This book gives an account of algebraic topology at the border of homotopy and homology, capturing some nonabelian aspects in dimensions 1 and 2. A main intuition is to use subdivisions and, conversely, compositions, of cubes.

The mathematics of knots has been a key part of Ronnie Brown's involvement in the popularisation of mathematics. He led the team which designed the "Mathematics and Knots" exhibition for the 1989 Royal Society PopMaths Roadshow.

Group Theory has been a major part of mathematics for centuries. Groupoids not only give a "spatial component" to group theory, modelling more general activities than "return journeys"; they also allow non trvial higher dimensional versions of groups, modelling complicated aspects of homotopy theory.

The diagram is part of a "two dimensional rewriting" proof that two maps are homotopic. Diagrams of this kind are fundamental to the proofs of the main theorems.

"Out of Line" is the title of a talk given in 1992 as a Royal Institution Friday Evening Discourse, and to the British Science Association as the Presidential Address to the Mathematics section, in Southampton. It is addressed to a general audience. These web pages are an illustrated record of the talk.

B. May 5th 1935, d. April 6th 2007. John Robinson left his life as a sheep farmer in Australia to pursue a new career as a sculptor. He came to specialise in symbolic sculpture with mathematical and geometric character. After coming across some of his work at an exhibition in a London art gallery, Ronnie met John and began a friendship and collaborative association.