I awoke this morning wit a iterative formula in my mind, based on what i have been meditiating on, and discussing in recent posts on Newton. It requires the modern technology of a 3d Fractal generator that uses Quaternions as a reference frame.
It requires the Euler Cotes identity utilised as a trochoid line and surface generator , and it requires a little imagination
It is written for Quasz, by Terry W Gintz at Mystic Fractals.com
z= e^(x)*( e^(iy)+e^(jz)+ e^(kw))+0.1*c
In fact i am experimenting with 1/x in place of x to keep things within bounds , and gradually increasing the iterations .
Everything will be varied, and the julia mode used first.
In these experiements the bailout is analogos to the Reynaulds number.
By ensuring that the coefficients are measured in arc lengths i can , by logos Analogos, consider this formula as a Twistor formula as opposed to a vector formula. The difference is not in the form, but the magnitude!
a full Twistor would be defined as
T- radius*e^(arc magnitude*i)
Where i is the direction of rolling of the centre for radius. Imagine a disc rolling along a surface in the direction i, defined by the direction in which the centre of the disc moves.
Often we can ignore the direction of rolling if we want to simulate a fixed rotation, but the notation actually is a motive formula and this should not be forgotten.
At some stage, early in my exploration of the 3d mandelbrot, i made this exact smae analogy between the formula and a description of motion. Now i will make a formal definition in terms of the Newtonian idea of Motive, a type of fluid energy which in Quasz shows up as black space for high energy, or high acceleration or high velocity or for low Reynauld number. The notion of viscosity and density are analogous and so a analogos logos description is apt.
I have now generated an expected result, in accordance with the philosophical meanderings hithwerto. Thus i feel confident that Newtonian Motive is a viable system for modelling Spacematter, the aether, spacetime Manifolds or whatever we want to cal it. The Logos Analogos method of Eudoxus is responsible for this computational ability to model things we only dream of .
The Arithmoi that was later developed will produce a SIMILAR result, because Arithmos is a muultiple form based on configured monads or metrons as opposed to configured straight or circular lines, but the important relationship is the logos analogos one. 
Newtons corpuscles as analogies of Maxwell's vortices, are vindicated in this extraordinary sculpture.
http://userpages.monmouth.com/~chenrich/Trochoids/Trochoids.html
T = (1-w)e^iθ + ρwe^i(1-1/w)θ .
This is the relation i want to explore.
Thus algebraic formulation is based on polar coordinates. The logos Analogos is constructed in this type of mosaic.
Mandelbrot formula is
z = exp(2*(lnr +iø))+c
in polar
variations in the theme are
z=exp(m*(lnr +iø)) + exp(n*(lnr +jΩ))+exp(q*(lnr +kç))+c
z = exp(m*z)+c
z = exp(m*(lnr +iø))+exp(n*(lnr +iç+jΩ))+c
etc, varying every parameter, fixing the arc length proportions to mimic the trochoid formula ;
varying the proportional {logos Analogos} in the definition of Z;
using points, line segments, plane segments, solid segments a la Grasmann Algebras, and Geometric algebras in the Hamilton quaternion reference frame. Twistors and plane twistors and solid twistors;
logarithmic form of the relationship a la Cotes De Moivre formulae.
I want to try
z=exp(m*(lnr)*exp(m*(iø))*exp(n*(jΩ))*exp(p*(kç))/(i+j+k)^(mnp)
Also z= (exp(m*(lnr+i*ø))+exp(m*(ln(v*(1-r)) +(1-1/1-r)*i*ø))+(exp(m*(lnr+j*Ω))+exp(m*(ln(v*(1-r)) +(1-1/1-r)*j*Ω))+(exp(m*(lnr+k*ç))+exp(m*ln((v*(1-r)) +1-1/1-r)*k*ç)) + c
and variations
i programmed this into Quasz
<exp(1*(rl+y#*i))+exp(1*(1*rs+rf*y#*i))
[rfun
s=imaj(z)
w=imak(z)
rl=ln(3-x#)
rs=ln(1*x#)
rf=3-3/x#
rend
Ti=(exp(1*(rl+y#*i))+exp(1*(rs+rf*y#*i))
Tj=(exp(1*(rl+s*j))+exp(1*(rs+rf*s*j))
Tk=(exp(1*(rl+w*k))+exp(1*(rs+rf*w*k))
z=Ti+Tj+Tk+0*c>
