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Author Topic: Transform for R^3  (Read 3733 times)
Description: Yet another Transform for R x R x R inspired by the operators i and j
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jehovajah
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« on: November 29, 2009, 04:23:01 PM »

I started a new topic only because if this right hand formulation provides interesting results i will derive the left hand formulation and attempt some exposition.

For (a,b,c) and (d,e,f) \in R^3

(a,b,c) transforms (d,e,f) = (d,e,f) transforms (a,b,c), that is commutativity is a design feature.

Similarly associativity is a design feature.

The additive transform is as standard

(a,b,c) + (d,e,f) = (a+d,b+e,c+f)

 Calling the transform "vx"

(a,b,c) vx (d,e,f)=(a2d2 + b2e2 + c2f2 + 2cbef - 3abde - 3acdf  - a2e2/2 - a2f2/2 - d2b2/2 - d2c2/2  ,
                            a2e2/2 + d2b2/2 + a2de + a2bd/2 -a2df/2 - b2de - e2ab + abd2/2 + abde - acdf - bcdf - acef - c2d2/2 ,
                              a2cd/2 +  a2ce/2 + a2df + a2ef + d2bc + d2ac/2 - c2df  - f2ac + abdf + acde - bcde - abef)






This gives  (x.y,z) vx (x,y,z) = (x4 + y4 + z4 + 2y2z2 - 4x2y2 - 4x2z2
                                                   2x3y  -  2y3x - 2z2xy + 2x2y2- 2x2z2 ,
                                                       2x3z - 2z3x - 2y2xz + 4x2yz)


So for programming

newx = x4 + y4 + z4 + 2y2z2 - 4x2y2 - 4x2z2

newy = 2x3y  -  2y3x - 2z2xy + 2x2y2- 2x2z2

newz = 2x3z - 2z3x - 2y2xz + 4x2yz


and of course for any C =(a1, b1, c1)
we can write

x = newx + a1
y = newy + b1
z = newz + c1

The design is based on being able to analogise

Z = Z vx Z + C

And answering the computational signal given by ij.

If this proves interesting then it will warrant further investigation to see if it forms a field or a group or a ring with R^3.
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bugman
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« Reply #1 on: December 07, 2009, 05:03:34 AM »

I am not getting consistent results for your squaring function based on your multiplication function. The first term looks right, but the second and third term don't seem to agree (as far as I can tell).
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David Makin
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« Reply #2 on: December 07, 2009, 12:52:55 PM »

I'll try it tonight - I'll also check your conversion from the multiply version to the power version given Paul's comment smiley
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bugman
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« Reply #3 on: December 07, 2009, 09:58:49 PM »

This is what I get using your vx function. But I'm not exactly sure what it's based on.


* Mandelbrot-vx.jpg (43.41 KB, 280x280 - viewed 529 times.)
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David Makin
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« Reply #4 on: December 08, 2009, 01:33:03 AM »

I'll try it tonight - I'll also check your conversion from the multiply version to the power version given Paul's comment smiley

Apologies - I got side-tracked and it's now 12:30 am and I'm knackered so I'll get on it tomorrow - the good news is that I'm now free from "real work" until the 4th of January so I'll be doing some serious fractalling for a while smiley
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David Makin
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« Reply #5 on: December 08, 2009, 03:41:32 PM »

Starting from the multiplication definition I get:

(x,y,z)^2 = (x^4 + y^4 + z^4 + 2.0*y^2*z^2 - 2.0*x^2*y^2 - 2.0*x^2*z^2,
                 3.0*x^2*y^2/2.0 + 2.0*x^3*y - x^3*z/2.0 - 2.0*y^3*x - x^2*z^2 - 2.0*x*y*z^2 - z^2*x^2/2.0,
                 2.0*x^3*z + 9.0*x^2*z*y/2.0 - 2.0*z^3*x - 2.0*x*y^2*z)

So something is definitely off smiley
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David Makin
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« Reply #6 on: December 08, 2009, 04:25:46 PM »

(x,y,z)^2:

http://www.fractalforums.com/gallery/?sa=view;id=1163

(x,y,z)^2 using the definition of (a,b,c)*(d,e,f):

http://www.fractalforums.com/gallery/?sa=view;id=1164
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kram1032
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« Reply #7 on: December 08, 2009, 04:38:01 PM »

Although they're definitely not what you where searching for, I really like those shapes:
They look a lot like some futuristic atempts on rebuilding an old huge sailing ship!
Well one is... the other one rather looks like already motorized.
But I prefer the one with sail..., so the one you defined for (x,y,z)^2
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jehovajah
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« Reply #8 on: December 10, 2009, 05:29:18 AM »

Thanks guys! I will recheck my manipulations to see where i have gone wrong and also derive the left handed version. It may be that i need both to give the full transform!
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May a trochoid of ¥h¶h iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
jehovajah
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« Reply #9 on: December 11, 2009, 05:01:52 AM »

In R^2 there is a transform for (a,b)*(c,d) which is defined as (ac - bd, ad + bc).


ForR^3 I designed a transform based on the R^2 transform in the following way:

(a,b,c) vx (d,e,f)  

(ad - be - cf, ae + bd, af +cd, bf + ce) using R^4 as a manipulation /construction space.

This i reduce to

(A,B,C,D) =(ad - be - cf, ae + bd, af +cd, bf + ce)

Using pairs from the construction bracket in the R^2 transform i obtain 6 building blocks

1 (AA -BB, 2AB)

2 (AA -CC, 2AC)

3 (CC - BB, 2BC)

4(AA - DD, 2AD)

5 (BB - DD, 2BD)

6 (CC -DD, 2CD)

THE unary OPERATORS i and j are used to inform the manipulations so that i2 = j2 = -1 and (ij)2 =+1.


Now my intention was to rotate the planes xy, xz, yz by this construction and i assumed that was what was happening until i rechecked the construction principles. The yz plane is not the same as the other two planes with the unary operators i and j operating on the axes. Under the R^2 transform the yz plane is sheared to the xij plane whatever that is. It may be a vortex surface.

so the first constructed transform is mistaken in two counts. The manipulations were faulty and i will show the correct manipulations; but the design was mistaken as it was not tranforming to a map of geometrical space.


The expansions are as follows for the right handed form

AA = (ad)2 + (be)2 + (cf)2 - 2abde - 2acdf + 2bcef

BB = (ae)2 + 2abde + (bd)2

CC = (af)2 + 2acdf + (cd)2

2AB = 2(a)2de + 2ab(d)2 - 2ab(e)2 - 2(b)2de - 2acef - 2bcdf

2AC = 2(a)2df + 2ac(d)2 - 2ac(f)2 - 2(c)2df - 2abef - 2bcde

2BC = 2(a)2ef + 2bc(d)2 + 2abdf + 2acde

Now the first posted construction was based on combining blocks 1,2,3

supposedly giving

1\over 2 *{AA - BB + AA - CC, 2AB + BB - CC,  2AC + 2BC}

=> {AA - BB/2 - CC/2, AB + BB/2 - CC/2, AC + BC }.                                                    


 [ in fact it should be {AA - BB/2 - CC/2, AB - BB/2 + CC/2, AC + BC } due to an error in the original formulation of block 3]

So clearly (when i expand it) my original manipulations were wrongly copied from page to page to screen.

But now i realise i have not combined like with like and so have to construct the following transform from blocks 1and 2 which i fear will be even less interesting than my mistaken one

{AA -BB/2 - CC/2. AB, AC}
« Last Edit: December 14, 2009, 02:51:35 PM by jehovajah » Logged

May a trochoid of ¥h¶h iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
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