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).
The IFS for relative 131 is
$$\begin{array}{l} {f_1}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} {0.5} & 0 \\ 0 & {0.5} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} 0 \\ {0.5} \\ \end{array}} \right] \\ {f_2}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} {-0.5} & 0 \\ 0 & {-0.5} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} {0.5} \\ 0.5 \\ \end{array}} \right] \\ {f_3}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} {0.5} & 0 \\ 0 & {0.5} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} {0.5} \\ 0 \\ \end{array}} \right] \\ \end{array}$$
The IFS for relative 234 is
$$\begin{array}{l} {f_1}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} {0} & -0.5 \\ 0.5 & {0} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} 0.5 \\ {0.5} \\ \end{array}} \right] \\ {f_2}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} {-0.5} & 0 \\ 0 & {-0.5} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} {0.5} \\ 0.5 \\ \end{array}} \right] \\ {f_3}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} {0} & 0.5 \\ -0.5 & {0} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} {0.5} \\ 0.5 \\ \end{array}} \right] \\ \end{array}$$
The IFS for relative 333 is
$$\begin{array}{l} {f_1}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} {-0.5} & 0 \\ 0 & {-0.5} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} 0.5 \\ {1} \\ \end{array}} \right] \\ {f_2}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} {-0.5} & 0 \\ 0 & {-0.5} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} {0.5} \\ 0.5 \\ \end{array}} \right] \\ {f_3}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} {-0.5} & 0 \\ 0 & {-0.5} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} {1} \\ 0.5 \\ \end{array}} \right] \\ \end{array}$$
The IFS for relative 432 is
$$\begin{array}{l} {f_1}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} {0} & 0.5 \\ -0.5 & {0} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} 0 \\ {1} \\ \end{array}} \right] \\ {f_2}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} {-0.5} & 0 \\ 0 & {-0.5} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} {0.5} \\ 0.5 \\ \end{array}} \right] \\ {f_3}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} {0} & -0.5 \\ 0.5 & {0} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} {1} \\ 0 \\ \end{array}} \right] \\ \end{array}$$
