Occluding pillars
other posts

math

Airy tiling patterns that obstruct all lines

Almost two decades ago, while playing about with graph paper creating maps for games, I stumbled upon the following pattern:

This arrangement of filled squares in a grid has the property that every straight line that passes from one side of it to the other touches some filled square. If these represented pillars, the wall of would completely obscure all lines of sight.

For some reason that property tickled me, and from time to time over the following years I would play around with the concept of sight-occluding pillars. That pattern only worked in one direction, and had an asymmetry that felt less than elegant. Could I do better?

A few years later I came across this six-by-six tile:

When tiled to fill space, it occludes in every direction: every straight line at any angle touches a colored tile.

That arrangement of square tiles has
 8 36
=
 2 9
≈ 22% of the tiles filled, and the longest line segment that does not touch any filled tile is just under 5√10 (about 15.8) square-widths long. It is symmetrical, which appeals to me, and it also is minimal More formally, the convex hulls of the filed regions cannot shrink and maintain their function. in the sense that shaving off any corner of any filled tile breaks the occluding property.

For some reason, this pattern continues to grab my fancy more than fifteen years later. It appears in doodles I have created across that time, has been my desktop wallpaper, has been used as the foundation of mazes I’ve put players through in tabletop roleplaying games, and is one of my go-to backgrounds when trying out new rendering engines. I enjoy it, for no reason beyond its simply being.

Throughout those years I would occasionally seek out other patterns with the same sparse-but-occluding property, but never very earnestly nor with much success. However, just last month I found another such tile, even sparser and based on a hexagonal tile—or more accurately, a triangular tile with clusters of six filled triangles, but I find the hexagonal superimposition to be easier to remember even though the filled regions are not themselves full hexagonal tiles.

Like the square tile, if this is tiled every straight line touches a filled region.

This pattern is only
 1 6
≈ 17% filled,
 3 4
the density of the square pattern; the longest line segment that does not touch a filled region is a bit under 23 hexagon-widths long; and like the square tile, shaving off any corner breaks the occluding property.

I now know of two of these symmetric tiling airy occluding pillar fields, in addition to several symmetric and asymmetric pillar walls (I don’t find those as interesting so I’m only including the one in this post). More hours than I care to consider have gone into finding and analyzing them. And so far as I know, they have no purpose at all.

Just as a bonus, here are some of the render tests I’ve done using these tiles:

Square
Hexagonal