The steps above ground at Piccadilly Circus are crowded – strewn with fast-food cast-offs and pocked with the grubby permanence of chewed gum. Hardly a suitable plinth for Eros – the statue at the heart of a city that, by some estimates, is capital of the world. Lovers of travel should descend instead on the opposite stairs, where the world opens up within the handsome circumference of Piccadilly Circus Tube station. It is the starting point for a great little railway journey that carves a slice through London, leading to the most expensive building in Europe.
Plenty has already been made of this month's 150th anniversary of the original – and still the greatest – underground railway. However, there's also a more modest claim due to the westward extension of the Piccadilly Line, which has just turned 80 years old. In January 1933, the dark blue arrow first flew west from Acton Town, stretching the boundaries of both the capital and architectural innovation.
To witness the eternal westward drift of daylight, you need venture no further than the ticket hall at Piccadilly Circus. Between exits 1 and 4, the creamy marble of the inner circular wall is interrupted by a large mahogany frame announcing "The World Time Today". A map of the Earth has a band across the middle showing the progress of the sun, with white indicating where daylight prevails and black showing the parts of the globe enduring night. Five cities are highlighted, three obvious (London, New York, Sydney), the other two baffling (Victoria and La Plata, in Canada and Argentina respectively).
Glide beneath Piccadilly, pause at Green Park, then get off at Hyde Park Corner. The Tube station here is now a prosaic affair. But above ground, near the south-western corner, you can witness a fine example of the Underground look as created by the Great Northern, Piccadilly and Brompton Railway. When it opened in 1906, the station was above ground, with a regal fascia decorated in glossy oxblood tiles.
Behind the venerable frontage, you now find The Wellesley, one of the most luxurious hotels in London, characterised by a sense of light and space, and a nightly rate of £320. Down the hill, life for shoppers would be made easier if the next stop were named "Harrods" – but that would erase the Tube trivia nugget that Knightsbridge is the only London Underground station name with six successive consonants.
The line's subterranean scenery is unrewarding, yet a pleasing aspect of the Underground is that most of it is overground. You emerge from the darkness at Barons Court – a baronial castle beside the A4, all unnecessary flourishes and emerald tiles. On the platform, the station name is spelled out on the high-backed benches – the only location on the Underground, Tube enthusiasts will tell you, with this feature.
Aboard one of the dozen trains an hour to Heathrow, you will whizz through Ravenscourt Park, Stamford Brook, Turnham Green and Chiswick Park at high speed: a rare example of an Underground "express" service, with intermediate stations served by District Line trains.
Mind the gap. The suburbs pause for the transit of the M4 and Grand Union Canal in quick succession. You then scythe through Wyke Green Golf Course towards the journey's highlight: Osterley station. Both of them.
Alight at the 1933 Tube station to appreciate this model of English Art Deco, with a superfluous brick tower topped by a soaring, illuminated spire that resembles a landing beacon for UFOs. Turn left along the Great West Road, walk for a couple of minutes to Thornbury Road. Go left again. Soon you find yourself on a bridge over the Piccadilly Line. On your left is the original Osterley station, now a second-hand bookshop.
Forty-four years ago, Tony Vesely and his wife left art school and sought studio premises to produce posters for the underground movement. What venue could be better than an Underground station, albeit an overground one? "We moved in, then we were told by the council that the property had to have a retail outlet. So we went to a few jumble sales and set up a bookshop. All in all, it works."
The Tube extension west to Heathrow burrows below the surface, reappearing briefly to cross the River Crane. The final hop to Terminal 5 is the one journey on which you can travel free – but only if you possess an Oyster stored-value card. A Dunlop umbrella and Spider-Man hat had continued to the end of the line without their owners. (If they're yours, I handed them in from the train that arrived at 4.30pm on Friday 4 January.) Inside the gleaming £3.4bn structure, function and form unite as elegantly as Piccadilly Circus.
From here you can head back into town (if you're in a hurry, opt for the Heathrow Express), or follow the sun west to Canada or California. The Piccadilly Line is the capital's conduit to the world.
Underground cash fares are punitive, eg £4.30 for a Zone 1 journey compared with £2 for pay-as-you-go Oyster Card. To order a card online, see bit.ly/oys4u.
Hyde Park Corner: The Wellesley (020-7235 3535; thewellesley.co.uk); £320 double, excluding breakfast.
Heathrow Terminal 5: Sofitel (020-8757 7777; sofitel.com); £210 double, excluding breakfast.
Harrods: 020-7730 1234; harrods.com; Osterley Bookshop: 168a Thornbury Rd; 020-8560 6206.
London Transport Museum events: see bit.ly/Tube150.