This article is mainly concerned with visual mathematics: the delight of regular colourings of patterns and the way in which shapes (some of them strange) fit together. A regular tiling, such as the tiling of squares or of equilateral triangles, is made up of regular shapes fitting together in a regular way. It is sometimes possible to combine a number of regular tilings to make a regular compound tiling: imagine the tilings drawn on transparent sheets and laid on top of each other. There is a close connection between regular compound tilings and tilings in which the tiles are coloured to form an orderly, regular pattern. But regularly coloured tilings, usually called perfect colourings, are not always associated with regular compounds in the normal sense. The author’s discussion of the surroundings of the hyperbolic plane encourages the reader to contemplate tilings made up of tiles that are not regular, or made up of tiles that are regular but infinite and much more complex than the regular polygons of elementary geometry.