Nowhere-Neat Tilings

If no two tiles in a tiling have a full side in common, the tiling is called nowhere-neat, a term introduced by the author of this Demonstration in his 1994 book Nutts and Other Crackers on geometrical problems in the plane, and more deeply researched in his 1999 book New Mosaics.

The main mathematical problem connected with nowhere-neat tilings is:

(a) to find nowhere-neat tilings of -gons with -gons, and

(b) to find all nowhere-neat tilings of the plane using only very few prototiles.

Many interesting results surfaced from research on (a) and (b).

Part (a):

This Demonstration shows a few examples of part (a).

Related games: Zillions games "Square the Square", "Square the Square II", "Square the Rectangle".

The main mathematical results for (a) are on tiling a square or rectangle with smaller squares in a nowhere-neat way:

Theorem on Nowhere-Neat Tilings of Squares with Squares
A square has a nowhere-neat tiling if its sidelength is 11, 16, 18, 19, 20, or greater
than 21. The tiling can always be chosen to contain the unit square 1×1 as a tile.
Additionally, the tiling can be chosen to be fault-free, except for the case = 22.

(By Karl Scherer; proof published in The Journal of Recreational Mathematics, 32(1), 2003–04 pp. 1–13. Patrick Hamlyn solved the cases = 18, 22, and 24. The proof is also attached to the Zillions games "Square the Square" and "Square the Square II".)

In 2005 the author also proved a similar theorem for squaring rectangles:

General Theorem on Nowhere-Neat Tiling of Rectangles
There is such a nowhere-neat tiling as long as the rectangle is sufficiently large. To be precise, there is a fault-free nowhere-neat tiling for each with and . Additionally, each such solution can be chosen to contain the 1×1 square and hence is not an enlargement of a smaller tiling.

(Karl Scherer, proof attached to his Zillions game "Square the Rectangle".)

Similar theorems (by the same author) hold for no-touch tilings.

It is the author's conjecture that all rectangles and squares of size 22×22 or larger have a fault-free nowhere-neat tiling that is not an enlargement of a smaller tiling. This conjecture is based on the fact that all rectangles of size 22×22 up to 50x50 have such a tiling.

Part (b):

Several hundred interesting nowhere-neat tilings of the plane using only one or two prototiles are shown in the Zillions games "Floor Tilings" and "Floor Tilings II" by the author of this Demonstration. These tilings are a subset of results published in his book New Mosaics (1999, privately published), where they first appeared.