Four Symmetric Sierpinski Relatives
There are 8 symmetric Sierpinski relatives. The four videos below illustrates the construction of those symmetric relatives using just rotations (transformations 1, 2, 3, and 4) where the lower left piece is rotated by 180° (transformation 3).
Relative 131
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The IFS for relative 131 is
$$\begin{array}{l} {f_1}({\bf{x}}) = \left[ {\begin{array}{cc} {0.5} & 0 \\ 0 & {0.5} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{c} 0 \\ {0.5} \\ \end{array}} \right] \\ {f_2}({\bf{x}}) = \left[ {\begin{array}{cc} {-0.5} & 0 \\ 0 & {-0.5} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{c} {0.5} \\ 0.5 \\ \end{array}} \right] \\ {f_3}({\bf{x}}) = \left[ {\begin{array}{cc} {0.5} & 0 \\ 0 & {0.5} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{c} {0.5} \\ 0 \\ \end{array}} \right] \\ \end{array}$$
Relative 234
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The IFS for relative 234 is
$$\begin{array}{l} {f_1}({\bf{x}}) = \left[ {\begin{array}{cc} {0} & -0.5 \\ 0.5 & {0} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{c} 0.5 \\ {0.5} \\ \end{array}} \right] \\ {f_2}({\bf{x}}) = \left[ {\begin{array}{cc} {-0.5} & 0 \\ 0 & {-0.5} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{c} {0.5} \\ 0.5 \\ \end{array}} \right] \\ {f_3}({\bf{x}}) = \left[ {\begin{array}{cc} {0} & 0.5 \\ -0.5 & {0} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{c} {0.5} \\ 0.5 \\ \end{array}} \right] \\ \end{array}$$
Relative 333
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The IFS for relative 333 is
$$\begin{array}{l} {f_1}({\bf{x}}) = \left[ {\begin{array}{cc} {-0.5} & 0 \\ 0 & {-0.5} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{c} 0.5 \\ {1} \\ \end{array}} \right] \\ {f_2}({\bf{x}}) = \left[ {\begin{array}{cc} {-0.5} & 0 \\ 0 & {-0.5} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{c} {0.5} \\ 0.5 \\ \end{array}} \right] \\ {f_3}({\bf{x}}) = \left[ {\begin{array}{cc} {-0.5} & 0 \\ 0 & {-0.5} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{c} {1} \\ 0.5 \\ \end{array}} \right] \\ \end{array}$$
Relative 432
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The IFS for relative 432 is
$$\begin{array}{l} {f_1}({\bf{x}}) = \left[ {\begin{array}{cc} {0} & 0.5 \\ -0.5 & {0} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{c} 0 \\ {1} \\ \end{array}} \right] \\ {f_2}({\bf{x}}) = \left[ {\begin{array}{cc} {-0.5} & 0 \\ 0 & {-0.5} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{c} {0.5} \\ 0.5 \\ \end{array}} \right] \\ {f_3}({\bf{x}}) = \left[ {\begin{array}{cc} {0} & -0.5 \\ 0.5 & {0} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{c} {1} \\ 0 \\ \end{array}} \right] \\ \end{array}$$
