Step Into the Impossible: M.C. Escher arrives at Somerset House

We all know an Escher when we see one. The staircase that climbs forever and gets frustratingly nowhere. The hand drawing the hand that's drawing it. The shoal of fish that turns, tile by tile, into a flock of birds.

His work has seeped so far into culture – onto album covers, into comic books, across that famous endless stairway in Labyrinth and into games like Monument Valley – that you've almost certainly fallen down one of his impossible worlds without knowing whose it was. Which is fitting, really: for most of his life, the art world overlooked Escher, even in his home country, the Netherlands. He was 70 before he got a proper retrospective. The crowds came later, and they haven't stopped since.

From today, you can meet the originals, step inside some of them and trace the entire journey from the Dutch artist's early Italian landscapes when he was just getting started, through to the tessellations he worked out by hand after a trip to the Alhambra, to the mind-bending paradoxes that made his name.

M.C. Escher. The Exhibition at Somerset House opens at a moment when machines can conjure an image in seconds, so there's something genuinely humbling about remembering that every one of these artworks was cut by a human hand, on a woodblock or a copper plate, by one extraordinarily patient person. The wonder was never just how is this possible… It's that somebody actually made it.

The show is organised as a journey through eight chapters of Escher's life, curated by Federico Giudiceandrea, the world's leading authority on the artist's work. It opens with the realistic landscapes and Italian coastlines he sketched in his twenties – an artist still working within the rules, making the kind of accomplished prints you'd expect from that era. Then comes the turning point: a 1936 trip to the Alhambra in Granada, where the geometric tilework of Moorish craftsmen sent him home, where he filled a notebook with 137 watercolours, teaching himself the rules for filling a flat surface without leaving a single gap. Swap the abstract tiles for lizards, fish and birds, and you have the Escher the world fell for.

There's a lovely thread running through it for London audiences in particular, because two of the encounters that unlocked his most famous prints happened thanks to British minds. Escher always insisted he had no head for maths – he worked by "eye and instinct" – yet some of the sharpest mathematical minds of the century sought him out. The Cambridge-trained geometer H.S.M. Coxeter posted him a diagram in 1958 that finally gave him the maths for infinity, and led directly to the Circle Limit prints. And it was the Penroses, father and son Lionel and Roger, working in London, whose little paper on "impossible objects" handed Escher the seed for Ascending and Descending and Waterfall, two of the staircase-and-water puzzles we still can't quite get our heads around.

It's not all walls with framed originals, either. Alongside his most famous prints – Hand with Reflecting Sphere, Relativity, Day and Night, Drawing Hands, and many more – the show has illusions you can actually climb inside. You can stand on a checkerboard floor between two giant reflecting spheres while his enormous, skull-pupilled Eye watches from the walls; lose yourself in a mirrored room where his birds multiply into infinity; or take a turn in a forced-perspective chamber that shrinks one friend to doll-size while another towers over the room.

There's the joy of looking at an Escher up close, and then there's the chance to briefly live inside one, which makes the whole thing a genuinely good shout for families. Kids tend to get Escher instantly, and they get him even faster when they can walk into one of his worlds. It's the grown-ups left frowning at the staircases.