Friday, April 27, 2012

Geometry in the Tiling Patterns All Around Us

I took this picture of a section the floor of the hospital where I volunteer and fortunately I wasn't dragged to the psych ward. Students see tiling patterns every day yet rarely think of applying their knowledge of geometry.

Assume each white square has side length 2 and that the shaded square is obtained by rotating one of the white squares 45 degrees.

Show that the overlap is a regular octagon of side length 2√2 - 2.

Sent from my Verizon Wireless 4GLTE Phone


Mr. Chase said...

It also makes me wonder if the person tiling the floor was able to cut the four white tiles in a perfect way, such that the little triangles that were removed could be used in the open corners.

Dave Marain said...

Interesting question...
Now I'm curious about how the tile pattern was manufactured to begin with. Perhaps these weren't individual tiles at all, rather it was vinyl linoleum flooring. When I go back next week, I'll inspect the floor more carefully and get back to you. Does the picture show where the seams are? My vision is not strong!

Also I should have stated that the rotation was about the center of the square even if "obvious ".