Marjorie Senechal Marjorie Senechal Marjorie Senechal is a mathematician at Smith College who studies patterns, especially tilings. Tilings are patterns made of shapes, which we call tiles, fitted together to cover a surface such as the infinite plane or a curved one, such as a sphere. For example, the clay or linoleum tiles with which we cover walls and floors are usually either squares or hexagons. You can explore these questions with cardboard and scissors, or with software such as Geometer's Sketchpad. There are also some good programs for exploring tilings, such as Kali .You can also learn about tiles and tilings in Marjorie Senechal's chapter in On The Shoulders of Giants .
Tilings span all dimensions, and stretch our imaginations.
Suppose we require that all the tiles in a tiling must be arranged in parallel position. In two dimensions, only quadrilaterals and hexagons can be arranged in this way. The answer to this question was found over one hundred years ago by the Russian crystallographer E. S. Fedorov. Once we have constructed a tiling, with whatever number of tile shapes, parallel or not, in any dimension, there are still many questions to answer. The Penrose tilings use two shapes that must be fitted together according to certain rules.