A Man, A Plan, A Periodic Pentagonal Pavement

Your wife is remarkably indulgent of your mathematical fancies.

Knowing her, this doesn’t surprise me.

Thanks for posting these! Like everyone else, that part of your last post caught my interest.

OK – I admit it does look nice, much better than I imagined.

Squashed hexagons with a pleasing, symmetrical, four-part subdivision.

One of the legacies of Early Islam, is a fascination with different tilings. They were diligent at not portraying animals, etc, lest they be tempted to idol worship, and so they excelled at periodic tillings of all kinds and sorts, with every kind of periodicity imaginable. And some that were surprises.

May I have your permission to use your ‘pentagon shower picture’ (with due credit to you of course) on my website?:
http://www.tess-elation.co.uk/cairo-tiling/miscellaneous-1
This pentagon tiling shown is known as the ‘Cairo tiling’, a paving of pentagons (so named after it originating in Cairo), with picturesque properties, and of interest to mathematicians:
http://en.infogalactic.org/wiki/Cairo_pentagonal_tiling
Whether you know of this attribution is not clear. All things ‘Cairo tile’ are one of my interests, hence the desire to add this somewhat unusual sighting on my page.

Welcome, Mr. Bailey. The photo above is of a bathroom floor. A tub is visible on the left, and a closet flange for a toilet can be seen. This tessellation is a little different from the Cairo tiling. The Cairo tiling is said to have pentagons with two 90 degree angles and three 120 degree angles, and the base of the pentagon is a different length than the other four sides. The top figure above shows the geometry of my tiles: They are equilateral, and instead of 120 degree angle corners, the two on each side of the base are approximately 114.3 degrees, and the angle opposite the base is approximately 131.4 degrees. So, I think this bathroom floor would be considered Cairo-like, but not quite Cairo.

You are welcome to use my photos of this pentagonal paving on your website. Let me know if you wish any other information about my floor.