Questions 51-75

In which we consider tessellations

51: What is tessellation?

Imagine taking an infinite flat plane. Like an impossibly big bathroom floor. We could imagine it covered with tiles. Tessellation is the mathematical property that a shape can be fit together to cover an infinite plane.

Squares and triangles tessellate the plane, as do regular hexagons. My grandparents had a bathroom with hexagonal tiles on the floor, as was fashionable when they bought their house. Regular pentagons do not tile the plane - if you put three regular pentagons meeting at a corner, each pentagon will contribute 108 degrees, for a total of 324 degrees - to fill up the space, you'd need exactly 360 degrees.

Other shapes tessellate the plane - the artist M. C. Escher has a lot of interesting artworks based on the idea of tessellation.

52: Can you tessellate the plane with pentominoes?

Yes! Each pentomino, taken by itself, can tessellate the plane in a number of different ways. And, if you assemble groups of pentominoes together into larger shapes, those larger shapes might tessellate the plane. As an example, you can assemble 12 pentominoes into a rectangle, and all rectangles tessellate the plane.

53: How many tessellations are there for different pentominoes?