[1] actually, the plaine of complex numbers.
[2] a so called iteration formula.
[3] e.g. if the series "jumps in circles", i.e. reaches a point it has already visited. If this happens, the series can only repeat the same subsequence again and again.
[4] Zi+1 = Zi * Zi + C, whith C being the starting point and Z starting with zero.
[5] picture 52, for example, has the forumla Zi+1=(Abs2(Zi)+P1)^atan(Abs2(Zi)+P2)+C; with Abs2() being a function which assignes the real- and imaginary part its respecive absolute value, and P1 and P2 being constant complex numbers.
[6] actually rather "a structure".
[7] constructivism or platonism?