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 "e_{1}", "e_{2}" or "e_{3}".
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 e_{1}e_{2, }meaning e_{1} x
e_{2}, and you then reverse the order, the +
must be changed to a –, so you get e_{1 }x e_{2}
= – e_{2} x e_{1}. 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 "e_{4}". 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 e_{1}’s e_{2}’s,
e_{3}’s and e_{4}’s. I pointed out how
you could now have a "quadri-vector", like "e_{1}e_{3}e_{4}e_{2}".
Nothing particularly new or different was happening
other than that. The +/– orientation stone kept flipping
when two adjacent stones were reversed. Two "e_{4}"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 19^{th} 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).