I was at Westminster Bridge on a Sunday morning, and tourists were crowding on to the Greenwich boat. I had chosen this spot because of the London Eye, the giant observation wheel on the South Bank. A walk from here to Greenwich would be a good opportunity to catch up on London's millennial architecture and the fast-changing face of the capital's bustling skyline.
Big Ben was striking 11 as I crossed the bridge to County Hall. In the 1980s, this was the symbol of defiance to Thatcherism, as Ken Livingstone posted the unemployment figures on a giant hoarding looking across the river to Westminster. These days it is the start of the Millennium Mile, a marketing conceit stretching to Tower Bridge.
County Hall is now a hotel, complete with an aquarium in the basement, and if Ken Livingstone should ever become mayor his new office will be in a headlight-shaped building designed by Norman Foster opposite the Tower of London.
Day trippers were gawping at the millennium wheel, but I pressed on beneath Hungerford Bridge to the Royal Festival Hall. Built on derelict land in 1951, when London last reinvented itself, this was the centrepiece of the Festival of Britain. Now it is all graffiti-scrawled concrete, surrounded by some of the ugliest buildings anywhere in London. Dome merchants take note: there are some uncomfortable parallels to be drawn here.
The sun came out as I arrived at Gabriel's Wharf, a villagey little shopping mall with an arty, alternative feel. This was the site of a celebrated London battle, as locals fought to save the area from development. Now the Oxo Tower, an Art Deco canning factory, has been converted into low- cost housing, whose residents have the finest view in London from their balconies.
I took the lift to the viewing gallery, where brunch at Harvey Nichols was in full swing and a notice advertised a millennium-eve dinner at pounds 1,000 a head. I settled instead for a duck and hoisin sandwich at the riverside cafe on the ground floor.
By now it was one o'clock, and I had barely started my walk. It was time for some serious foot-slogging if I was to reach Greenwich before it got dark. The next stretch is the most popular riverside walk in London, passing the Bankside power station (yet another millennium project, soon to be the Tate Gallery of Modern Art) and the thatched Globe Theatre with views across the river to St Paul's.
Soon I was under London Bridge, all dark railway bridges and tunnels, a cathedral shrouded in scaffolding and the alleyways of Borough Market. This is London's oldest working market; my grandfather, Albert Newton, had a stall here. I looked in vain for Stubbins & Newton, but found potato merchants and market pubs among the tapas bars, fish restaurants and Neal's Yard Dairy. Has gentrification saved the market, or is it an inevitable stop on the road to extinction? In 20 years' time, will there still be barrow boys or will this be another Covent Garden?
After Tower Bridge, everything changes as you leave the familiar tourist landscape behind and step into the unknown. Suddenly, all the day-trippers have disappeared. Mystery abounds. What was Maggie Blake's Cause (the name of an alley in Butlers Wharf)? Is that really called Posh Spice Quay, or is it merely a developer's joke? Is that a nautical sculpture, or a giant pair of breasts? And why on earth is someone walking along the riverbank on a Sunday afternoon with a bag of golf clubs over his shoulder?
I crossed the swingbridge at St Saviour's Dock and entered a different world, a village London of wharves and docks and tidal flats. Seagulls swooped over the water. A heron dived. Children played ball in the street. In Bermondsey and Rotherhithe, council flats and village pubs were gradually being invaded by luxury housing developments with enchanting names such as Lavender Dock and Riverside View. Canada Tower, Britain's tallest building (and home to this newspaper), blinked at me across the river from Canary Wharf.
At Surrey Docks Farm, children were peeking at turkeys through the fence and the path led me down to a muddy, impassable foreshore. There were more farm animals at Barnards Wharf, this time in the form of wooden playground sculptures for children. I passed a family shivering over Sunday roast on the deck of their houseboat, then came to an unlikely tower block whose walls were adorned with gargoyles of an array of local characters.
There was Sir Francis Drake, whose slave ships left from here; there was his patron, Elizabeth I; but there too were Tommy Martin and Pete Pope, residents of the Pepys Estate. They were grinning down from the concrete walls and playing their part in London's history. The juxtaposition seemed to sum up the whole appeal of this varied walk along the Thames.
At Deptford, the path turned inland through a jumble of unfinished developments. I cannot honestly say that I recommend this last bit of the walk. I got lost in Millennium Quay (who will want to live there in 20 years' time, I wonder?) and eventually found my way into Greenwich along the main road.
Back at the waterfront, the dome came into full view - that is, the small green dome of the foot tunnel beneath the river, now made redundant by the extension of the Docklands railway.
The real Dome was visible too, less than two miles away, but I had no idea how to get there. I pressed on in the half-light, passing gasworks and silos on my way to the Blackwall Tunnel. After crossing a footbridge, the Thames Path was diverted but I could see the Dome half a mile away to my left.
I walked the final stretch along a main road, without as much as a pavement for protection. Only later did I discover that I should have continued across the peninsula, picking up the footpath and approaching the millennial area from the other side. It may be different when it opens, but for now it does not seem to me that walkers are being encouraged.
Iain Sinclair, in his book Sorry Meniscus, compares the Dome to a genetically modified mollusc. My first impression was that it was much too small. I simply don't believe you could stand Nelson's Column up in there, or that there's room for 18,000 double-decker buses. Perhaps what it needed was some space; it should have been surrounded by piazzas and parks, but instead it was buried amid the clutter of chimneys and cranes. But then maybe the Dome was a symbol of the London through which I had walked for the previous six hours - messy, chaotic, unfinished, half-planned, yet endlessly fascinating and strangely beautiful.
ALONG THE THAMES TO THE DOME
In theory the Thames Path is waymarked for the length of its route, but you need to look out carefully for the signs. The path mostly sticks to the river, but is forced inland in several places by factories and private housing developments. In most cases you can simply skirt the obstruction and return to the riverbank, but there is a prolonged stretch away from the river in Deptford and again on the Greenwich peninsula. The path from Greenwich to the Dome is very poorly signed and much of it is affected by building works and security around the Dome itself.
The pace of building in East London means that maps and books are quickly out of date, but The Thames Path by Leigh Hatts (Cicerone, pounds 7.99) is a useful guide, available at the Southwark Information Centre by London Bridge (tel: 0171-403 8299). Sorry Meniscus: Excursions to the Millennium Dome by Iain Sinclair (Profile, pounds 3.99) is an entertaining diatribe against the Dome itself.
After the walk, you can return to central London on the Jubilee Line extension from North Greenwich, just outside the Dome. The Jubilee Line station at Westminster has not yet opened, but is expected to do so later this month.