# Name and number of “equilateral tessellations with same angles on all vertexes”

Longer background, shorter questions below:

Tessellations of 2D plane consisting of regular polygons are usually described with vertex configurations such as "3.4.6.4" 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): "3.3.3.3.3.3", "4.4.4.4" 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: "3.3.3.3.6", "3.3.3.4.4", "3.3.4.3.4", "3.4.6.4", "3.6.3.6", "3.12.12", "4.6.12", "4.8.8". Sometimes the 9th semi-regular tessellation is separated as the "3.3.3.3.6" 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 "3.4.6.4" 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:

1. Do these kinds of equilateral tessellations with same angles on all vertexes have a better name already used?
2. 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)?
• As long as the shapes surrounding the vertex add up to 360 degrees, they may form a tessellation. – Puffy Jun 29 '18 at 11:15
• @Puffy- not sure, if I understand Your comment - many counterexamples of potential angle combinations can be found where they add up to 360, but don't form a tessellation (with equal length lines) – Krišjānis Nesenbergs Jun 29 '18 at 11:55

Tilings where the vertex arrangements (i.e. each vertex with its incident edges) are all the same are called monogonal. This is analogous to the term monohedral for tilings that use a single type of tile shape. Note that the edges do not need to be all the same length, but their arrangement around each vertex must be the same.

A tiling is called isogonal if the tiling as a whole has symmetries that map one vertex onto any other vertex. This is analogous to isohedral, and similarly the term corresponding to 2-isohedral is 2-isogonal, where the symmetries of the tiling as a whole splits the vertices into two equivalence classes, etcetera.

So the tilings you are interested in are monogonal and equilateral (and presumably periodic).

In Grunbaum and Shephard's brilliant book Tilings and Patterns, they classify all isogonal tilings, finding 93 types, two of which need extra markings on the faces or edges to distinguish their symmetries from the others. I don't know how many of these can be equilateral.

I don't think much is known about other monogonal tilings (i.e. those that are k-isogonal for k>1), and it seems to me that the situation is much the same as with monohedral tilings in that it is hard to find them all without a lot of computer assistance.

Edit:

All 93 aforementioned isogonal tiling types are topologically equivalent to semi-regular tilings, so if you impose straight edges of the same length, most of them revert to those semi-regular tilings. A few however have a degree of freedom remaining that allow some of the faces to be distorted into non-regular shapes. The one in the question corresponds to type IG88, and if I have not overlooked any, there are 15 other periodic equilateral isogonal tilings, as shown in this drawing:

Note that in some cases it was easier to use angles of $60/120/180$ in the drawing, but be aware that any angle could be used.