Either move the white point in the middle of the linkage or drag the entire diagram.
Being able to write this animation is the ultimate goal of the paper: Associate to every point in the hyperbolic plane a unique equilateral pentagon. This should be done in a way that is as smooth as possible.
Whenever you cross the border of a line of the (5,4)-tiling the combinatorial type of the linkage changes.
The article creates a conformal and highly symmetric way to achieve this smooth mapping.