Three days in Innsbruck: a Habsburg capital with the Alps at the end of the street

The historic centre is a handful of arcaded lanes and you can see the point of it in an afternoon. At the end of the main one, a late Gothic balcony juts out over the square under a roof of gilded copper tiles, put there around 1500 to mark Maximilian I's marriage to Bianca Maria Sforza and to give the couple somewhere to sit while tournaments went on below. It is small, and photographs make it look bigger than it is, and the painted and arcaded houses around it are half the pleasure anyway.

Two minutes away is the thing most people miss. The Hofkirche was built to hold Maximilian's tomb, and around the empty sarcophagus stands a double row of bronze figures well over life size, ancestors and relatives real and invented, blackened by four centuries and known to everyone here as the black men. King Arthur is among them, cast in Nuremberg, and he is one of the finest bronzes of the Renaissance in any country. Maximilian himself is not in the tomb: he was buried at Wiener Neustadt and the monument here was finished long after he died.

End on the tower. A hundred and forty eight steps up the old watchman's tower gets you onto a platform above the roofs with the whole northern wall of rock filling the view, which is the best possible orientation for the following day. The tower was the city prison in the Middle Ages, which explains the barred windows on the lower floors.

Go up early, when the light is on the south faces and the valley haze has not built. The funicular starts underground beside the Hofgarten, comes out on a bridge over the river and climbs to Hungerburg in eight minutes; the stations are Zaha Hadid's, curved white shells that look like snow drifts and were built to replace a line running since 1906. Two cable cars carry on from there, and about half an hour after leaving the old town you step out at the top station just under the summit ridge.

The last stretch is on foot: ten or fifteen minutes up a gravel path to the top of the Hafelekar, at a bit over two thousand three hundred metres. From it the whole main chain of the Alps stands to the south, the Inn valley is a thin grey line straight below, and the city looks close enough to drop something onto. Take the terrace at the middle station for lunch on the way down, which has the same view and windbreaks.

In the afternoon go the other way, out to the hill on the southeast side of town. Archduke Ferdinand II built Ambras in 1563 and lived there until he died in 1595, and he filled it with the most famous collection of curiosities in Europe: armour, coral, clockwork, portraits of people with rare conditions, a bronze cabinet of automata. It survives in its original rooms, which almost nothing of its kind does, and the argument that this is the oldest museum in the world is a serious one. Upstairs, the Spanish Hall has an inlaid wooden ceiling and twenty seven full length portraits of Tyrolean rulers ranged along the walls.

The Imperial Palace is the third of the great Habsburg residences after the two in Vienna, and it is the least visited of them by a wide margin. Building started around 1460 and Maria Theresa gave it its present baroque face between 1754 and 1773, so what you walk through is largely hers: white and gold state rooms, an ancestral gallery of family portraits, and the apartment kept for Empress Elisabeth. It takes ninety minutes and the crowds are nothing like the ones in Vienna.

Then south, on tram 1, to the Bergisel. The ski jump is a Zaha Hadid design of 2001 that replaced the plain one used for the 1964 and 1976 Winter Olympics, and it is a tower with a café stuck on the top of it like a bird's head. You can take the lift or climb four hundred and fifty steps. From the terrace you look straight down the landing slope with the city, the valley and the whole Nordkette lined up behind it, which is the view that explains the place better than anything in a museum.

The station is a short tram ride back down the hill, so the jump works as the last thing you do rather than something squeezed in.