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
–Two wrongs may not make a right, but two Wrights made an airplane.
Are you talking actualities here or theories? In actuality, most tile configurations have an intermediary connecting material (grout). Even if there was no grout, most linoleum floor patterns have tiles of one color overlapping - which causes a gap between tiles of the other color.
Based on your other questions, I’ll assume you’re talking theories:
Let’s assume we’re talking ideal tiles. These tiles come together at a point. A point is infinitessimally small; so small that a physical measurement has no meaning. As you bring the four adjacent corners of each tile together at this point, the adjacent corners make point contact. This holds for horizontally, vertically, or diagonally adjacent corners.
If you want to try and visualize what’s going on, envision that each corner ends with a single atom and these four atoms can touch each other simultaneously…
Of course, if you look close enough, even at the atomic level, we find a gap because atomic structure is less than ideal… but if you’re looking that close, I’d have to point out that none of the individual atoms within a single tile are touching either…
…but, yeah, I would say all four tiles touch each other–at a single point (assuming everything is “perfect”).
In reality, either the two white squares touch, or the two green squares touch, but not both pairs are touching, because the alignment won’t be “perfect”. In reality, I don’t think it’s possible for two tiles to “touch at a point”, there will inevitably be some (small) positive length of contact, and that length of touching between, say, the two white tiles will prevent the two green tiles from touching.
We’re assuming an idealized notion of “perfect” alignment, however, and in that case it is possible for all four tiles to touch at a point (which has zero length).
Do Utah and New Mexico touch; similarly, do Colorado and Arizona? This brings up mapping and the four-color theorem (sp?).
With more lines, even more areas could be touching at the same point a la the “Four Corners” states. With only 4 colors, obviously some touching areas would have the same color. So only areas whose border is not infinitessimally small are considered bordering for the four-color theorem.