WOLFRAM|DEMONSTRATIONS PROJECT

Nowhere-Neat Tilings of the Plane

​
tiling
NN-3-3
(number
1
of
228
)
select
<<
<
-
>
>>
do
-
default
copy
paste
point grid
iso
on top
snap
​
zoom
x 1
x 2
x 4
x 8
←

↑
↓
axes
outline
opaque
show
part+
combined
stored tiling
grid of tiles
1
of
3
, using
3
-gons
action
-
save
=+
dbl+
r
d+
del+
1+
color
■
■
■
■
■
■
■
■
■
■
■
tiles from
-3
to
3
tile displ: h
9
v
3
rows from
-6
to
5
row displ: h
0
v
6
--------------prototile--------------
group
1
rotation+
90
size+
*2
*3
*4
/2
/3
/4
​
prototile+
-
⟷
↕
←

↑
↓
drag+
pivot
use origin
point size
create, delete vertex with (command+click)
If no two tiles in a (polygonal) tiling have a full side in common, the tiling is called nowhere-neat.
Related mathematical problems: to find nowhere-neat tilings of
n
-gons with
m
-gons and to find nowhere-neat tilings of the plane with only a few prototiles.
The Demonstrations "Nowhere-Neat Tilings" and "Nowhere-Neat Squaring the Square" by the same author were dedicated to the first of these two problems.
This Demonstration deals with the latter case of covering the infinite plane in a nowhere-neat way while using only one or two prototiles. In fact, it only deals with tilings on the integral grid.
There are over 200 tilings shown in this Demonstration; maybe you can find a few more.