True 3D mandelbrot type fractal

[lycium]

and while the thread is getting derailed, i might as well play the trump card of all useless (and harmful) human activities: religion!

[pnorthover]

"useless (and harmful) human activities: religion!"


There are advantages at the individual level, like a membership that helps each other out a little more than can be expected of dog eat dog. 

I'd have said war was the most useless and harmful of human activities. 

[cKleinhuis]

@pnorthover: indeed!

and, isnt fractalism ( or science in general ) just another religion ?

in the meaning that you need to believe in something ... 

[David Makin]

I can't resist the temptation

nor can i... what a strange, strange selection of examples! the over-arching theme in my replies is that one should try to see the world beyond one's own chosen set of values, and that just because something is useless to you or me, doesn't mean it's intrinsically useless.

Climbing mountains

some like to make fractals, some like to climb mountains.

Racing fast cars

much as i dislike racing, not all recreation needs to be intellectual.

Having sex with no intention of having children

this one is by far the strangest; there's nothing wrong with consensual sex, and the last thing our poor planet needs is more people.

Looking for weapons of mass destruction in Iraq

this serves as a convenient excuse for all sorts of dodgy manoeuvres... no moral explanation however.

Using sanctions to stop countries nuclear weapons testing

it should somehow be strongly disincentivised, and limiting the growth of a militaristically-inclined nation (e.g. north korea) is difficult to logically argue against.

Watching TV

that's a personal judgement, but i'm with you 100%. it's a first class, fully addictive and mind-altering drug. then again, so is alcohol, which is notably absent from this list...

Sunbathing

personal judgement, and i agree, but some people want to attract others; instead of blaming the effect, why not find fault in the cause - that people find tanned skin attractive.

Finding the solution to Fermat's last theorem

so some purely intellectual pursuits (making fractals) are definitely less useless than others?

Calculating pi to more than 100 significant figures

as above, try not to be so biased

They were all meant humorously - though not all in the same way
In no case was I necessarily giving my own beliefs - except....sorry I'll stop because it's already way OT

Edit: I guess that in most cases I was just making the point that anything recreational could be seen as "useless"

[David Makin]

OK back on topic - I assumed that getting the derivative of the "True 3D" formulas I've been using would require some complicated maths involving the Jacobian for 4 functions of 4 reals but after considering the Jacobian for complex and quaternionic z^2+c it appears that only the first column from the Jacobian is required to get the derivative - is this correct or am I missing something ?

If you consider complex z^2+c as:

f1(x,y) = x^2 - y^2 + cx
f2(x,y) = 2*x*y + cy

Then the Jacobian is:

| 2x -2y |
| 2y  2x |

Take the first column gives (2x,2y)  i.e. 2x+flip(2y) i.e. 2*z

Similarly for quaternionic q^2+c:

f1(x,y,z.w) = x^2 - y^2 - z^2 - w^2 + cx
f2(x,y,z,w) = 2*x*y + cy
f3(x,y,z,w) = 2*x*z + cz
f4(x,y,z,w) = 2*x*w + cw

Then the Jacobian is:

| 2x -2y -2z -2w |
| 2y  2x    0     0 |
| 2z   0   2x     0 |
| 2w  0    0    2x |

Again taking the first column gives (2x,2y,2z,2w) == 2*q

So given that the first column of the Jacobians for the "True 3D" hypercomplex-like family are the same as the quaternion do I take it that for all members of this family then the derivative of f^2+c is simply 2*f ?

[pnorthover]

"isnt fractalism ( or science in general ) just another religion ?"

Not for me, but if you want to worship fractals or science, go ahead! 

"in the meaning that you need to believe in something ...  "

You don't have to believe in anything. 

Anyway, I was just providing a counter example to the claim that religion is useless and harmful. 

[lycium]

Anyway, I was just providing a counter example to the claim that religion is useless and harmful.  

we'll have to weigh that (admittedly valid, although human brotherhood shouldn't be based on fictitious beings) counterexample against the crusades  

anyway back on topic, although it is far from "clean" i think the full jacobian would be required and that it is only a numerical coincidence that a factor of 2 appears throughout the first column.

furthermore, all that's involved is differentiation, which is easy enough (mechanical, unlike integration), and getting a determinant (which is also pretty easy and mechanical). hence my suggestion to use a symbolic algebra package to do the grunt work

by the way i think matlab should be able to do this, in which case the free clone called "octave" should be good for the job.

[David Makin]

Anyway, I was just providing a counter example to the claim that religion is useless and harmful.  

we'll have to weigh that (admittedly valid, although human brotherhood shouldn't be based on fictitious beings) counterexample against the crusades  

anyway back on topic, although it is far from "clean" i think the full jacobian would be required and that it is only a numerical coincidence that a factor of 2 appears throughout the first column.

furthermore, all that's involved is differentiation, which is easy enough (mechanical, unlike integration), and getting a determinant (which is also pretty easy and mechanical). hence my suggestion to use a symbolic algebra package to do the grunt work

by the way i think matlab should be able to do this, in which case the free clone called "octave" should be good for the job.

I thought it might be more complicated...unfortunately higher calculus (partial derivatives, differential equations, multivariable differentiation and integration) are the parts of my education that I seem to have lost (in any case I only did maths to Further Maths "A" level and even then only attended about half the lessons and consequently just scraped a pass grade "E" in Further Maths).
I need to get a book on multivariable calculus from the library....doing it for specific cases with Matlab/Octave etc. doesn't cover allowing general cases of 4 functions in 4 variables which is something I'd like to include in my 3D+ formulas.
So it may be several days before I have a working version using DE.
Incidentally while it was easy to do I tried a modification of Ron Barnett's 3D raytrace formula (in reb.ufm) such that the running derivative was calculated as if the number form was quaternionic and the iteration was performed using the "roundy" "True 3D" method and was surprised to find that it worked to some extent - then I tried "correcting" the calculation of the running derivative to use the roundy True 3D multiplication instead of quaternionic multiplication and the result was a resounding mess

[lycium]

I thought it might be more complicated...unfortunately higher calculus (partial derivatives, differential equations, multivariable differentiation and integration) are the parts of my education that I seem to have lost (in any case I only did maths to Further Maths "A" level and even then only attended about half the lessons and consequently just scraped a pass grade "E" in Further Maths).

my maths education mostly stops at 2nd year and i usually got poor grades, except in courses of genuine interest like numerical methods. fortunately, with some effort and the right materials we can eventually master it

I need to get a book on multivariable calculus from the library....

i can recommend thomas' calculus (pictured above with erwin kryzig's differential geometry) for calculus 2; tomorrow i hope to get my new books from amazon, and begin a nice fractal-oriented weekend of study

doing it for specific cases with Matlab/Octave etc. doesn't cover allowing general cases of 4 functions in 4 variables which is something I'd like to include in my 3D+ formulas.

yes, that's ultimately the right way to do it (expand the parameter space), as it enlarges the artistic possibilities of a given method. flame fractals are awesome because they are high-dimensional in this vein it shouldn't be too difficult to work with a slightly more general cayley table, where the products don't always have a coefficient of +/- 1.

Incidentally while it was easy to do I tried a modification of Ron Barnett's 3D raytrace formula (in reb.ufm) such that the running derivative was calculated as if the number form was quaternionic and the iteration was performed using the "roundy" "True 3D" method and was surprised to find that it worked to some extent - then I tried "correcting" the calculation of the running derivative to use the roundy True 3D multiplication instead of quaternionic multiplication and the result was a resounding mess

that sounds like quite a battle you had; the results are great though, especially the fly-around animations. i'd really like to see that mess btw, one of the coolest things about computer graphics is the weird error images you get!

[pnorthover]

"fictitious beings"

God has neither been proved nor disproved. 

"the crusades"

Not a religious function.  I say unto you, "Go Forth and KILL!!"  I don't think so.   

[lycium]

God has neither been proved nor disproved.

Nor have the gnomes that come in the night and steal my socks so as to leave odd numbers in my drawer Of course the onus is on everyone else to prove my theory wrong, it's perfectly reasonable when looked at logically - some unseen force of considerable ingenuity and complex character must be responsible...

Anyway, Marcus Aurelius long ago summed up everything there is to meaningfully consider about higher powers: http://www.goodreads.com/quotes/show/63966 I will do my best to leave my heretical thoughts on the matter at that  

[David Makin]

I need to get a book on multivariable calculus from the library....

i can recommend thomas' calculus (pictured above with erwin kryzig's differential geometry) for calculus 2; tomorrow i hope to get my new books from amazon, and begin a nice fractal-oriented weekend of study

Thanks - I'll investigate those.

doing it for specific cases with Matlab/Octave etc. doesn't cover allowing general cases of 4 functions in 4 variables which is something I'd like to include in my 3D+ formulas.

yes, that's ultimately the right way to do it (expand the parameter space), as it enlarges the artistic possibilities of a given method. flame fractals are awesome because they are high-dimensional in this vein it shouldn't be too difficult to work with a slightly more general cayley table, where the products don't always have a coefficient of +/- 1.

It also allows the possibility of 4 functions of 4 variables where some of the functions are like this:

f1(x,y,z,w) = log(x) + sin(y) + y*z + z^2 + 3*x*z + tan(w)

etc.

Incidentally while it was easy to do I tried a modification of Ron Barnett's 3D raytrace formula (in reb.ufm) such that the running derivative was calculated as if the number form was quaternionic and the iteration was performed using the "roundy" "True 3D" method and was surprised to find that it worked to some extent - then I tried "correcting" the calculation of the running derivative to use the roundy True 3D multiplication instead of quaternionic multiplication and the result was a resounding mess

that sounds like quite a battle you had; the results are great though, especially the fly-around animations. i'd really like to see that mess btw, one of the coolest things about computer graphics is the weird error images you get!

Well actually all the renders I've posted were done using my mmf.ufm:Solid-3D Quaternions formula to which I've added the 3 interesting not-quite-standard-hypercomplex variants.
That formula is just solid on iteration count but performs reasonably well as I included every trick I could think of to improve speed and accuracy.
Testing the "roundy True3D" in Ron's DE raytracer only took about an hour altogether

[pnorthover]

"Nor have the gnomes that come in the night and steal my socks so as to leave odd numbers in my drawer "

God is a unique case.  If there is a supreme deity, there can only be one.  That's the meaning of supreme.  Do you claim your gnomes are God?  If so, then it becomes a question of naming.  If not, then your statement is meaningless.  What properties do you claim for your gnomes (or flying toasters, or spaghetti monster etc.) that match the properties claimed for God? 

"Of course the onus is on everyone else to prove my theory wrong"

Since God's existence cannot be proved or disproved, onus is irrelevant. 

" - some unseen force of considerable ingenuity and complex character must be responsible..."

In the case of "God", *may* be responsible.  Or not.  There is no answer.  Which is why off topic religious potshots should be avoided.  Where did you think it was going to get you? 

[twinbee]

I suppose the discussion of whether God exists is actually pretty related to this thread in a funny roundabout kind of way. Because it takes a certain amount of 'faith' (for want of a better word) to believe the 3D mandelbrot (in the form I'm hoping for) actually exists. We have the 2D version, and it's a strange kind of judgement to ascertain if there's really a 3D analogue. As usual, it comes to probabilities, guesswork, and uncertainties with wide margins of error. It's out there. Maybe. Maybe sort of. Sort of maybe.

Bring on more pseudo 3D mands in the meantime - or hit the jackpot whilst trying!

[bib]

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