de Bruijn tilings

I came across this page On de Bruijn Grids and Tilings years ago, maybe 10, and always thought it was very cool. Basically, draw 5 groups of equally-spaced parallel lines, 72{}^\circ from each other, and rhombuses drawn on the points of intersection can be arranged into a tiling – Penrose’s quasi-5-fold-symmetry tiling using 2 rhombuses.

Picture 7
Picture 8
They also give a version with lines in 7 directions, using 3 rhombuses.
Picture 9
Last week I wrote a program to construct tilings, given the lines. I tried it with de Bruijn’s pentagrids and multigrids, and soon realized it works with lines in any direction, even with lines of random angle and position. So far I haven’t found any mention of people using de Bruijn’s method with anything but parallel, evenly-spaced groups of lines. Martin Gardner wrote about the Penrose tiling in the mid-1970s, with contributions from Conway, and later about de Bruijn’s methods to construct them, which date from the early 80s. It’s discussed in Branko and Grunbaum’s book on tilings from the 80s. So I’m not sure what to call it – mostly it’s called de Bruijn’s multigrid method, but the ‘grid’ means evenly-spaced lines, and this isn’t that. “de Bruijn tilings” for now, I guess – as far as I know, the idea of making tilings from the crossings of straight lines is his.

Some examples of what I’ve thought of trying so far:

3-fold symmetry
3fold - 90 lines
quasi-5-fold symmetry
quasi5fold symm

simple 5fold - 40 lines
10fold - same as 5
5-fold symmetry, 80 lines
5fold 80l

7-fold symmetry

7fold 2
7fold 3
7-fold symmetry with 110 lines (3307 shapes)
7fold - 110 lines - 3307 shapes






Grid, using 3 random angles
rand 3 angles

non-parallel lines

random lines

20 random lines
20 rand lines
20 rand lines 2

30 random lines
r30l 10
r30l 5
r30l 6
rand 30 lines 2
rand 30 lines 3

rand 30 lines
r30l 7

r30l 8
r30l 9
r30l 4
50 random lines
r50l 4
r50l 5
r50l 7 - 762 shapes
r50l 2

r50l first attempt
r50l 10 - 812 shapes
r50l 8 - 834 shapes

r50l 9 - 779 shapes

r50l 3
r50l 6

80 random lines
r80l - 2034 shapes

r80l 2

r80l 3




irr 3 symm









7-fold with shapes planted (i.e. lines meeting at a point)

decagon planted in centre
7fold - with decagon planted in centre
6-gon planted
7fold - 6gon planted at centre
8-gon planted
7fold - 8gon planted in centre
7fold - planted face

7fold - planted

[Dec 2015]

One thought on “de Bruijn tilings

  1. This is really great!

    In 2016 Steinhardt and Boyle wrote a couple of papers together where they use groups of parallel lines, but those lines are spaced according to a quasiperiodic sequence, instead of spaced evenly. However, they also choose alignments such that more than 2 lines meet at most intersections; so they end up with more complicated polygons rather than just rhombuses.

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