Move the central linkage and whatch which faces are highligted in the 3D solid.
This anamation may be hard to follow at first glance and require multiple layers of investigations.
It illustrates an isomorphism between the 24 hyperbolic pentagons that form the realization space of equilteral pentagons
and the Dodecadodecahedron, a self intersecting 3D solid with 24 pentagons (half of them realized as pentagon stars).
We resemble the linkages by their associated juzus (the edge directions), since this makes it a little easier to see the combinatorics in each cell.
In both objects each pentagon edge has exactly two neighboring pentagons. Moving the point in the hyperbolic tiling highlights the corresponding faces in the Dodecadodecahedron. The vertices of the 3D solid are colored in 5 different colors. The vertices of a single color forms the vertex set of an octahedron.
There is an isomorphism between the order of the colors in the juzu corresponding to a face in the hypervolic tiling and the ordering of the colors in the faces of the 3D solid. If the vertices of a face are traversed along its edges, the corresponding "skip-1" sequence is the the sequence of the associated juzu.
Hence if in the face the order of the colors while following the edges is red-green-purple-yellow-blue,
the corresponding juzu is colored red-purple-blue-green-yellow. The orientation in which the faces are traveresed alternates in a checkerboard like pattern.