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:
1 Answer
$\begingroup$I haven't found anything about this shape and I'll try to answer my own question here. It is not so stricly mathematical, but shows the concept geometrically.
One possible explanation about the speciality is related to planar periodic structures, and in this case it is worth examining first the properties of the simple grid which is formed by this shape.
I take two axes (y, z) as on the picture, and perform reflection operation over these two axes and see what properties the structure achieves in each case.
Reflection over Y
The resulting structure is a highly ordered periodic structure, but with some "new" information, namely it has more shapes in its periods. And what is noticable, the new rectangular period (marked as T1 on both images) has exactly the shape of a "two-square" so it is 2:1 rectangle.
Reflection over Z
The resulting structure is again a highly ordered periodic structure, and the new rectangular period (T2) has the shape of 6 squares so it is 6:1 rectangle. Relation between area of T1 and T2 is:
s(T1) / s(T2) = 3/5
It will be too hard to show that no other parallelogram has same properties. However this all hardly shows exactly why this phenomenon is so unique.
Apart from formal classification attempts, it is of course most interesting what structures actually arise in this grid. And indeed, one such structure is very interesting and has outstanding quality, namely I would describe it as a "first class citizen" among possible planar periodic structures and here it is, with its periods shown:
