Transform for R^3

[jehovajah]

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)

(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 .

[bugman]

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).

[David Makin]

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

[bugman]

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

[David Makin]

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

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

[David Makin]

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

[David Makin]

(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

[kram1032]

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

[jehovajah]

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!

[jehovajah]

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


For I designed a transform based on the transform in the following way:

(a,b,c) vx (d,e,f)  

(ad - be - cf, ae + bd, af +cd, bf + ce) using 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 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 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

*{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}