06 April 2012

Pythagorean tiling


The painting is Street Musicians at the Doorway of a House, by Jacob Ochtervelt (1665).  The pattern on the floor is an example of "Pythagorean tiling."
In geometry, the Pythagorean tiling or two squares tessellation is a tessellation of the plane by squares of two different sizes, in which each square touches four squares of the other size on its four sides. A tiling of this type may be formed by squares of any two different sizes.  It also is commonly used as a pattern for floor tiles; in this context it is also known as a hopscotch pattern...

This tiling is called the Pythagorean tiling because it has been used as the basis of proofs of the Pythagorean theorem by the ninth-century Arabic mathematicians Al-Nayrizi and Thābit ibn Qurra, and by the 19th-century British amateur mathematician Henry Perigal. If the sides of the two squares forming the tiling are the numbers a and b, then the closest distance between corresponding points on congruent squares is c, where c is the length of the hypotenuse of a right triangle having sides a and b. For instance, in the illustration the two squares in the Pythagorean tiling have side lengths 5 and 12 units long, and the side length of the tiles in the overlaying square tiling is 13, based on the Pythagorean triple... By overlaying a square grid of side length c onto the Pythagorean tiling, it may be used to generate a five-piece dissection of two unequal squares of sides a and b into a single square of side c, showing that the two smaller squares have the same area as the larger one.

4 comments:

  1. Lovely! I've seen such tiling patterns but, being a Maths anti-genius, I never made the connection to Pythagoras and his theorem.

    ReplyDelete
  2. I went mad over the internet, trying to find more information about mathematically interesting tiling. Here's a wiki article that links tiling to a previous TIWKIWDBI on quasicrystals:

    http://en.wikipedia.org/wiki/Penrose_tiling

    Your blog never stops being amazing.

    ReplyDelete
  3. As a wall and floor tiler I love reading about the history of the patterns I use regularly. Thanks for sharing this, my colleagues will appreciate it too

    ReplyDelete

Related Posts Plugin for WordPress, Blogger...