fractals with real scators

Submitted by Jesus Hernandez on Tue, 05/17/2016 - 15:22
English

Real scator algebra is a non distributive extension or complex algebra to higher dimensions. The square of a scator $\overset { o }{ \varphi  } =s+x\hat { { e } } _{ x }+y\hat { { e } } _{ y }$, is

\begin{equation}\overset { o }{ \varphi  } ^{ 2 }=s_{ \diamond  }+x_{ \diamond  }\hat { { e } } _{ x }+y_{ \diamond  }\hat { { e } } _{ y }=s^{ 2 }\left( 1-\frac { x^{ 2 } }{ s^{ 2 } }  \right) \left( 1-\frac { y^{ 2 } }{ s^{ 2 } }  \right) +2sx\left( 1-\frac { y^{ 2 } }{ s^{ 2 } }  \right) \hat { { e } } _{ x }+2sy\left( 1-\frac { x^{ 2 } }{ s^{ 2 } }  \right) \hat { { e } } _{ y }.\label{eq:sca squ fact} \end{equation}

hun:corte c2i0E+1+2(-0.5;0,.3)(-0.5;-1.6,.38)3200x1800-00slice,y=0,z=0
hun set c2i0E+1+2(-0.5;0,.3)(-0.5;-1.6,.38)

The quadratic mapping for the scalar component in real scator algebra is,

\begin{equation} s_{ m+1 }=s_{ m }^{ 2 }\left( 1+\frac { x_{ m }^{ 2 } }{ s_{ m }^{ 2 } }  \right) \left( 1+\frac { y_{ m }^{ 2 } }{ s_{ m }^{ 2 } }  \right) +s,\label{eq:iter scalar} \end{equation}

and for the director components, the recurrence relationship is

\begin{equation}x_{ m+1 }=2s_{ m }x_{ m }\left( 1+\frac { y_{ m }^{ 2 } }{ s_{ m }^{ 2 } }  \right) +x,\label{eq:iter dir1} \end{equation}

\begin{equation}y_{ m+1 }=2s_{ m }y_{ m }\left( 1+\frac { x_{ m }^{ 2 } }{ s_{ m }^{ 2 } }  \right) +y.\label{eq:iter dir2} \end{equation}

The square magnitude of a real scator $\overset{o}{\varphi}$ is

\begin{equation}\Vert \overset { o }{ \varphi  } \Vert ^{ 2 }=\overset { o }{ \varphi  } \overset { o }{ \varphi  } ^{ * }=s^{ 2 }-x^{ 2 }-y^{ 2 }+\frac { x^{ 2 }y^{ 2 } }{ s^{ 2 } } .\label{eq:sca mag squ} \end{equation}

With this preamble of real scator algebra, it is then possible to perform the quadratic iteration. It is then possible to visualize the bound sets in two or three dimensions as well as the escape velocities of unbound points.

An introductory article about fractals with imaginary scators can be obtained in the following link http://148.206.32.135/mfg/arti/11-15/hypersca-1+2aaca.pdf