Alan Turing (1912–1954) has been increasingly recognised as an important mathematician who, despite his short life, developed mathematical ideas that today have led to foundational aspects of computer science, especially with respect to computability, artificial intelligence and morphogenesis (the growth of biological patterns, important in mathematical biology). Some of Turing’s mathematics and related ideas can be visualised in interesting and even artistic ways, aided using software. In addition, a significant corpus of the historical documentation on Turing can now be accessed online as a number of major archives have digitised material related to Turing. This paper introduces some of the major scientific work of Turing and ways in which it can be visualised, often artistically. Turing’s fame has, especially since his centenary in 2012, reached a level where he has had a cultural influence on the arts in general. Although the story of Turing can be seen as one of tragedy, with his life cut short, from a historical viewpoint Turing’s contribution to humankind has been triumphant.