As Achim Borchardt-Hume, director of exhibitions at Tate Modern, points out, an Alexander Calder mobile has long been an almost essential fitting for the modernist dream home; in renders as much as the realised, where they are instantly recognisable, and bring colour and movement to hypothetically elegant, airy spaces. Architects love Calder as he loved them (he knew all the important ones) because he created wonderful objects that move in space and make spaces feel more wonderful. (His mobiles are also as much about what isn’t there – their transparency and fluttering lightness – which also appeals to modernist architects.)

Of course, its not just architects. Everyone loves a Calder. The Calder mobile is the friendly, accessible, unpretentious and playful face of modernism. There is nothing difficult about a Calder. And, as happens, Calder’s reputation has suffered from this warm over familiarity.

'Alexander Calder: Performing Sculpture', a new show at Tate Modern and the largest ever UK show of Calder’s works, reminds us how serious Calder was about his art. And how, as it evolved over time, much Calder removed to get what he wanted.

Sandy Rower, Calder’s grandson and president of the Calder Foundation, was in London for the opening of the show – and was quick to point out that this wasn’t a simple retrospective. The show starts with Calder’s early output from his time in 1920s Paris and finishes with works from the early 1950s. Calder carried on working, and at a good pace, for almost three decades, during which his work got bigger and more ambitious (the Tate Modern show doesn’t include any of the more monumental stabiles that architects loved to put in their modernist plazas). But it was in these three decades that Calder got the essentials right.

The show begins with his wire sculptures, clever 3-D sketches including one of Josephine Baker and others of the circus performers and acrobats who were a continuing fascination to the artist.

By the 1930s, and following a career-changing visit to the studio of Piet Mondrian – 'it was like a baby being slapped to make its lungs stop working,' Calder said – he had moved on to mechanised ‘mobiles’, a term coined by his friend Marcel Duchamp. (When A Universe, amongst the best known of these motorised pieces, was first exhibited at MoMA, Einstein apparently spent forty minutes watching it complete its full cycle of movements).

Later in the decade he would lose the motors and relay on precise balance and air flow to create movement. These fluid abstracts, constantly shifting and ‘performing’, were a radical break from the past, taking sculpture to entirely new places – Calder was from a long line of figurative sculptures so he knew where he was heading away from. These pieces – Morning Star, Constellation, Vertical Foliage, Gamma, Snow Flurry are all on show – are cosmic systems or natural events. 'The idea of detached bodies floating in space, of different sizes and densities, some at rest, while others move in particular manners, seems to me the ideal source of form,' Calder said.

The show ends with Black Widow, a 3.5m mobile donated to the Institute of Architects of Brazil in São Paulo in 1948, while Calder was travelling and exhibiting in Brazil.

Calder’s influence on young Brazilian artists of the time, such as Lygia Clark and Hélio Oiticica, is obvious. And Calder remains an influence. Witness Sarah Sze’s marvelous mobiles, shown at Victoria Miro this year. Sze is just one the artists to enjoy a residency at the Atelier Calder, built by Calder near Saché in France in 1962. Run by Rower and the Calder Foundation. It offers artists the chance to work in what was Calder’s huge, light-filled studio, live in the Calder residence and enjoy the views across the Loire Valley.