Mathematics into pictures – Linking and knotting (1978)

Christopher Zeeman

Christopher Zeeman kicks off the very first CHRISTMAS LECTURE on mathematics by exploring the complex properties of linking and knotting.

Watch time: 57:51
Christopher Zeeman speaking at the Ri theatre
Credit: Royal Institution

Lecture 1 – Linking and knotting

Across the lecture, Professor Zeeman looks at a branch of mathematics called Topology, a form of geometry in which we "imagine things are made of rubber: we can bend them and twist them, but not cut or glue them".

As Zeeman states, it leads us to study basic things like "links and knots and holes and curves and surfaces and insides and outsides and surfaces with only one side (not only in 3-dimensions but also in higher dimensions)."

By looking at linking numbers, Zeeman unravels how a mathematical theorem can be used to describe the number of times two closed curves are linked. He also defines knotting numbers which are used to prove how various mathematical knots are different and takes a look at the impossibly shaped Möbius Strip.

We are also introduced to the concept of a mathematical proof and shown how to generate a 'theorem'. Zeeman demonstrates that once a theorem has been proved true, it can be used as the basis to prove further truths. Unlike science which sets out to disprove its truths, mathematical proofs stand true for all time.

Finally, if all this seemed a little abstract, Zeeman applies these concepts to the field of genetics, where mathematics has been used to provide fresh perspectives on the structure of DNA.


Professor Christopher Zeeman presents the 1978 CHRISTMAS LECTURES on one of the oldest and most splendid endeavours of mankind.

As he explains in his original introduction to the series maths is at its core a paradoxical subject: "We are never quite sure whether it is an art or a science, whether we invent it or discover it, whether it is a man-made toy or a truth so universal that it is independent of the universe."

True to the tradition of the CHRISTMAS LECTURES, Zeeman places practical demonstrations at the heart of his presentations, using diagrams and pictures to reveal the nature and beauty of mathematical theorems and illustrating different types of mathematical modelling.

The first three lectures start with a pure point of view to ask 'What is the nature of mathematics?' before shifting to an applied point of view to explore the mathematics of nature.

These were the first CHRISTMAS LECTURES in its then 149-year history to be presented on the subject of mathematics and the series is still lauded as inspiring a new generation of mathematicians. Amongst the live audience was a budding young mathematician called Marcus du Sautoy who went on to present the Lectures in 2006 entitled ‘The num8er my5teries’.

Furthermore, the enthusiasm generated by the series led Professor Zeeman to establish the Ri’s Mathematics Masterclasses programme in 1981. They continue to enable thousands of young people across the UK to participate in the hands-on and inspired learning that is the hallmark of the CHRISTMAS LECTURES.