I suppose this geometric shape is something very 'special'. I cannot clarify in short about being 'special', but I think this shape stands together with such special shapes like the square and the regular hexagon. Here it is:
So it is a parallelogram based upon a square. Its height is equal to the side of square and the long side is 3/2 of the side of the square.
Please note, that the 'specialty' of this shape is a hypothesis and makes sense in context of creating low entropy 2d structures. One category of such structures shows a period close to this shape. But it can take much effort and time to prove or disprove this.
Is there a name coined for this shape? If not, what name would you choose for it?
Probably someone knows something interesting about this shape?
My thoughts for the name:
1) Proportional parallelogram -- This could be an official term. It reflects the presence of proportionality, which 2d stable structures have.
2) Quadrogram -- Somewhat shorter, but still hard to pronounce.
3) Tooth -- This is the best name for it: simple and clear.
Updated:
The name of the shape is of course not so important here. I summarize the question, concerning the properties of plane tiling.
Questions
- What is known about this shape in context of tilings?
Namely what tiling properties can be compared to tilings made up by other shapes:
- Which periods has the tiling apart from the shape self?
- What properties the tiling has after e.g. reflection or rotation are performed onto itself?
- How those periods relate to other shapes?
The relations of periods to the square shape should probably give some connection to "planar balance", which however I cannot explain, so it is only mental image, as well as many things about tilings and structures.
Some related terms are briefly described here:
http://www.uwgb.edu/dutchs/symmetry/tilterm.htm