In this lecture Roger Penrose uses M.C Escher's work to illustrate and explain important mathematical ideas and their connections to the visual arts. Roger Penrose's work has ranged across many aspects of mathematics and its applications from his influential work on gravitational collapse to his work on quantum gravity. However, Roger has long had an interest in and influence on the visual arts and their connections to mathematics, most notably in his collaboration with Dutch graphic artist M.C. Escher. Oxford Mathematics hosted this special event in its Public Lecture series during the conference to celebrate the 20th Anniversary of the foundation of the Clay Mathematics Institute. After the lecture Roger was presented with the Clay Award for the Dissemination of Mathematical Knowledge. The Oxford Mathematics Public Lectures are generously supported by XTX Markets.