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Author Topic: The space time manifold tensor candidates  (Read 8371 times)
Description: Iteration statements that capture the strain tensor of Einsteins space time
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jehovajah
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« on: May 04, 2010, 07:42:27 AM »

So, not complete but accessible and any of you are welcome to use them. The fabric of my thought turns out into these statements, and i frankly do not comprehend them.
I have not decided what is the best format for them or whether they need to be in all formats: these being iterated function systems, ordinary formulaic fractal generation, 3d graphing programme format or anything else.

So this is a work in progress,

I have not even checked the excellent QuaSZ Mac to see if Terry has not done this stuff already!  embarrass

« Last Edit: May 05, 2010, 01:06:39 PM by Nahee_Enterprises » Logged

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« Reply #1 on: May 04, 2010, 07:50:45 AM »

Conical and Cylindrical Helix

* conical helix.txt (25.63 KB - downloaded 448 times.)
« Last Edit: May 14, 2010, 09:45:44 AM by jehovajah » Logged

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« Reply #2 on: May 04, 2010, 08:15:01 AM »

So my first interesting candidate and truly fractal as I will show in developmental posts.

                          z=e^z+z*j+c+z

This moment at iteration 5 is where the underlying vortices are revealed for the first time:



A post in the Mystic Fractal Gallery details the creation of an atom from vortices interacting, so I look closer to see if one pops out here:



A closer look at the tip shows no atom,but reveals the complex motion that is being shaped here by |z|<20


 
« Last Edit: May 05, 2010, 01:10:25 PM by Nahee_Enterprises » Logged

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« Reply #3 on: May 14, 2010, 10:25:19 AM »

I have updated the text file with a major addition, in particular Terry Gintz correspondence with math doctor re quaternion hyperbolic and trig formulae. This is of significance to quasz mac users as the e^z formulae are quad formulae. Quasz sculpts  in the quad block although it visualises in the tri block. The images above are so complex because of these additional factors and certain shapes that are intuitively expected do not arise as expected through these factors also.

The extension of hyperbolic formulae to quaternions leads to the speculation

e^q= e^(x+i*y+s*j+w*k)≈e^x*e^(i*y)*e(j*s)*e^(k*w)≈e^x*y^3*(cosy+i*siny)*s^3*(cos(s)+j*sin(s))*w^3*(cosw+k*sinw)/[sqrt(y^2+s^2+w^2)]^3  

or some variation.

In any case the conditions for equality are clearly laid out in the reply.

* conical helix.txt (63.8 KB - downloaded 355 times.)
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« Reply #4 on: May 17, 2010, 02:37:53 PM »

For example, exp (i + j) = cos sqrt 2 + (i + j)(sin sqrt 2)/sqrt 2,
while exp i * exp j = (cos 1)^2 + (i + j)(sin 2)/2 + k(sin 1)^2, and
exp j * exp i = (cos 1)^2 + (i + j)(sin 2)/2 - k(sin 1)^2.  However,
whenever ab = ba (for example, whenever a is real), then
exp (a + b) = exp a * exp b does hold in the quaternions


That fact is the secret to calculating exp (a + ib + jc + kd).  Since
a commutes with ib + jc + kd (when a, b, c, and d are real),
exp (a + ib + jc + kd) = exp a * exp (ib + jc + kd). Now, you can't
break exp (ib + jc + kd) into exp ib * exp jc * exp kd, but you don't
need to.  Simply divide this vector by its magnitude. The quantity
L = (ib + jc + kd)/sqrt (b^2 + c^2 + d^2) satisfies L^2 = -1 (check it
for yourself).  And you can go back to the infinite series to see that
exp Lx = cos x + L sin x, whenever L^2 = -1 and x is real.  So let
M = sqrt (b^2 + c^2 + d^2), so ib + jc + kd = LM.  Then

exp (a + ib + jc + kd) = exp(a)*(cos M + L sin M)=exp(a)*(cos M + {ib+jc+kd}*sin M/M)
                                  =exp(a)*(cos M + {i*bsin M + j*csin M + k*dsin M } /M) 
 
So the speculation will be way off base but some weird conection may show up in the sculptures embarrass     

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« Reply #5 on: June 04, 2010, 08:02:28 AM »

Further study and a confluence of ideas leads me to the following set of motion statements for the spacetime manifold. This time reflecting Classical Newtonian conceptions

z=z^(-2)+n*Z+c^(-1)

Where n is some real constant.

http://csep10.phys.utk.edu/astr161/lect/history/newtonkepler.html

Although not properly worked out i note

{x^2\over k}+ {y^2\over n}-1=f(x,y) is analogous to

z^2 -2*(x#*y#*i-y#^2)+c =z which is analagous to

z^2 -2*y#*e^(i*)+c =z where is a real radian measure and z is the polynomial numeral x+iy

Using z=r*e^(i*) as a general form i posit
z^2 -n*z+c =z

And because i am interested in an inverse square law i suggest

z=z^(-2)+n*Z+c^(-1) as a protoform.

Alternatively

z^2 +2*y#*(flip(conj(z)))+c =z

suggesting again

z=z^(-2)+n*flip(Z)+c^(-1) as a protoform.

Of course for Quasz the z is a quad which is a polynomial numeral z= x+iy+zj+wk.

These kind of analogous manipulations are not rigorous but creative. the rigour comes later if anything interesting pops out! grin

One of the straight jackets of classical maths was the emphasis on finding a solution, that is... an arithmetic driven scheme! From this we derived cartesian geometrical forms which gave solutions but did not explore the form of the statement in any other way. This gave a solution form in which the functions in more than one variable were inexorably plotted against a "z" axis to give a 3d form. However this device is confusing the function statement with a geometrical surface. The function statement is describing a relationship between the parameters which is intuitive or derived. This relationship does not necessarily have a geometrical form. Iteration is a more general way to explore functions in many variables and the use of visualisations rather than only geometry can help to develop an intuition about the function statement . I mean in this instance the use of a colour scheme and or a sound scheme to represent the variable values as well as the extensions along orthogonal axes or the use of non orthogonal axes which we do every day in 2d diagrams! kiss
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« Reply #6 on: June 05, 2010, 09:37:38 AM »

In quasz Terry has an in built function number 37 which corresponds to the second manifold tensor candidate with n set to 0 and c instead of 1/c.

1/z^2  +  c

The first sculpture is an image of a Minkowski torus with a gravity well  structure that is complex



the second looks into the gravity well structure



the third shows the torus shape followed by 2 other views





I then take a look at the gravity well structure which passes completely through the torus but not symmetrically and the structure appears to be the result of antisymmetric action from the other side of the torus






I then show the spatial structure with the wells




The sculptures are sensitive to the seed points with the real coefficient of the seed determining the toroidal shape while the i and j  coefficients seem to determine the well structure. The anti symmetry is due to rotation not reflection which is a translation in relativistic motion i am currently cogitating.

The mandelbrots are informative especially mandelbrotp but i have no cosmological referent for the mandelbrot boundary conditions yet in mind.

Each sculpture relies on the quad motion statement in quasz and the cutting boundary |z|≤4. This parameter has much to do with the eventual form and surface structures of the sculpture and i am exploring form and structures in spaciometry.
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« Reply #7 on: June 15, 2010, 07:34:16 AM »

i*h*d(z)/dt=(-h^2/m)*(d2(z)/dt+c*z :a working  note on schroedinger's equation, not formulated yet. The time element for example is going to be replaced by an iteration counter, and the specific constants by some more general ones hopefully preserving the relationships

(b*m*c^2+(a1*p1+a2*p2+a3*p3)*c)*z=i*h*d(z)/dt : a working note on diracs equation.
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« Reply #8 on: June 15, 2010, 08:59:08 AM »

A couple of sculptures using the protoforms

z=z^(-2)+0.1*Z+c^(-1) 


z=z^(-2)+0.1*flip(Z)+c^(-1) 

These suggest the conjecture that black holes form in pairs linked by a gravitational wormhole. embarrass
And notice the black holes are not symmetric or equal in cross section, and can have a distorted event horizon.

I am just glad the form sculpts out elliptical shapes!
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« Reply #9 on: July 05, 2010, 04:12:10 AM »

Some sculpts from the schroedinger equation. Very rough translation to Quasz.



http://www.fractalforums.com/gallery/2/410_05_07_10_3_56_22_3.png 



And finally   



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« Reply #10 on: July 08, 2010, 10:23:21 AM »

So i refine the laplace to reflect its operation better and i get this



This is a sculpture of the d shells possibly.

Then i get this as an initial iteration with baiout 4



Which turns out to be what is left of this which is bailout 1000   

Which seen in context looks like this

which on iteration begins to morph like this

which when seen close looks like the bailout 4 sculpture .

Over time then the s shell fades to a weak p shell? Is the s shell correctly identified here or is it showing 2 strong p shells fading?

Is this fractal sculpture in any way near the Schroedinger equation outputs?

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« Reply #11 on: September 19, 2010, 09:34:58 AM »



Nice molecular, crystal lattice, with possible s,p and d shells showing. Inspiring!
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« Reply #12 on: October 07, 2010, 01:35:02 PM »

An archimedian spiral requires a logarithmic increase in radius per iteration, while a golden ratio type spiral requires an exponential increase.

So z= r*e^z+c and z= r*ln(z)+c should be interesting julias.
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« Reply #13 on: January 19, 2011, 10:11:11 AM »

I watched a programme called Beautiful Equations today , and it gladdens my heart to see artists and mathematicians and scientists once more united, reunited after a centuries long process of abstraction and dissociation . As human animates we are connected more than we are distinguished, and although distinctions are important they are not barriers between us, but stepping stones to different vantage points from which to view our common experience.
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« Reply #14 on: January 19, 2011, 08:29:05 PM »

In the beautiful  equations Matthew Collings is given a glimpse of a positron in a cloud chamber!

the Dirac equation was presented as
  i \gamma \rho \Psi =m\Psi

\Psi being a "twistor".

I will have a go at sculpting this.

Interestingly the cloud chamber reminded me of the dynamic magnitudes around and in me, and was the first glimpse of a dynamic relativity measure. It showed me how measurement is dynamic instantaneous, vectored, and aggregations of plethorate units which are dynamic.The particles in the cloud chamber were illuminated, drifting in Brownian motion, and their geometric relation was what was altered vectorially to show the positron path. The slight change in aggregational density was visible vectorially, a tensor relationship made visible in the clouds for an instant.
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May a trochoid of 去逸 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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