**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. **

Galleries:

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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