Longer background, shorter questions below:
Tessellations of 2D plane consisting of regular polygons are usually described with vertex configurations such as "18.104.22.168" meaning that there are a regular triangle, two squares and a regular hexagon meeting at each vertex.
There are only 3 regular tessellations (consisting of only the same regular polygon): "22.214.171.124.3.3", "126.96.36.199" and "6.6.6".
Next step is semi-regular or Archimedean tessellations (consisting of multiple different regular polygons). It is usually assumed there are 8 of these: "188.8.131.52.6", "184.108.40.206.4", "220.127.116.11.4", "18.104.22.168", "22.214.171.124", "3.12.12", "4.6.12", "4.8.8". Sometimes the 9th semi-regular tessellation is separated as the "126.96.36.199.6" can be laid in two mirroring ways.
My interest is the "next stage" after this. I currently call them equilateral tessellations with same angles on all vertexes but I would like to know if maybe someone has already named them and You could point me to existing research.
For formulating this problem it is not fair to use the previous naming convention on these tessellations, as they do not consist of regular polygons, only equilateral ones. Thus a new naming convention of describing the vertex angles is needed, e.g. the "188.8.131.52" tessellation can be called "60-90-120-90" describing the angles surrounding each vertex.
It can be shown, that such tessellations with each vertex consisting of the same angles and each line being of equal length are a superset of semi-regular tessellations, and include more examples, such as "90-120-150", which looks like this:
In short my questions are:
- Do these kinds of equilateral tessellations with same angles on all vertexes have a better name already used?
- Is there a known limit of how many types of these exist? (Are there more than the ones I have mentioned?) And if not, what methods could I use to determine more existing configurations (or are there infinite amount of them)?