Some years ago, while staring at a linoleum tile floor in English class, a question with no real world application struck me. I haven’t slept since.
The tiles were squares and laid out like a checkerboard with alternating white and green tiles. My question is this:
Given four tiles that are perfect squares and are layed out in a perfect 2 x 2 formation (like a checkerboard) do the diagonal tiles touch each other?
If yes, how can both pairs (green and white) touch each other?
If not, what keeps them from touching? Empty space?
Please, someone give me the straight dope. I’m tired.
Scott Stevenson
–Two wrongs may not make a right, but two Wrights made an airplane.
No.
The floor tiles are not perfect squares. The edges are ever so slightly blunted during manufacturing, transport and installation. Hence, there is a small gap where the four corners meet. Also, the tiles are tipped into place, which will cause a gap when the tile that’s being installed rocks away from the adjacent tiles and into place. Contact cement, along with air, may separate the tiles after it squeezes out from underneath.
Yes.
While good, the people who installed the tiles are not perfect. The floor tiles may not be laid out in a perfect pattern, and occasionally, two tiles of the same color might meet. Alternately, there may be enough of a burr at the corner of some tiles to bridge the gap (though I doubt it.)
Looking down at the vinyl-asbestos tiles in my living room, I see the dark line of a gap between any two tiles, be they prefectly aligned or not. The tiles may touch below the crud in these gaps (old floor, not dirty), but I won’t excavate to find out. I have limits on what I will do in the name of knowledge.
In the world of geometry, as opposed to the real world, the corners of the squares are infinitely small, i.e. single points. (I’m using 2-d tiles, since it doesn’t make a difference.) So what you’re asking is, can four points all touch each other at the same time?
I don’t know. Infinitely small points are funny things. This sort of question is why pure mathematicians are weird.
