The Chaos Game: Different Rulesets

The so-called chaos game [1, 2] refers to a type of iterated map that can be used to approximate fractal patterns. In the video four popular examples, i.e. the Sierpiński carpet and hexagon as well as the fractal starfish and the fractal fern, are given. In general, the iterative procedure can be written as a set of affine transformations. From this set of transformations (ruleset) a specific one is chosen (per iteration) at random. Given a suitable ruleset, the resulting set of points lies dense in a fractal set.

[1] R. L. Devaney, Chaos Rules!, Math Horizons 12(2), 11 (2004).

[sagular]

The formula looks pretty interesting and kind of feels like it would yield an infinite spiral.
For T and for r, what matrices and what vectors are being taken into account?