One of the obsessions of my teen years.

When I was in my mid to late teens I spent a lot of time working on various geometric problems for no obvious reason. As I’ve grown and learned how to think more rigorously, some of these come back to mind and I think “‍How quaint;‍” others still intrigue me.

One that I still find interesting is the following tile:

This tile has the property that every straight line drawn over the tiling touches a black portion of a tile at least once in 2.5 tile lengths. Or, as I put it in my youth, “‍If the filled-in squares are pillars your vision is obstructed in every direction no matter where you stand.‍”

I spent many hours in my youth trying to find other “‍open‍” tilings with this same property. I hoped to find one that filled in less than

2

9

of the space, or that had the filled in squares further apart, or that worked on a hexagonal tile, etc. I never found another.

With academic maturity, I can pose what I was hoping for more crisply. “‍Given a tiling of black-and-white tiles, let b be the maximum distance between two points in the tile contiguously connected by black and w be the minimum distance between two black points that are not contiguously connected. For what ratios r = 

w

b

are the tilings such that every straight line drawn over the tile touches some black?‍” The tiling above satisfies this for r ≤ 0.7071…. Are the tilings with larger ratios?

There are other interesting questions here too. for example, I proved the absence of an all-white line via exhaustive analysis of all angles. Is there a cleaner way to prove it mathematically? What would a proof that r is the maximum ratio above which no such tiles exist look like? Are there tiles with this property for every r? For a given r, how small can the longest white line segment be forced to become? How small can the tile be?

I have no real reason to care about this problem. There is no obvious application, outside of maybe a strange game environment. And yet it continues to interest me, in a passive sort of way, and that one tiling that I have found has brought me a smile each time I see it for more than half my life.