fractals

Submitted by mfg on Sun, 05/08/2016 - 18:49
English

The quadratic mapping can be performed with hypercomplex scator numbers in $1+n$ dimensions.

 

ix set
ix set c2i0S-1+2(-.67;.007,-.25)(-1.91;-.002,.3)

Notation used to label visualizations:

\[ \underset{\textrm{confined quadratic iteration}}{\underbrace{c2i}}\overset{\textrm{parameter/dynamical space}}{\overbrace{0/\left(s_{i};x_{i},y_{i}\right)}}\underset{\textrm{algebra and dimension}}{\underbrace{\mathbb{S}_{\pm}^{1+2}}}\overset{\textrm{fractal location}}{\overbrace{\left(s;x,y\right)}}\underset{\textrm{viewpoint}}{\underbrace{\left(p_{0};p_{1},p_{2}\right)}} \]

  • $\textbf{c2i}$  $\textbf{c}$onfined $\left\{ \boldsymbol{2}\right\} $quadratic $\textbf{i}$terations, (that can be generalized to $\textbf{cpi}$ for a pth power polynomial or p$\rightarrow \textbf{func}$ for other $\textbf{func}$tion's mappings)
  • followed by $\textbf{0}$ if the initial value of the variable is set to zero (set depicted in parameter space) or the initial value $\left(s_{i};x_{i},y_{i}\right)$ at which the constant is fixed (set depicted in dynamical space)
  • followed by the number system: $\mathbb{R}$ real, $\mathbb{C}$ complex, $\mathbb{H}$ hyperbolic, $\mathbb{S}_{-}^{1+n}$  imaginary scators or $\mathbb{S}_{+}^{1+n}$  real scators (in $1+n$ dimensions), etc.
  • followed by the fractal location or plane in 2D $\left(s;x,y\right)$ that is being depicted.
  • followed, if necessary, by the viewpoint $\left(p_{0};p_{1},p_{2}\right)$.
  • in 3D renderings, fractal location and viewpoint are stated.

There are two distinct fractal family sets: one produced with imaginary scator algebra and the other generated with real scator algebra.