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.
Reposted from 2012 to accompany a related post.
Lovely! I've seen such tiling patterns but, being a Maths anti-genius, I never made the connection to Pythagoras and his theorem.
ReplyDeleteI 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:
ReplyDeletehttp://en.wikipedia.org/wiki/Penrose_tiling
Your blog never stops being amazing.
Thanks, Paulo.
DeleteAs 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
ReplyDeleteThis is nothing to do with the current post but I couldn’t find any other way to get it to you. I saw the guy commenting “Codex” a while back and I think you were also curious as to why. Well, believe or not Behind the Bastards just did a whole look into it: https://overcast.fm/+AA4cUOOdlNM
ReplyDeleteThought you might be interested …
Thank you, Tindrumfire. I went to the link, and started to listen, but podcasts are so linear without a transcript and you didn't offer a timestamp of the location, so I tried to listen at 1.8X with some 30 second skips, but still had to give up. But the subject matter was the Turing test, so I asked AI (!!!) and received this response: "In the world of AI, Codex was the foundational OpenAI system trained on both natural language and billions of lines of public code, which powered tools like GitHub Copilot. The architecture for autonomous coding and development continues to advance in models like GPT-5.3-Codex, blurring the lines of human-machine boundaries."
DeleteI presume that it the type of content you were referring to.
BTW, my blogs email address is listed in the right sidebar of the front page in the "About me" section. You may not be able to view the right sidebar if you're visiting on a small screen phone. Try a fullscreen laptop or desktop.
Thank you for the heads-up.
Stan
I'm going to try this pattern on a patio I have to do. Unfortunately, after a few beers, I can just see myself explaining the layout and the Pythagorean theorem
ReplyDeleteCodex: Not sure if it has anything to do with the username I picked that refers to old manuscripts. However, I am concerned about current technology and using old analog names for discardable digital technology.
ReplyDeleteAmazing! Thanks!
ReplyDelete