Sittin' and Thinkin'

Karen and I enjoyed a pre-Anniversary getaway last weekend. I won't bore you with the details or a droning travelogue, but I did want to share one photo that I found particularly thought-provoking:

That's the tile in the floor of our hotel bathroom. As I had the opportunity to contemplate it, I got to thinking about that extra filler tile (marked in green in the photo below), and wondering how you'd lay out tiles to avoid that awkwardness. As a result, I figured out something that I imagine every professional tile layer learns their first day on the job, but it was news to me.

Or it's possible I totally knew this before, forgot it, and rediscovered it. That seems likely. Doesn't matter; forgetfulness keeps things fresh.

The square tiles are all the same size. Those on the inside of the pattern are turned 45 degrees from those on the outside (while others are cut in half or quarters to fill the gaps). They don't space out evenly because the diagonal of a square is (naturally) longer than the length of its sides. In fact, as we all remember from Pythagoras, they're wider by the length of their side times the square root of two.

Since the square root of two is an irrational number whose decimals never repeat or end, the straight and diagonal tiles can never come out exactly even. However, we craftsmen in the tile-laying trade know that a little extra grout covers a lot of sins. How close can we get?

Because the square root of 2 is about 1.414, the first obvious answer is that if you laid out 141 straight tiles next to 100 diagonal tiles, you'd be off by four-tenths (0.4) of a tile. Increasing the separation between each of the 100 straight tiles by only 0.004 of a tile--just a skosh--would fill in the difference.

But we can do better.

Reduce the fraction from 141.4/100 to 14.14/10. How's that look? Pretty good!

You're left with just a little piece of diagonal tile hanging over the edge--again, a pretty easy imbalance to fix simply by spreading out the 14 straight tiles a bit.

Now the thrill of discovery was making me light-headed. Could I do even better? Dare I try?!

Simplify 14.14/10 to 7.07/5. What have we got?

If anything, that's even better! It's . . . it's so elegant! So beautiful! I began to weep. Luckily I had two-ply nearby.

Now here's the part that blew my mind and convinced me this had to be a blog post. I counted how many tiles wide the actual bathroom was. Six. Six straight tiles wide, plus that filler tile. The poor blighters almost made it! But wait . . . There's that border of skinny tiles on the right. And the more I looked at that border, the more convinced I was that if they'd just omitted it, they could have laid a seventh tile and gone all the way:

NOOOOO! So close! They were headed for the end zone and fumbled on the 1-yard line!

Karen and I have been married long enough that she barely even rolled her eyes when she saw our photos uploading and asked why I'd taken a picture of tile. She muttered something about knowing what she was in for when she married me. And she married me anyway.

I'm a very lucky man.