Explanation of using wallpaper groups with vertex figure for k-uniform tilings In my project I need to implement various uniform tiling of a 2D-plane, so some time ago I started to dig a little bit into sources related to subject.
From what I understand, any k-uniform tessellation can be represented as a combination of vertex configuration (vertex figure by extension) and one of wallpaper groups. I understand the concept of vertex configuration, which is a sequence of convex regular polygons "around" one vertex I suppose. I also "kind of" understand the wallpapers group on paper. But with all examples that I found in internet, I really understand the implementation of wallpaper groups only with some sort of triangle/quadrangle pattern.


What I refer to as vertex figure (red frame), vertex configuration (blue frame) and wallpaper groups (green frame)

What exactly I need to translate/rotate/reflect and how? How is determined the length by which I translate my tiles up and down on grid? I just need to take an image of vertex figure (that created by vertex configuration), cut off empty spaces and consider the center by this picture? Or I should consider that pivot point and symmetry line of tile is center of one of the figures of vertex figure? Or maybe I need to create a polygon/planigon with sides that match the values of vertex configuration, then tesselate it with wallpaper group, and put the appropriate shapes at the vertices of those tiled polygons? And if I have to tile the planigons, then I just don't know what I'm supposed to do with 2+ uniform tilings.
I'm afraid that my whole understanding has turned into a mess and I would be very pleased to see a step-by-step example of this whole implementation on base level.

*

*I will be very glad if you can explain to me with example all the steps that need to be done to get tiling (what exactly to tile, where is the center of the vertex figure, how much should I translate, how much should I turn, etc). For example, with (4.6.12) vertex configuration and [p6m] wallpaper group.

*I will be especially glad if you do the same with any of 2+ uniform tiling.

 A: To quote from the paper An integer representation for periodic tilings of the plane by regular polygons:

In this symmetry approach to representing tilings, the key to
satisfying the requirements above is to balance the amount of
geometric and topological data with the complexity of the replication
process, to ensure that a given region of the plane is covered without
repetitions. Moreover, finding the symmetries of a tiling
algorithmically requires a computational representation of the tiling.
Our approach avoids this dilemma and strikes a balance by using the
simplest topological elements, the vertices of the tiling, and the
simplest symmetries, the translations of the tiling. Then the only
geometric data required are coordinates for [seed] vertices and translation
vectors.

In other words, it's much simpler to represent periodic tilings of regular polygons using translations only, even if some data would follow from other symmetries, because representing and using those symmetries is not that simple.
The paper comes with a large database of tilings of the plane by regular polygons that is the state of the art as far as we know. Get the paper, the data, and play with a demo here. See an independent demo here.
[Full disclosure: I'm a coauthor of that paper]
