Cologne Cathedral audio guide,
five stops, free.

Stand far enough back on the Domplatte that both towers fit into one look, and then keep going back, because from most of this square they will not. The south tower, on your right, is 157.22 metres. The north tower, on your left, is 157.18. Four centimetres between them, over a front sixty-one and a half metres wide.

Look at the spires first. They are not solid. They are open stone tracery, eight-sided and pierced right through, and each ends in a great finial, a stone flower with its petals turned up. One of those finials stands full size on the square in front of you, so you can see what sits on top of a spire 157 metres up. The shape comes from goldsmiths' work: the medieval designer drew these towers as if the building were a reliquary, which, as you will hear inside, is exactly what it was meant to be.

Now the surprise. Most of what you are looking at is nineteenth-century, and it is still a medieval design. Building began here in 1248, under a Master Gerhard of whom almost nothing is known but his name. After 1520 the work slowed, and around 1560 it stopped altogether, with the choir finished at the far end and the south tower a stump. It stayed that way for three centuries, and Cologne had a saying for it: when the cathedral is finished, the world will end.

What rescued it was a drawing. The medieval elevation of this front, known as Riss F, is just over four metres high, twenty sheets of parchment glued edge to edge. It went missing and came back in two halves: the north tower in 1814, in the loft of an inn in Darmstadt, and the centre and south tower in 1816, at a dealer in Paris. Put together, it told the nineteenth-century builders exactly what to build. From 1842 they did, nave and transepts first; when the towers went up from the 1860s it was with more than five hundred workers, winch wagons running on rails across the site and a steam engine, and on 15 October 1880, six hundred and thirty-two years after the foundation stone, the towers were finished. For four years this was the tallest building in the world.

The architect Richard Voigtel announced shortly before his death in 1902 that the cathedral was complete at last. In May 1906 the wing of a stone angel fell from above the main portal after Sunday Mass, and an umbrella broke its fall. The workshop has been repairing this front ever since, one weathered block at a time, on a building that has now been declared finished twice.

Come in through the main portal, take a few steps, and stop. From here the vault runs east for the whole length of the church, and it is 43.35 metres high all the way. Nave, crossing and choir share one height and one width, twelve and a half metres between the piers, and every pier is the same round cluster of twelve shafts, one every seven and a half metres. That regularity is the strange thing. This space took more than six centuries and five campaigns to build, and it looks as though it went up in one.

It did not. For most of its life this was two buildings. The choir at the far end was finished in the fourteenth century and sealed off by a wall. Where you are standing was a half-built shell under a makeshift roof. The wall between them stood for five hundred and sixty years. It came down in 1863, when this nave was finished, and neither the Prussian king who had laid the stone for the completion nor the architect Ernst Friedrich Zwirner lived to see it: both had died in 1861.

Now look to your right, into the south aisle, at the big painted windows. These are the Bavarian Windows. King Ludwig the First of Bavaria gave them in 1848 for the six hundredth anniversary of the foundation stone; they were made at the Royal Glass Painting Institute in Munich and unveiled on 14 August that year. Then turn to the north aisle on your left. Those windows are three hundred and forty years older, made between 1507 and 1509, among the last things done before medieval building slowed after 1520 and then stopped, and the people who paid for them are in the glass: archbishops, the dean, the city of Cologne.

Two hard times left their marks on this room. Under the French occupation the cathedral was closed and used as a store and a prison camp, and in the winter of 1797 the prisoners burned the wooden furnishings of this nave to keep warm. In the Second World War it took fourteen heavy bombs, and most of the high vaults above you fell in. The medieval glass survived because it had been taken out in time, some of it to a bunker under the north tower. The nave was not whole again until 1956.

There are about ten thousand square metres of window in this building, roughly as much glass as there is ground under it, and of all the great cathedrals Cologne has the most window for its length. Let your eyes run up the nearest pier to where the light comes in, forty-three metres overhead.

Out of the cathedral, head for Roncalliplatz and find the steps going down beside the south tower. The climb starts below ground, in a passage through the tower's medieval foundations, and then it is a spiral: 533 steps to the top, on a stair so narrow that two people can only just pass. Take it slowly. Everyone does.

This is the oldest thing you will climb in Cologne and also one of the newest. The lower part is medieval. By the middle of the fifteenth century the tower stood at 56.14 metres, and there it stopped. On top of the stump the builders left their crane, a revolving timber machine more than twenty-five metres tall, and for four hundred years it was the emblem of the city, and a symbol of a city left behind, until it was taken down in 1868. In 1842, when the completion began, the new foundation stone was hauled up onto this unfinished tower and walled in. Most of what rises above the bell chamber went up in the eighteen-sixties and seventies.

After 291 steps, at about fifty-three metres, you come into the bell chamber, where eight of the cathedral's twelve bells hang. Two of them, Pretiosa and Speciosa, were cast right here on site in 1448 and 1449. The giant is the Petersglocke, in Cologne dialect the Decke Pitter, fat Peter: twenty-four tonnes, 3.22 metres across, with a clapper of about six hundred kilograms. It was cast in Apolda in May 1923 to replace the Kaiserglocke, which had been made from captured French cannon and was cut up by hand for war metal in 1918. Hauling the new bell up to this level in December 1924 took ten ropes, two hand winches and eight men, at about five minutes a metre. The first attempt to ring it, on Christmas Eve 1924, failed; Cologne heard it for the first time on 28 October 1925. Until the end of the twentieth century it was the largest free-swinging bell in the world, and it still rings only on a handful of great feasts.

Keep going. The spiral ends at about seventy metres and a metal staircase takes you up the last stretch to the platform, about a hundred metres above the square. Now look. The Rhine is right there to the east, with the Hohenzollern Bridge carrying six tracks across to Deutz and the main station just beyond the north tower. Then look across at the north tower's spire, close enough to read the tracery: four centimetres lower than the one under your feet, and four centuries in the making.