Nowhere-neat Tilings of the Plane
using only one or two prototiles
Belegungen der Ebene mit ein oder zwei Proto-Kacheln)
Copyright Karl Scherer 2001
Definition of a "nowhere-neat
You are given polygonal tiles (i.e., tiles with
In a 'nowhere-neat' tiling two tiles never have
a full side in common.
This is the total opposite of most classical tilings
found in churches etc, where the tiles always fit side to side (neat
We are especially interested in those tilings which
only use one or two types of tiles (prototiles).
We will see that some very interesting patterns
emerge from the unusual conditions of 'nowhere-neat'.
There seem to be not too many types of nowhere-neat
tilings of the plane using one or two prototiles, hence it might
be possible to list them all.
Links to my associated Zillions games : Floor
Link to my associated Wolfram demonstrations: Nowhere-Neat
Tilings of the Plane, Nowhere-Neat
Tilings of the Plane
Note that only those tilings are presented here
which use the integral grid plus diagonals or the 60-degree latice.
More examples with unusual angles along with some research can be
found in my book 'New Mosaics'.
(See also "NUTTS And
Other Crackers" and my pages on nowhere-neat tilings of squares
and rectangles. )
Solutions have been found for polygonal prototiles
with the following number of sides:
- (3,3), (3,4), (3,5),
(3,2+4*n), (3,7), (3,8*n), (3,9), (3,10),
- (4,5), (4,7), (4,2*n)
for all n > 1.
- (5,5), (5,6), (5,7),
Can you find some more types? It is unknown
whether there are any.
Type (5,7) was found as late as March 26, 2007 !
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