Problem #DES-tilings

Descriptions
 10–14

Problem

Today we will be looking at tilings. We define a plane tiling as a covering of the entire plane, without any gaps or overlaps, using identical geometric shapes that can be rotated and symmetrical to each other. Usually, it is sufficient to cover a small portion of the plane with a particular pattern that can be extended to cover the entire plane.

We will also look at some tilings of finite shapes, normally rectangles. You can imagine this as tiling a floor. With a huge number of \(1\times2\) tiles, we can investigate how to tile a floor in a rectangular room if no tile can overlap the other. It’s easy to tile the floor in a \(6\times8\) room.

image

We can notice that if the floor in a room of size \(p\times q\) is tiled with \(1\times2\) tiles, then \(pq\) is even (can you explain why?). The reverse is also true; i.e. if \(pq\) is even, then the floor can be tiled with \(1\times2\) tiles in a similar way to the picture above.

However, this tiling can be cut from one side to another by a grid line without splitting any tiles. Such constructions are impractical, and this type of floor can easily become uneven. That’s why in practise irreducible tilings are used.

A tiling of a rectangle by small identical rectangles (tiles) is called irreducible if any straight cut from one side of the big rectangle to another goes across at least one of the tiles.