Nowhere-neat Tilings of the Plane
using only one or two prototiles

(nicht-passende Belegungen der Ebene mit ein oder zwei Proto-Kacheln)
Copyright Karl Scherer 2001

Definition of a "nowhere-neat tiling"
You are given polygonal tiles (i.e., tiles with straight edges).
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 tiling).

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 Tilings, Floor Tilings 2
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:
   - (4)
   - (6)
   - (3,3),
(3,4), (3,5), (3,2+4*n), (3,7), (3,8*n), (3,9), (3,10), (3,12),
(4,5), (4,7), (4,2*n) for all n > 1.
   - (5,5), (5,6), (5,7), (5,8) 
   - (6,6)
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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