Computing Accurate 3D/4D Mandelbrot's
Laszlo Vecsey
attemped to use Quatorian numbers instead of Complex ones in an attempt
to gain a variable for the third dimension. Unfortunately the outcome is
not a Mandelbrot that 'grows' in all dimensions.. instead it is one that
is revolved about the real axis.
I've been told that there is a formula to compute this 3D Mandelbrot that
I'm thinking of, and that it was in fact given a new name. Does anyone
have any information about this?
- Lester
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Thomas Marsh
>I'm interested in computing a real 3D Mandelbrot (not extruded 2d). I've
>attemped to use Quatorian numbers instead of Complex ones in an attempt
>to gain a variable for the third dimension. Unfortunately the outcome is
>not a Mandelbrot that 'grows' in all dimensions.. instead it is one that
>is revolved about the real axis.
According to my only book (Chaos and Fractals; Pietgen, Jurgens 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. However, the attempt
by the Irish physicist and mathematician William R. Hamilton to create a space
of numbers with three components failed. Instead he had to resort directly to a
space of numbers with four components. This space, invented in the year 1843m
is called the space of quaternions H. A quaternions x element of H can be
represented by the symbol
"x = x + x i + x j + x k
0 1 2 3
"Where j and k denote additional imaginary units and x0 to x3 denote the four
components of the quaternion."
As with complex math, quaternion math is quite different, haveing
non-commutative multiplication and such. One thing to note: If you test
prisoner of escape values in quaternion math for z -> z*z + c you will get a
quaternion Julia, not a Mandelbrot. In fact, I don't think I have ever seen a
quaternion mandelbrot. Anyone?
>I've been told that there is a formula to compute this 3D Mandelbrot that
>I'm thinking of, and that it was in fact given a new name. Does anyone
>have any information about this?
>
Well, it would not be so different from computing a 2D fractal aside from the
math laws of quaternions and the fact that you now have a point (x, y, z, w)
where w is commonly regarded to be time and x y z to be the cartesian
coordinates of each point.
The only book I have on the subject (mentioned above) has only two pages on
this subject. Is there any source code (C?) out there or papers discussing the
technicalities of quaternion generation so that I might give it a shot? All
out there seems to be either DOS or else binary only...
On a side note, I have rendered very nice Mandelbrot's in 3D with height
corresponding to distance if you are interested in that.
--thomas
--
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Thomas Marsh, ama...@bga.com http://www.bga.com/~amadeus
Oz
> The only book I have on the subject (mentioned above) has only two
pages on
> this subject. Is there any source code (C?) out there or papers
discussing the
> technicalities of quaternion generation so that I might give it a
shot? All
> out there seems to be either DOS or else binary only...
You might like to look up John Baez on sci.physics.particle. He
offers a thread "This weeks find in mathematical physics" or
similar. He has been discussing quarternions in depth in them.
Actually another poster there commented on the non-existance of true
3-D fractals, and I had been under the impression that they existed
from threads here, and said so. It would appear that I was wrong.
John Baez is a mathematician, writes text books and is very
approachable. I suspect that he would be interested to give advice
if someone wished to produce a good quarternion fractal generator.
--
-------------------------------
'Oz "When I knew little, all was certain. The more I learnt,
the less sure I was. Is this the uncertainty principle?"
Jon Noring
As an interesting aside, and one that maybe is more relevant to sci.physics,
James Clerk Maxwell originally formulated his electromagnetic equations using
quaternions and not with the more familiar vector calculus. A little later,
Heaviside, a practical yet eccentric engineer, recast Maxwell's equations
into vector form, mainly to making it easier to solve the equations for
antenna design and the like. Towards the end of his life, Heaviside returned
to quaternions, thinking they may hold the answer to why the vector form of
Maxwell's equations seem to "break down" for things like rail guns and other
high energy electromagnetic phenomena (these deviations from Maxwell's
equations are often conveniently swept under the rug, or dealt with
empirically without a second-thought as to the deeper implications of the
disparity).
There is controversy as to whether the quaternion form of Maxwell's equations
will generate different solutions if one is able to solve them (probably
numerically). A professor at one of the Texas Universities (sorry, I don't
recall his name or college) recast the fundamental equations of fluid
mechanics into quaternion form. He did find solutions to unusual situations
that differed from the vector form, but it is not known to me if such
solutions better fit experimental data or not. Of course, fluid turbulence,
which to me is a classic example of fractals in nature, has always been an
intractable problem to solve from the fundamental equations other than
empirically. Who knows, maybe solving the quaternion forms of the fluid flow
equations might be the key to understanding turbulence.
And of course, quaternions is a subset of a more general form (Hamiltonians?).
Of course, what I wrote above is sort of speculative. It's been a while
since I've studied the above subjects and no doubt I'm a little rusty. I
can't even remember the name of the fundamental fluid equation. Must be
Mr. Alzheimers seizing me today. :^)
Back to fractals.
Jon (Ronald Reagan) Noring
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Alex Feinman
>> [Discussion deleted]
>
>You might like to look up John Baez on sci.physics.particle. He
>offers a thread "This weeks find in mathematical physics" or
>similar. He has been discussing quarternions in depth in them.
>Actually another poster there commented on the non-existance of true
>3-D fractals, and I had been under the impression that they existed
>from threads here, and said so. It would appear that I was wrong.
>
>John Baez is a mathematician, writes text books and is very
>approachable. I suspect that he would be interested to give advice
>if someone wished to produce a good quarternion fractal generator.
>
There are a number of good web sites on this topic -- following the advice of
some I have managed to code a (very simple) 3-D Mandelbrot set renderer, with
the function still being X' = X^2 + C but where X is a quarternion. Once place
to start looking is:
http://www.dtek.chalmers.se/Datorsys/Project/qjulia/index.html
(the QJulia web page) -- there is also information here on how to program the
equations into POV-Ray.
Hope this helps the original poster.
Alex
----------------------------------
Alex Feinman
John C. Hart
: I'm interested in computing a real 3D Mandelbrot (not extruded 2d). I've
: attemped to use Quatorian numbers instead of Complex ones in an attempt
: to gain a variable for the third dimension. Unfortunately the outcome is
: not a Mandelbrot that 'grows' in all dimensions.. instead it is one that
: is revolved about the real axis.
There's a proof in...
Ke & Panduranga, ``A Journey into the Fourth Dimension'' Proc. of
the IEEE Visualization '90 conference, IEEE Computer Society
Press, 1990, pp. 219-229.
-John
(email: ha...@eecs.wsu.edu, in case my inet domain thingy is still broke)
---
John C. Hart, Asst. Prof.
School of EECS, Wash. St. Univ.
Pullman, WA 99164-2752
(509)335-2343 fax:(509)335-3818
Stephen Harris
>From: hart@PROBLEM_WITH_INEWS_DOMAIN_FILE (John C. Hart)
>Subject: Re: Computing Accurate 3D/4D Mandelbrot's
>Date: Mon, 2 Oct 1995 21:41:33 GMT
>Laszlo Vecsey (mas...@future.internexus.net) wrote:
>: I'm interested in computing a real 3D Mandelbrot (not extruded 2d). I've
>: attemped to use Quatorian numbers instead of Complex ones in an attempt
>: to gain a variable for the third dimension. Unfortunately the outcome is
>: not a Mandelbrot that 'grows' in all dimensions.. instead it is one that
>: is revolved about the real axis.
>There's a proof in...
> Ke & Panduranga, ``A Journey into the Fourth Dimension'' Proc. of
> the IEEE Visualization '90 conference, IEEE Computer Society
> Press, 1990, pp. 219-229.
Mildly on this topic... the web page:
http://www.krs.hia.no/~fgill/quatern.html
Anyone know why they don't carry Pickover books much?
Antonio Cardoso Neto
where j=sqrt(-i) ?
Some suggestions?
Cheers
Cardoso
Andrew Dalton
>From: Antonio Cardoso Neto <acar...@spider.usp.br>
>Subject: Re: Computing Accurate 3D/4D Mandelbrot's
The problem is that sqrt(-i) is another complex number:
[-sqrt(2)/2 + i*sqrt(2)/2]^2 = 1/2 -2(1/2)*i -1/2 = -i
- Andrew Dalton
- asda...@umich.edu
--
--- "Faith, n. Belief without evidence in what is told by one who
----- speaks without knowledge, of things without parallel."
-------- --Ambrose Bierce
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John C. Hart
: What about using z=a+bi+cj
: where j=sqrt(-i) ?
: Some suggestions?
: Cheers
: Cardoso
sqrt(-i) ~= 0.707 - 0.707 i
-John
---
John C. Hart, Asst. Prof.
School of EECS, Wash. St. Univ.
Pullman, WA 99164-2752
(509)335-2343 fax:(509)335-3818
David Byrne
The equation for the four demensional Madelbrot is Z= Z^2 + C + D
Where Z, C, D, are all complex numbers and the initial value of Z is zero
(just like the 2-D Mandelbrot). C= x + y * i and D= z + t * i
The interpretation for D can vary, but it should work best to have one as
the Z coordinate and one as a set constant, or as time in the case of a
Mandelbrot animation. Because the Mandelbrot is based on complex
numbers, I think that each term would have to be complex. Thus there
would be no 3-D Mandelbrot set unless you set z or t to zero in the D
term of the 4-D set. However this would be more of a cross section of
the 4-D set.
I am currently in the process of designing a 4-D Mandelbrot program for X
windows that, hopefully, will be able to rotate 3-D cross sections of the
4-D set realtime, as well as take real time cross sections of the 3-D
cross sections. Obviously this program will take a lot of machine
resources. Any sugestions on making fast code would be greatly appriciated.
David Byrne