Five days in Mexico City: a sinking city with an empire underneath it

Land, drop the bags, and go and stand in the main square. It is one of the largest city squares anywhere, it is entirely paved, and it has almost nothing in the middle of it except a flag the size of a tennis court. Soldiers lower that flag at six in the evening, fold it in a long ceremony, and march it into the palace. It is free, it happens every day, and it is a better introduction to the country than any museum.

The cathedral on the north side took two and a half centuries to build, which is why the front is one style and the inside is several others. Look at the floor: the whole building has been sinking unevenly into the old lakebed for four hundred years, and the engineers who stabilised it in the 1990s left the tilt visible.

Finish above it all. The tower two streets west has an observation deck near the top, and the hour when the lights come on and the volcanoes are still just visible on the horizon is the one to be up there.

Start at the palace on the east side of the square, and start early. Entry is free, you need photo identification, and the queue at nine is a fraction of the queue at eleven. What you have come for is the staircase: Diego Rivera spent years covering it with the history of Mexico from before the conquest to the revolution, painted as one continuous crowd scene with the villains identifiable if you look.

Two minutes north is the hole in the ground where the Aztec capital surfaced. Electrical workers hit a carved stone disc here in 1978, and the dig that followed pulled a temple out from under a city block. What makes it strange rather than merely old is that there is not one temple but seven, each built over the last, so you walk down through the layers on a raised walkway.

Then west along Madero for the afternoon. The white marble concert hall at the far end is the city in one building: an Italian architect, an art deco interior finished decades after the marble exterior was started, and a stained glass stage curtain of the two volcanoes. It has sunk four and a half metres into the mud since it was begun.

The pyramids are fifty kilometres northeast, and the buses go from the northern terminal roughly every twenty minutes for about an hour's ride. Get the first one you can. The site opens at nine, has no shade worth the name, and by eleven the coach parties and the hawkers selling jaguar whistles have arrived together.

Nobody knows who built it. The city was already ruined and abandoned when the Aztecs found it centuries later, and the names we use, the Pyramid of the Sun, the Avenue of the Dead, are the names they guessed at. At its height this was one of the largest cities on earth, and the grid of it is still legible from the top of anything you are allowed to climb.

Whether the pyramids themselves can be climbed changes; ask at the gate rather than counting on it. The walk up the avenue between them is the point either way, and there is no shade on any of it.

One building holds the physical evidence for everything you have been looking at all week, and it is arranged so that you walk around a courtyard through three thousand years. Give it three hours and accept that you will not see it all. The Aztec sun stone, at the far end of the central hall, is what most people come for and it does not disappoint.

The labelling is mostly in Spanish and thin even then, so treat it as an art gallery rather than a lesson. The Maya and Oaxaca rooms on the ground floor are the ones people wish they had left more time for.

The castle is a twenty minute walk away, on a rock in the middle of the park. It has been a viceroy's house, a military academy, the home of a Habsburg archduke installed as emperor of Mexico by the French, and the presidential residence. The road up is a kilometre of switchbacks or a tram every twenty minutes, and the terrace at the top looks down the length of the great avenue.

Coyoacan is half an hour south and does not feel like the same city: low houses, cobbles, two linked squares full of dogs and balloon sellers, and a coffee shop with a queue out of the door on a Sunday morning.

The blue house on Calle Londres is where Frida Kahlo was born, painted, and died, and it is kept much as she left it, down to the studio and the bed with the mirror above it. Tickets go a week or more in advance and the museum shuts on Mondays, so this is the one thing on the trip to book before you fly.

Ten minutes away is the fortified house where Trotsky spent his final year and was killed with an ice axe in 1940. The bullet holes from an earlier attempt are still in the bedroom wall. Between the two of them the neighbourhood tells a strange amount of twentieth century history in about six streets.