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BA-sponsored Clifford walks, Exeter, September 2004

by Matthew Watkins

Back in the early summer of 2004, my friend Phil Smith, an Exeter-based playwright and performance artist, mentioned to me that the British Association for the Advancement of Science were undertaking their first ever arts commission. Their annual Festival of Science was to be held in Exeter that September and it had been announced that an arts/science crossover piece was to be commissioned jointly with Phoenix Media (an entity affiliated with the local arts centre).

Having a shared interest in the Situationist International, ‘psychogeography’ and other related, obscure 20th century art movements, Phil and I had been undertaking ‘mytho-geographic’ explorations of the city during the previous few years. He’d been involved in a lot of site-specific and walking-based performance art over the years; I had some training in mathematics, having completed a PhD at the University of Kent under the supervision of Roy Chisholm, initially looking at the use of Clifford algebras in gravitational modelling. Phil suggested that we join forces, and the resulting proposal which we dreamt up and submitted somehow managed to combine these threads. It was to be a walk – a sort of mixture of heritage walk, maths/physics lecture and experimental street theatre, exploring the life and works of the remarkable mathematician William Kingdon Clifford (who, I happened to know, had grown up in Exeter). Some weeks later, we were pleasantly surprised to learn that our rather unusual proposal had been accepted.

Due to limitations of time and the general public’s average walking speed, our walk couldn’t stray too far from its starting and finishing point, which clearly had to be Clifford’s childhood home on Longbrook Street. This meant that we had to draw our ‘raw materials’ from whatever architecture, imagery, etc., we could find in this small (and not particularly beautiful or inspiring) region of Exeter – presenting an interesting challenge to us both. It also meant that we were working very much in the area that young William would have played as a child (which at that time was on the very edge of the city).

People participating in the walks (held on the evenings of 5th, 6th and 7th September) were asked to meet outside the distinctive-looking house at 82 Longbrook Street. Phil and I looked on from a distance until they’d assembled, then wandered up the pavement towards the group. We purposefully ignored them, as I pointed out the house to Phil and began to tell him about William Clifford and the revival of his algebras in the 1980’s by my PhD supervisor. In full expository flow, I suddenly froze dramatically in mid-sentence, at which point Phil turned to address the group and explained that we had just re-enacted the occasion a year or two earlier when I had pointed the house out to him towards the end of a long, semi-random walk around the city.

He then mentioned an email he’d received from me a few days later.

I produced a piece of paper from a pocket and read out the email:

"A few nights ago, I dreamed I was walking through a city with my friend Amanda…We encountered a busload of kids, and one kid ran up to a lamppost, sort of jumped sideways in the air, grabbed the post, and spiralled downwards, his body remaining horizontal. When I woke up I remembered having read *years* ago in a little biographical piece on Clifford that he was an athletic child who invented a thing he called a "corkscrew" (the thing the kid in the dream did). So Clifford *as a child* appeared in my dream, not the bearded professor - he lived in Longbrook Street as a child, and it was presumably in Exeter that he practiced his "corkscrews")."

The "little biographical piece" was a preface which Prof. Chisholm had written in a book of conference proceedings. From that (a book I’d often referred to in the initial phase of work on my thesis), I’d remembered the fact that William Clifford had grown up in Exeter. So, when I moved there in 2000, I contacted Roy and Monty Chisholm, curious to know exactly where his childhood home was. But what I’d forgotten for the intervening fifteen years was this reference to the "corkscrew". This had been lurking somewhere in the back of my memory and having discussed WKC with Phil during that walk, it had somehow slipped into the unfolding of my dream a night or two later.

The helix – the mathematical curve described by a corkscrew motion – was one of the main themes of the walk, used to evoke the spirit of young William playing in that very neighbourhood, as well as to symbolise his contributions to geometry. Fortunately for us, just behind the Clifford house is the King William Street Carpark, a multi-storey structure featuring a large, helical concrete ramp, which we were able to make full use of.

But first, after briefly introducing ourselves and explaining what we were about to undertake, we crossed Longbrook Street to make use of another carpark – an ordinary two-dimensional carpark this time, rather a large one, at the foot of Howell Road. For a very up-and-down town like Exeter, this is a remarkably large flat area. And it’s painted with a lattice of white lines at right angles – exactly what we needed.

Using chalk, string and a copy of WKC’s The Common Sense of the Exact Sciences which Phil had found in the stacks of Exeter Central library, we proceeded to enact passages from the book where Clifford explains basic arithmetic operations geometrically – addition in terms of taking steps along a line, for example.

While I was busy chalking out marks for the positive integers along an existing white line, Phil read, and partially acted out, WKC’s children’s story "The Giant’s Shoes" (from an anthology called The Little People), pointing out that even in this playful context, geometry played a significant role in his descriptive language.

Once addition had been explained geometrically, the analogous steps-along-a-line explanation of subtraction then led naturally to the idea of negative numbers, and of a complete number line. Multiplication and division were described in terms of magnification and dilation of this line. At this point, Phil asked the seemingly absurd question "But, what happens if you step off the line?", as he stepped off the line he’d just spent the last few minutes walking up and down.

This took us deeper into the carpark. While I prepared a pair of coordinate axes to explain the 2-dimensional plane of complex numbers, Phil used a length of string (which I was about use) as a prop in telling the audience an anecdote about Clifford’s failed attempt to fly an elaborate kite he had constructed in Wales, the strings having become entangled in a flock of sheep. Phil was also able to point out the railway bridge over which the scientifically inquisitive 12-year-old Clifford would almost certainly have seen the first steam train ever to arrive in Exeter in 1857.

I was then ready to lead the walkers through a geometric explanation of how it’s possible to have "two dimensional numbers" – represented as points in our car-park plane – and how the addition, multiplication, etc. of these "numbers" these can be understood in terms of geometric motions: walking a certain number of steps in a certain direction, rotating around the origin (point where the axes cross) at a fixed distance (with the help of my string), dilating or contracting that distance, etc.

Assured that they wouldn’t need to remember the details, and with some well-placed questions by non-mathematical Phil, the audience were able to leave the carpark a few minutes later with an understanding of the existence of vectors in one, two or three dimensions, and an awareness that ordinary numbers on a number line, or these new ‘complex’ numbers in their ‘number plane’ could be thought of as examples of these. I’d explained how a 3-dimensional vector space would work by using nearby lampposts and the prominent view of a corner of the fortuitously nearby, very large and rectilinear Debenhams building. I explained that these 3-dimensional ‘numbers’ (or vectors, or points) could easily be added, exactly analogously to their 1- and 2-dimensional equivalents, but for a long time, how you would go about multiplying them was a total mystery.

We headed back down Howell Road, Phil mentioning that Howell was one of the regional Celtic kings, which led on to a mention of the best-known Celtic contribution to geometry – knotwork patterns – the branch of mathematics known as knot theory getting very quickly described. And it was arguably the "King of Celtic mathematicians", William Hamilton, the greatest of Irish mathematicians, Phil told the audience, who discovered a way to multiply vectors beyond two dimensions. But they’d have to wait to find out more…

When explaining addition and multiplication on the number line, I’d made a point of emphasising the property of commutativity – that a + b = b + a and a x b = b x a. I explained that the big breakthrough (it has been joked by the maverick philosopher R.A. Wilson that only an Irishman would have thought to do this) was that the multiplication Hamilton introduced wasn’t commutative.

I’d also explained how a plane was two dimensional because through any point, a maximum of two lines mutually at right angles could be drawn – never a third. Three dimensional space allows three such lines through any point, but no more. The idea of a 4-dimensional space can be at least considered (although not imagined directly) according to this way of describing dimension.

Across Longbrook Street, we walked up a bit and into an alley along one side of the Bishop’s Move building. This is the moving company – their slogan is "Better Across the Board" which appears over a chessboard logo. We stopped next to a large sign affixed to the wall and examined the design. Phil and I were very lucky that this was here for us to use, for the chessboard is presented at an oblique angle to create an irregular quadrilateral subdivided by non-parallel grid lines (imagine looking at a chessboard from an oblique angle). I explained how, when we look at this, we hardly notice that we’re looking at a flat design of diamond-shapes because we’re able, instantly, to reconstruct mentally an image of an undistorted chessboard (I produced one from my bag as a prop). The mechanics of visual perception are very much tied in with Euclidean geometry – Phil read a passage from one of Clifford’s books which related to this.

I was also able to use the idea of the chessboard as an extremely helpful tool to convey the exactitude of geometry. If we were to examine it with sufficient precision, the actual wooden board in my hand would be revealed as only a crude approximation of a true planar surface, I explained, the lines only approximately straight, only approximately parallel. This would be true of any physical chessboard, however crude or seemingly precise. But we can all imagine the perfect, archetypal Platonic chessboard which is exactly planar, whose angles are exactly right angles, whose lines are exactly straight and parallel. That’s the kind of world which Euclidean geometry takes place in.

Having wanted to say something about recent applications of Clifford algebras in robotics, but not knowing much about this, I’d looked in the University library and found Geometric computing with Clifford algebras: theoretical foundations and applications in computer vision and robotics (edited by Gerald Sommer). Leafing through this, I’d found a photographic image showing a robotic imaging device carrying out image recognition exercises involving a chessboard viewed from an oblique angle. So I was able to produce this book from my bag, immediately showing the audience the picture in question, held up next to the very similar Bishop’s Move logo. Then, once they’d all grasped that this was a book explaining the mathematics behind electronic image recognition, I showed them the cover, with "Clifford algebras" prominently in the title.

Up along the pavement which abuts the multi-storey carpark, we stopped to look up at the corner of the Debenhams building again. I wanted everyone to think of the three edges as coordinate axes in three-dimensional space, I explained. Having the audience all facing this direction allowed Phil to slip away, through a door and up some stairs to a landing on the other side of a large window just above, and behind the audience. He put on a mask I’d made, featuring a photograph of my own face, and waited with (non-permanent) magic marker in hand.

I then looked behind me, expressed exaggerated comic surprise and pointed to Phil in the window, ‘wearing’ my face. "Oh, look, I appear to have bi-located…" (much laughter). Using this ‘copy’ of myself, I was able to explain an important point about spatial orientation. In the carpark earlier, I’d explained that a line has a left-right pairing of possible orientations, and the plane a clockwise-anticlockwise pairing. I now demonstrated how these are relative to one’s point of view. "Your left is my right," I pointed out to the audience, who were facing me. "Now watch this." I got my chessboard out again, and on the back of it drew two arrows, at right angles from a common point, one to the right, and one up. I then drew an arcing arrow (anticlockwise) to indicate the rotation of the former into the latter. Mimicking me in the window, Phil drew the same figure on the glass. Except, from our point of view on the other side, his arrows were to the left and up. And his arcing arrow was clockwise. Clearly "Your clockwise is my anticlockwise."

But how does this work in three dimensions? We have these words: "left", "right" and "clockwise", "anticlockwise" – but what such concepts apply in 3-d?


To explore this, I then handed out small wooden cubes and pencils to each person. On each cube, I’d chosen a corner and then blackened the three edges emanating from this corner, drawing a little arrow at the far end of each of them. I asked the audience to simply label the three arrows "1", "2", and "3", however they wanted. They quickly did so. I then instructed them to hold their cube so they could see all three numbers, and to turn it so that the "1" pointed "up". I then separated the group according to whether their "2"s were pointing to the left or right. This conveyed the analogous duality of spatial orientation which exists in 3-d.

Phil then, having reappeared as himself, explained how this notion seemed unfamiliar because, unlike a line on the pavement, or a plane-like window, we can’t "go around the other side" of a 3-dimensional space.

We wandered on towards a small overhead footbridge which links the carpark to the shops on nearby Sidwell Street. I returned to William Hamilton, who, we’d mentioned earlier, had found a way to multiply vectors in dimensions beyond two, and how this had involved a violation of the usual ‘commutativity’ of multiplication. I told the story of how the key equations had come to him as he walked along the Liffey near Dublin, and how, in his excitement, he had spontaneously carved them into the nearest stone bridge (they’re still visible). By now, a set of equations had come into view, chalked on a wall (by me, earlier) below the footbridge. We expressed mock surprise, and I was able to use the display to reinforce the notion of "non-commutativity".

i x i = j x j = k x k = i x j x k = -1

i x j = k, j x k = i, k x i = j

j x i = -k, k x j = -i, i x k = -j

Phil and I then sat cross-legged on the pavement and began play a game involving some stones I’d brought along in a bag. As the game unfolded, I explained to the audience that they were witnessing the basic mechanics of Clifford algebra, and in the process, I was able to draw in most of the themes we’d dealt with – commutativity, vectors, spatial orientation, the possibility of four or more dimensions. The stones (waxy, black lens-shaped lumps of shale from Lyme Regis) were painted with "e1", "e2" or "e3". A separate stone had a "+" on one side and a "– " on the other.

The game was pointless – you couldn’t win or lose – but there were rules you had to follow. The stones stacked in a row, with the +/– stone at the far left, followed by some of the black stones. You could introduce a new stone to the right. If two stones with the same marking were adjacent, you had to immediately remove them from the row. Otherwise, you could interchange two adjacent stones, but you had to flip the +/– stone whenever you did this. Phil played along, with me occasionally pointing out that he’d broken a rule and correcting him. The audience looked on, encircling us.

A single black stone in a row represented a Clifford vector, I explained. A pair represented a Clifford bivector and three black stones represented a Clifford trivector. Any more than three could always be reduced to three or less by switching stones and removing them according to the rules. I explained how addition was possible, but what we were doing was basically multiplication. And this multiplication was "anti-commutative". If you start with e1e2, meaning e1 x e2, and you then reverse the order, the + must be changed to a –, so you get e1 x e2 = – e2 x e1. This flip of the +/- sign works in all dimensions, and it is the flip that changes left to right, clockwise to anticlockwise, and the 3-dimensional equivalent (as witnessed in the differing labelling of the cubes) when you look at reflections in a mirror.

Once the audience appeared to be grasping the rules, as well as the ‘pointlessness’ of the game, despite its correspondence with certain ideas we’d been talking about, I was able to symbolically demonstrate one of WKC’s big leaps of geometric thought. I produced a handful of new stones from another bag, this time labelled "e4". There was no reason I couldn’t introduce these to our game, I argued. The rules would all still work. So Phil and I continued our game, but now manipulating rows of e1’s e2’s, e3’s and e4’s. I pointed out how you could now have a "quadri-vector", like "e1e3e4e2". Nothing particularly new or different was happening other than that. The +/– orientation stone kept flipping when two adjacent stones were reversed. Two "e4"s next to each other "annihilated" each other. But here we were, looking at the workings of a 4-dimensional algebra!

Phil expressed comic surprise that we hadn’t gone through a time warp or into a parallel universe, things his non-scientific mind associated with something as "far out"-sounding as "four dimensions". No, I explained, it’s really not that mysterious. Even though we can’t imagine the geometry, we can describe it algebraically, and the rules are just the same. We just introduce one more symbol for the new dimension. And there’s nothing to stop us adding a fifth, a sixth…

We gathered our stones up and walked along a bit, while I explained how Einstein had used a four dimensional model (developed earlier by Minkowski) to describe space-time and thereby formulate his theory of special relativity. This then led to the idea that "space-time" might be curved. We can imagine a line or a plane being curved or warped, but it’s hard to imagine anything like this for three dimensions, let along four. But having seen how it’s possible to leap from three to four dimensions in an entirely algebraic way, the audience were largely prepared to accept that spatial curvature can be described mathematically, and that such descriptions could work just as well in 3 or even 4 dimensions as they do in 2 or even 1.

At this point we were able to mention the remarkable fact that Clifford had suggested the idea of gravitation being due to spatial curvature many decades before Einstein did.

As I went on to talk a little bit about the beautiful applicability of Clifford algebras to electromagnetic theory, I led the group into the centre of the large helical ramp which forms part of the multi-storey carpark. Phil had slipped away and was already visible, ascending the helix. Using my umbrella as a prop, I explained the relationship between electrical current, magnetic fields and helical coils of wire. I explained how Clifford bivectors make a description of such things so much simpler. And I brought in the idea of the duality of spatial orientation again. Phil played the role of an electrical current travelling around a coil, generating a magnetic field, the direction of which (up) I indicated with my umbrella. We then "reversed the polarity" in the electrical current, at which point Phil reversed direction and began descending and I flipped my umbrella to point down. I explained about Fleming’s left and right hand rules for wiring dynamos and motors, mnemonic configurations of thumb, index and middle finger on each hand – directly demonstrating how the orientation of sets of three mutually perpendicular vectors relates to polarity in electromagnetism.

Clifford was particularly keen on helices, I explained. If there's one iconic image for his life and work it would be this. Not only does it correspond to his "corkscrew" motion, it captures the duality of spatial orientation inherent in his "geometrical algebras". Using a gyroscope and a mirror, I was able to explain about how there are two ways you can orient a helix and how this can be related back to the duality between fermions and bosons in particle physics via the notion of subatomic spin.

I went on to mention Dirac’s theory of the electron, how his algebras were basically Clifford algebras in another form, then mentioned twistors and Sir Roger Penrose’s work in applying those curious, Cliffordian mathematical objects in fundamental physics. I produced from my bag of props his recent, hefty, book The Road to Reality and showed the group how many times Clifford’s name showed up in index – Clifford algebra, Clifford parallels, "Clifford-Dirac square root of a wave operator", even "Clifford bundle".

Emerging from the helix, we moved on a short distance to a small, but heavily contoured patch of lawn. Phil and I then did a trick to demonstrate how the curvature of space could be described mathematically. First, on a flat area of nearby paving, he walked four paces north and then 3 paces northwest, while I, starting at the same point, walked 3 paces northwest then four paces north. We ended up at the same spot, unsurprisingly, having each walked two edges of a parallelogram. This showed, I explained, how the addition of vectors in a plane is commutative – it doesn’t matter which order you add them in. We then repeated the same exercise on the contoured lawn, this time ending up at noticeably different locations. I explained how the curvature in the space had "warped the commutativity" – and how the size of the gap between us was information relevant to any mathematical description of exactly how that piece of ground was curved.

This "gap" was a theme which came up in various forms throughout the walk/piece and led to our naming it "The Gap". There were references to the gaps in our visual perception which we fill in unconsciously with the use of geometry, the gap between Phil’s naďve understanding and my mathematical knowledge, the unavoidable gap between pure geometry and the imprecise physical objects it is used to describe, leaps from one dimension to another, from pavement to lamppost…

At this point, I was able to mention Clifford’s awareness of, and interest in, the emerging non-Euclidean geometries of his time, which were also relevant to Einsteinian relativity. Euclidean geometry, I explained, had become an almost ‘religious’ dogma in some sense. So it was not surprising that WKC was so willing to question the absolute nature of Euclid’s postulates, I suggested, as Phil drew everyone’s attention to what would have been the Clifford family’s parish church when he lived in Longbrook Street – St. Sidwell’s, across the road. He then talked about WKC’s atheism, his leaving the church behind, the principled stand he took, his forced departure from Cambridge, etc. From this location, we were able to then literally "turn our backs on the church" and look down on a grid of lines – a world of geometry, the substance of Clifford’s adult life – in one of the open parts of the multi-storey carpark, a couple of metres below pavement level. Beyond this, we pointed out, we could see the back of 82 Longbrook Street.

The subject matter then moved further into speculative and philosophical matters as we led the group down some steps and under the cover of one of the storeys of the multi-storey carpark around which we have just walked. A colourful, abstract geometric mural (from the mid-80’s, I would guess) then served as a backdrop – as good as anything we we’re likely to find in this neighbourhood – for Phil and I to describe Clifford’s speculations on "mindstuff", and how this could be seen as anticipating the field of study known as "physics of consciousness" which didn’t appear until the 1990s. The same Roger Penrose who uses Clifford algebras so extensively in his theoretical physics (twistors, etc.), I informed our audience, is also (coincidentally?) now a prominent figure in this emerging, interdisciplinary field.

Passing though this space and back into Longbrook Street, Phil pointed out the "New Horizon", a Palestinian café just across from no. 82, and opined that WKC would have enthused about this, having studied Arabic during a visit to Algiers, and having written enthusiastically about the preservation of classical mathematics and science within Arabic culture during Europe’s Dark Ages (and he’d have probably liked the name too). Meanwhile, I crossed over to retrieve a mysterious bundle which the New Horizon’s proprietors had allowed us to store in the back of their café. We jokingly referred to it as our "Clifford bundle", before leading the group back to the front of no. 82 and ending the walk by removing the blankets wrapped around an intriguing piece of geometric sculpture.

One of the conditions for the BA commission was that it had to involve some permanent, physical ‘piece’ being somehow installed in the city. It is likely that the organisers were originally envisioning something more like a sculpture or a mural than a performance-walk! To satisfy this condition, we collaborated with local visual artist Tony Weaver, who put together a geometry-themed sculpture (in steel and coloured vinyl) intended to be affixed to a lamppost. It involved conic sections, skew planes and a small silhouetted figure of a boy in 19th century costume, positioned so as to look as if he were twirling around the post in question – a sort of multidimensional signpost-to-nowhere-in-particular to be installed somewhere near the Longbrook Street home, as an oblique memorial to WKC’s childhood there (supplementing the current blue plaque on the front of the house, which goes largely unnoticed due to its distance from the pavement).

Unfortunately, as a result of endless bureaucratic wrangling involving various branches of Exeter City Council, as of spring 2008, the piece has yet to be installed – despite our having found the ideal location, discussed it with the relevant city councillor, obtained letters of support from Sir Roger Penrose and Sir Michael Atiyah, and despite the persistent e-mail efforts of Phil Smith! All options with the Council now seem to have been exhausted, so we are seeking a private site to install the piece (Exeter University and the Phoenix Arts centre being two contenders).

 

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