single real number 0 would be
> represented by infinitely many different points (t, t, 0). Is
> that what you want?
Ah I see. Well, that's only true if you make the assumption that the
notation "(x, y, z)" implies addition, as in: x*1 + y*h + z*i.
Is there another choice of operation under which 1, h, and i would be
independent?
Addition would have to work as expected:
(1, 0, 0) + (0, 1, 0) = (1, 1, 0)
however,
(1, 1, 0) != (0, 0, 0)
But it still needs to be true that:
(2, 0, 0) - (3, 0, 0) = (0, 1, 0)
Hmm... I think this is where the idea falls apart.
Gerry Myerson
Pat <ns...@noemail.com> wrote:
> Ken Pledger wrote:
>
> > Well, for every positive real number t, you would have
> > t + ht = t - t = 0, so the single real number 0 would be
> > represented by infinitely many different points (t, t, 0). Is
> > that what you want?
>
>
> Ah I see. Well, that's only true if you make the assumption that the
> notation "(x, y, z)" implies addition, as in: x*1 + y*h + z*i.
>
> Is there another choice of operation under which 1, h, and i would be
> independent?
Not with h understood as negation, no, but a lot of work went on
in the 19th century trying to find a useful arithmetic for triples of
numbers to generalize the a + b i interpretation of pairs of numbers.
Turns out it can't be done, but it can be done for quadruples of
real numbers - these are called quaternions, q.v.
--
Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)
Daniel Royer
Tim Golden BandTech.com
wrote:
> In article <Xns9CC8BEA71138Cnonenonen...@69.16.186.8>,
Polysign numbers
http://bandtechnology/polysigned
generalize sign. It turns out that the complex numbers are the same as
the three-signed numbers, but in a different format. As Myerson states
above here, Hamilton attempted a 3D version, and he even looked at i
(the complex component) as a sign, but never truly generalized sign.
The four-signed numbers are 3D and algebraically behaved with
commutative and distributive properties. The quaternion does not
preserve commutative behavior. The four-signed numbers P4 are the
numbers you seek.
Developing graphics from the higher dimensional systems in the way
that the 2D images are generated is troublesome. I do have some slice
images on my site of the standard Mandelbrot function over the higher
dimensional systems from a simple projection:
http://bandtechnology.com/PolySigned/Mandelbrot/MandelbrotStudy.html
Oddly from these nonorthogonal systems comes a square shape, but this
is merely a small slice of a very large data set whose full graphical
representation will not be straightforward.
The polysign numbers do maintain algebraic properties familiar from
two-signed numbers P2(the reals) and traditional complex numbers (P3),
but in the higher systems one familiar behavior is not maintained:
| z1 z2 | = |z1||z2|
This equation is broken above P3. Consider the progression
P1 P2 P3 | P4 ...
We now have arithmetic support for spacetime with unidirectional zero
dimensional time and electromagnetic structural support to boot, from
a natural progression. This is structured spacetime. It breaks
assumptions and begs that we reconsider the fundamentals, signalling
coherencies to existing mathematics and physics, but suggesting that
we've overlooked some very simplistic technicalities.
- Tim
Tim Golden BandTech.com
wrote:
approaches ninety degrees so as a remnant this is probably not a great
mystery.