Theo van Doesburg was one of the most daring and influential artists of the avant garde, yet he is often overshadowed by his contemporaries. A new exhibition is set to change this, writes Simon Mawer

Everyone knows Piet Mondrian, so why is it that almost nobody knows the equally Dutch, equally abstract Theo van Doesburg, subject of a forthcoming exhibition at Tate Modern? Why is Mondrian the celeb while Van Doesburg is a mere footnote to 20th-century art, a name that keeps cropping up while never taking centre stage? It can't be the paintings, because if you're honest you'll admit that the two men – almost direct contemporaries (Van Doesburg was eight years younger) and, for a while, close friends – did virtually the same thing in paint, although I suppose tiny detail is significant in such arcane matters: according to some commentators, Van Doesburg's daring introduction of the diagonal into his work was enough to cause a rift between the two men. We live in days when an artist would have to bottle his grandmother in urine before anyone would be mildly shocked, so it's refreshing to know that in the 1920s an argument over the use of diagonal lines was sufficient to break off a relationship.

Presumably another reason for ­Mondrian's greater fame is those ­little dresses that Yves Saint Laurent ­created in the 60s, which elevated the reclusive Dutch artist to the status of a design icon. But perhaps the major reason is that Mondrian stuck to his guns. Both artists evolved out of the Dutch figurative tradition into complete abstraction at exactly the same time, but while Mondrian remained with his bleak, geometric painting throughout his life, Van Doesburg had other ideas, dozens of them. You reach out to grab Mondrian and what have you got? An abstract painter, rather solitary, rather austere. You try the same thing with Van Doesburg and he's as slippery as an eel. Painter, poet, art critic, designer, typographer, architect, performance artist – he was all those things and more. Proteus himself. A fox to Mondrian's hedgehog.

So who was he? Born Christian Emil Küpper in 1883 into an artistic family in Utrecht, he only became "Theo Doesburg" when he started painting – his adopted name being borrowed from his stepfather. The "van" was added later, in much the same way that Ludwig Mies added "van der" when he merged his mother's maiden name, Rohe, with his own paternal surname. One detects a similar feeling of social insecurity in the two men.

At the start of his career Van Doesburg was a competent figurative painter, his work reminiscent of Van Gogh in early, potato-eater mode; but he soon came in contact with non-figurative painting and in 1916 met Mondrian, newly returned from Paris. The devotion of both men to the creation of a purely abstract art led to the formation of the De Stijl group in 1917 and the publication of its magazine, De Stijl, which Van Doesburg edited and published from its foundation that year until its demise following his early death in 1931.

Van Doesburg's life may have been short but it was energetic. Throughout the 20s his saturnine features, often topped with a homburg and usually accompanied by a cigarette – think Humphrey Bogart – appear in photographs of divers artistic groups from Paris to Weimar, from Berlin to Zurich and Milan. Neo-plasticism, constructivism, suprematism, dadaism, elementarism – the "isms" of the time are bewildering to anybody but a specialist, but Van Doesburg was involved in all of them. Indeed he invented some. He was both gregarious and eclectic, a centripetal element in a diverse and chaotic artistic world. He lectured and published, talked and theorised, attended conferences and congresses and exhibitions, many of which he organised himself. So it is fitting that Tate Modern, in collaboration with the Municipal Museum of Leiden (where De Stijl was launched), has decided to give the man his due. Perhaps as a result of this exhibition he will begin to take his deserved place in the public imagination.

Of all the arts that Van Doesburg touched perhaps his greatest influence lay in the area of architecture and design. Together with the architects JJ Oud and Gerrit Rietveld, it was he who took the flat, geometric painting of the De Stijl group and burst it out into the third dimension. Indeed he even tried to inform his work with a fourth dimension, although with what success is a matter of debate. Certainly he was fired with a thrilling spatial imagination. His axonometric projections of ideal houses, created in conjunction with the young architect Cornelis van Eesteren, are crucial in understanding this concept so it is a shame that they do not form part of this otherwise comprehensive exhibition. A plastic model of one of the proposed buildings (the "Maison Particulière") gives some idea but a 3-D model is not as striking as the original drawings. A model is too literal. In the drawings perspective is ambiguous; walls are no longer supporting structures but floating, intersecting planes of primary colour; rooms are not static boxes but conceptual spaces hovering in the air. The volumes of the buildings seem to explode from an ­inner core, as though erupting into the third dimension and straining for that elusive fourth.

In 1921, armed with such architectural visions (he had been talking of the fourth dimension since 1917), Van Doesburg set off for Weimar, apparently with the intention of mounting an assault on the portals of Walter Gropius's newly founded Bauhaus. Whether or not he expected to be taken on to the staff of the Bauhaus is not clear; what is certain is that his presence was a yeast in the ferment that swirled around the design school. Some, such as Gropius himself, were alienated by Van Doesburg's dogmatic and aggressive views; others, such as the young Mies van der Rohe, were inspired. In June he was publishing De Stijl from Weimar and the next year he began his own De Stijl architecture course, poaching students from the Bauhaus itself. This was a crucial time in the development of the Bauhaus, when it was in the process of moving from its individualistic arts and crafts origins to embrace the uniformity and austerity of style that was soon to be given the epithets "modernist" or "international"; the first architectural style for almost a thousand years not to imitate something else. Van Doesburg's contribution to this shift in emphasis was crucial. He preached geometry and the use of primary colour and the submersion of the individual in the collective, things that later became an integral part of the Bauhaus philosophy.

The German period lasted for almost two years – of frenetic writing, publishing, lecturing and organising – but behind all this activity there is a love story: with Van Doesburg from the start was the redoubtable Nelly van Moorsel. Nelly was a pianist whom he met at an exhibition of the Section d'Or group of abstract painters that he organised in the Hague in 1920. He was 15 years her senior and already married but that did not stand in Van Moorsel's way: she abandoned her orthodox Roman Catholic family and went off with him to Paris. From the outset she considered herself married to him, although in fact Van Doesburg was not divorced from his previous wife until January 1923. They finally married in 1928. They were inseparable; whenever Van Doesburg appears in a photograph – and there are plenty in the exhibition – there is Van Moorsel beside him, often the only woman in the group, her mischievous grin a wonderful counterpoint to his solemn gaze. One can almost hear her laughter.

And there was a great deal to laugh about: besides the rational philosophy of De Stijl, Van Doesburg was actively involved in a movement that seems to embody the exact opposite: Dada. To understand Van Doesburg one must understand this marked polarity in his life: De Stijl on the one hand and Dada on the other. From the sublime to the ridiculous.

Nothing was more influential, or outrageous, or emblematic of its time, than Dada. "Dada is useless, like every­thing else in life," announced the founder of the movement, the poet Tristan Tzara. Dada swept aside traditions and all perceptions of what constitutes art. Its influence is felt right up to the present day. Tracey Emin's unmade bed is Dada. Randomly selected members of the public doing whatever they pleased on the fourth plinth of Trafalgar Square in London is Dada. The only difference is that Emin's bed and Antony Gormley's curation of the fourth plinth were rather tame, while Dada raised hell.

In Weimar, while pursuing his ideas at the Bauhaus, Van Doesburg had organised the Congress of Constructivists and Dadaists which included such luminaries as Hans Arp, Kurt ­Schwitters, Tzara and El Lissitzky, the Russian constructivist. In 1923, directly after returning to Holland from Germany, the Van Doesburgs partnered the wonderfully barmy Schwitters in a grand Dada tour of Holland. They'd done this performance in Jena the year before; now they performed in 10 different cities before a succession of suitably bewildered audiences. Nelly would play the piano – perhaps Rieti's "Wedding Breakfast of a Crocodile" or an Erik Satie piece they advertised as "Ragtime-Dada". On stage, wearing a monocle and with his face whited up, Van Doesburg would recite from his recent pamphlet Wat is Dada???, while Schwitters, incognito at the back of the auditorium, would interrupt the lecture by barking like a dog. Later Schwitters would be invited on stage where, indifferent to laughter or abuse, he would recite one of his mad Dada poems. In Utrecht a fight broke out after members of the audience invaded the stage and tried to present Schwitters with a wreath of dead flowers and a copy of the Bible. It was, everyone agreed, a most successful evening.

Although few knew it at the time, Van Doesburg's active involvement in Dada predates all this. In 1920 De Stijl magazine published a Dada poem by a certain IK Bonset. More Bonset poems followed in subsequent editions and the same poet also edited the short-lived Dada Magazine Mécano (1922-24) as well as contributing to other Dada publications. "IK Bonset" is actually a Spoonerism (surely the most Dada figure of speech) for "I am a fool" in Dutch (Ik ben sot): it is a nom-de-plume for Theo van Doesburg. To complete the picture there is a wonderful photograph of Nelly playing the part of IK Bonset, wearing false moustache, ­aviator's helmet and goggles, and smoking a pipe. All this is very Dada. The true identity of the poet was not revealed to most of Van Doesburg's friends until after his death.

Shortly after the Dada tour of Holland, like some maniacal product of its own imagination, Dada self-destructed, the Dada Soirée at the Théâtre Michel in Paris famously ending in chaos with the actors attacked on stage by an enraged by then-ex Dadaist ­André Breton. He broke one actor's arm with his walking stick, the poet Paul Éluard was knocked into the footlights, the audience rioted and the auditorium was wrecked. Finally the police were called. More "Rock Around the Clock" than the fourth plinth. Of course, Theo van Doesburg and Nelly van Moorsel were in the audience.

Perhaps the end of Dada was a signal to settle down. The Van Doesburgs re-established themselves back in Paris, living in the outer suburb of Meudon. He began painting again – he had done no painting at all in Germany – and design commissions came his way: the Flower Room in the modernist Villa Noailles in the south of France ­being the first. This was an opportunity, albeit a small one (the room is a mere 1.2m by 1.5m), to put his ideas into practice. Shortly afterwards came a collaboration with Arp and his wife Sophie Taeuber-Arp to redesign the ­interior of one wing of the Aubette building in Strasbourg as an entertainment centre. This work is Van ­Doesburg's masterpiece of interior design. For the first and only time, his powerful and dynamic diagonal blocks of primary colour march across the walls and ceilings of large, public rooms. Sadly the designs did not meet with public ­approval and were covered over in 1938. Only recently have the rooms been restored, finally being opened to the public in their entirety in 2009. The whole complex is now classified as a Monument ­Historique.

With money she had been left in her father's will, Nelly bought a plot of land in Meudon. There, at 29 Rue Charles Infroit, the couple built a studio-house to Van Doesburg's own design. Construction was slow, partly because of difficulties with the building material, which was "solomite", an insulating fabric of compressed straw used by Le Corbusier. Perhaps symbolically, Van Doesburg was building a house of straw: he died within a few months of completion, not in Meudon but in Davos, of a heart attack following a bout of asthma. He was just 47. Painter, poet, critic, architect, of all the dimensions of his short life the most important one was as intangible as the fourth dimension for which he searched: the influence that he had among the avant-garde of the 1920s, an artistic movement that has shaped our own world. This influence cannot be easily measured, but you can get the flavour of it through the more than 400 exhibits, from paintings by Mondrian and Arp to furniture by Rietveld and sculpture by Brancusi, that the Tate has brought together in this important and extensive exhibition.

Nelly van Doesburg, a tireless promoter of her husband's vision, lived on in the house in Meudon until her death in 1975; the house still stands, a modest and poignant memorial to a man who for 10 years was one of the major catalysts of the art world in the 20th century.

Van Doesburg and the International Avant-Garde: Constructing a New World is at Tate Modern, Bankside, London SE1 from 4 February to 16 May 2010. Ticket office: 020 7887 8888. www.tate.org.uk/modern