By Wendy Barnaby
It looked unlikely: twenty people over a 60-year age range bunched up in an ugly Exeter carpark to stare at chalk marks on the asphalt. One thin man in baggy shorts and a battered straw hat paced along a chalk line, peered at a protractor and drew arrows at more-or-less guesswork angles. Not, you might think, a spellbinding way to spend a hot September afternoon – yet the crowd was still, all ears.
Mathematician Matthew Watkins (he of the straw hat) and writer Phil Smith were initiating us into the four-dimensional world of Clifford geometry.
As a child, nineteenth-century mathematician William Kingdon Clifford lived at 82, Longbrook Street, in Exeter. We met there, and heard how he twirled his body round a pole in play, describing the pattern of a corkscrew that figures so largely in his work. Then we set off for the carpark. We were part of a Science in the city event at the BA’s annual Festival of Science.
Why the carpark? Well, it has a lot of asphalt, but you can also see the prison from there; and Smith wondered whether Clifford might have been able to see the public executions that were carried out just outside it. Not, perhaps, a very solid reason; but our 90-minute tour was to be full of similar sidelights and suggestions that linked Clifford’s maths with the buildings and spaces in the block around his childhood home.
“Places get imbued with ineffable essences of people that have lived there and things that have happened there,” said Watkins. It’s called “psychogeography”, and Watkins and Smith are converts.
It turned out that drawings on the asphalt were a great way for us to understand what Clifford meant by numbers: how “2” doesn’t just mean two of something, but an instruction to move two steps from where we are at the moment. From there we added a direction and made an acquaintance with vectors, and soon we were stepping off the chalk line into two-dimensional, and then three-dimensional space.
Smith looked up from the asphalt to remark that it made him feel that the buildings around the perimeter of the carpark were in particular orientations, not just plonked down on the ground.
Back up an ugly alley we crowded, to see how the line of the houses went up the hill. We thought about perspective and heard a theory that our brains have evolved in response to the geometric shapes in the environment.
Then it got complicated. Under the footbridge leading to the King William carpark (a multistorey affair, this – no asphalt in sight), we watched Smith and Watkins play a game with stones that showed how Clifford multiplied vectors to make bivectors and even trivectors. We briefly entered the fourth dimension – algebraically – and I thought I glimpsed understanding when the stones were clicking on the concrete, but don’t ask me to explain it now.
Watkins’ eyes lit up as he told us that Clifford imagined rotating four-dimensional hypercubes in five-dimensional space, although many of his contemporaries disapproved. “Years later,” said Watkins, “Einsten’s theory of special relativity involved an explanation of how the universe works in four-dimensional space. Decades after that, it was discovered that Clifford’s algebra was the perfect framework for discussing Einstein’s theory.”
We trooped off to stand inside the helix-shaped space made by the carpark’s circular access drive, and heard how the helix is at the heart of Clifford’s description of the motion of the electron. It’s all to do with mirror-images of reality. Smith said he was reminded of it when he threaded his belt from left to right instead of right to left.
It was hot, but everyone stayed. “It was fantastic,” enthused one of the group. “I was vastly impressed with the conciseness of the area we moved around in – it just revolved around the city block. It was fascinating.” “It was very demanding,” said another, “but you see how the whole theory of vectors relates to architecture.”
“I hope people will use it as a springboard to find out a bit about it themselves,” said Smith. Added Wastkins: “They might be quite mundane spaces, but we want to show people you can wander round some carparks in a few back streets and find all sorts of wonderful things there.”