# A class of tilings with amazing visual qualities

For more examples please see my related question on MSE: Interesting tiling with a lot of symmetrical shapes

This is achieved by rotation of square grid over itself by atan(3/4). Resulting grid is a base for a whole class of beautiful tilings, see examples in the linked question. Playing with this grid I can make up many tilesets and most interesting is, that they can act in 8 directions (4 axes). Namely I can represent contiguous equal-width lines in all 8 directions and respectively all shapes formed by these lines. Resulting shapes have smooth transitions, no gaps and are 'scalable'.
On the other hand such tilesets are very simple and with low amount of unique tiles. So in other words, it is a whole nugget of interesting applications, e.g. in vision and typography.

Still I was not able to find any concrete source related to this tiling class.

So can you recommend any good sources dedicated to this phenomenon?

Update:

It has some connection to Pythagorean tiling, namely a Pythagorean tiling with 1:2 square ratio can be used to construct this tiling.

Draw lines through centers of big squares in such manner:

After drawing lines all over the plane it results in the above described tiling. Here it is painted in two colors:

These multigrids can be used to construct Penrose tilings. Here we are overlaying five sets of parallel lines. And I will stop here because you can read the article, using the grids alone try to fill out the tiling.

http://www.ams.org/samplings/feature-column/fcarc-ribbons

now MathOverflow is a research site but I will advocate this is a research level topic

• Penrose tilings are mentioned as an example of a "non-commutative" space in the book of Alain Connes.

• The spectra of these grids also have some pecular and unexplored properties. Here is an example from 2015

• This paper combines properties of visible points and Penrose tilings (from 2014)

### Amman-Beenker Tilings (Tetragrid)

Beenker's original paper Algebraic theory of non-periodic tilings of the plane by two simple building blocks : a square and a rhombus is online. And he shows you how to use the tetragrid to build the squares and rhombuses one by one.

### Muslim Architecture and Design

Tiles with 8- and 10-fold symmetry appear in Muslim (?) architecture and design as well as Arabic-influenced Spanish architecture

• Thank you for good resource links. According to this can I suggest that there is probably nothing about the particular tiling I describe? Including other suggestions on linked MSE question, it seems to be so. – Mikhail V Feb 10 '17 at 0:25

This appears to be related to the topic of tilings with multiplicity, i.e. where almost every point is covered by fixed number $k$ of tiles, see https://arxiv.org/abs/1103.3163, https://arxiv.org/abs/1405.6928 and references therein.