An exhibition in Dulwich is the first in London devoted to the Dutch artist, though his work has been hung in students’ bedrooms everywhere, writes John Westbrooke

The best known of Maurits Cornelis Escher’s work doesn’t so much show you something as make you wonder what it is you’re seeing. Hooded men go up and down a staircase that turns out to be endless, circling round and rejoining itself. White geese fly east in formation against a black sky that turns into black geese flying west against a white sky.

He didn’t always aim to intrigue. He began training as an architect in Haarlem, until a teacher spotted his talent at making prints and persuaded him to switch to the graphic arts department. In the earliest work shown you can see him trying out styles: one portrait is a little art nouveau, a little cubist. He travelled a lot in Europe, recording – sometimes faithfully, sometimes with artistic exaggeration, sometimes from unusual perspectives – towns with towers, towns on clifftops.

In the Alhambra in Spain, Escher was struck by the patterns of Islamic tiling, in which white lines go under and over each other and break up the background into regular geometric shapes – or maybe it’s the “background” that’s breaking up the lines? He copied one, and spent much of his life finding pictorial ways of elaborating on it. Mere geometry wasn’t enough; he turned polygons into cunningly interlocking fish, butterflies or Chinese men in conical hats.

One approach was tessellation – breaking things up into tiles, like a mosaic. The gallery is showing Metamorphosis II, one of the most striking of these. Imagine a strip about four metres long. It starts grey; straight lines appear and form squares like a chessboard that fragment into odd shapes that transmute into lizards that resolve into hexagons that turn into a beehive.

Bees emerge turning into birds, while the spaces between them turn into fish that reassemble into squares that become the sugar-cube buildings of an Italian town whose tower becomes a chess piece on a chessboard that gradually dissolves into grey again… It could be hung in a circle as a single strip.

In Britain, Escher attracted the attention not of collectors (there’s only one work in a public collection, in Glasgow, and it seems to have got there by accident) but of mathematicians, fascinated by his puzzles: how does the ladder in Belvedere start inside and yet end up outside? How can you go up and down a staircase on both sides, as in Relativity? Not a scientist himself, he had to work hard to embody tricky concepts in a form that would be simultaneously clear and perplexing.

He was also admired by the counterculture of the 1960s, who helped carry his work into the mainstream, but it was not mutual. Mick Jagger wrote to commission some cover art for an album; Escher, annoyed that the letter began “Dear Maurits” instead of “Dear Mr Escher”, refused. None the less, Mott the Hoople used an image on a cover (they must have been more polite). And though he turned down an offer to work with Stanley Kubrick, of whom he’d never heard, you can see nods to his influence in films like Inception and Labyrinth, with David Bowie. Prints of his art have long hung in student halls and appeared on T-shirts.

As his painstaking work suggests, he was no hippy; indeed the curators suggest he may have been autistic, though no doubt a high-functioning one. He loved Bach’s music; and the birds turning into fish and frogs are evidence of elegant visual wit.

A related debate is whether he was an artist at all. The curators of Britain’s national collections clearly don’t think so. The major count against him is probably that – unlike his Belgian close contemporary René Magritte, whose visual deceptions are very similar – he didn’t paint in oils; he worked in lithographs and woodcuts and seldom used colour.

Nor does he deal much in people: his self-portraits, rather than involving self-analysis, are mostly an exercise in showing himself reflected in crystal balls and the like. A double portrait with his wife, Bond of Union, shows them as a long strip, leaving half their faces out. You can certainly detect some spiritual feeling in metamorphoses that indicate the circle of life or the calm tessellation that suggests everyone and everything has its place neatly fitting in with others; but they can still look like ingenious puzzles, created by an illusionist rather than an artist with an interest in human life.

The categories don’t really matter; his work is famous enough, whatever it is. One of the best known is Drawing Hands, showing just that – two hands drawing each other, and inviting you to think about the perennial artistic problem of representing three dimensions in two. Magritte might have done the same, though perhaps he’d have labelled it “These are not hands”. Escher doesn’t need labels, the work itself does the job.

The Amazing World of M. C. Escher, Dulwich Picture Gallery until 17 January 2016. Tickets: adults £14, seniors £13, unemployed, disabled and students £9 including voluntary donation.