Roger Penrose is well-known for his 1974 invention of Penrose tilings, which are formed from two tiles that can only tile the plane aperiodically. In 1984, similar patterns were found in the arrangement of atoms in quasicrystals.
He has written books such as The Emperor's New Mind where he argues that known laws of physics do not constitute a complete system, and that true artificial intelligence is impossible. In this controversial book, he argues this based on claims that humans can do things outside the power of formal logic systems, such as knowing the truth of unprovable statements, or solving the halting problem. These claims were originally made by the philosopher John Lucas of Merton College, Oxford.
Some mathematicians consider these claims to be mathematically incorrect. See the articles on Gödel's incompleteness theorem, the Church-Turing thesis and the halting problem for more on their reasoning.
According to Marvin Minsky, human beings can understand things to be true facts which are false, and therefore the process of understanding is not limited by mathematical systems of formal logic. Further, AI programs can also conclude that false statements are true, so this is not unique to humans.
Penrose and Hameroff have constructed a theory of human consciousness in which human consciousness is the result of quantum gravity effects in microtubules. Max Tegmark, in a paper in Physical Review E, calculated that the time scale of neuron firing and excitations in microtubules is slower than the decoherence time by a factor of at least 10,000,000,000.
See also: Penrose triangle, Penrose stairs, Penrose tiling, Stephen Hawking, Stuart Hameroff