Little rectangles hurry across the paper, arranging themselves into figures. Urgent and precise, they might fly apart at any moment. Coloured rectangles warp away from the horizontal and vertical, impelled by unseen forces. Dense strokes rush about the paper, looking for a form, while circles grow arms or sit at a bar or turn into masks or become boules in a game of pétanque. Sophie Taeuber-Arp’s retrospective at Tate Modern in London is low-key but full of delight and surprise. Organised with the Kunstmuseum Basel and New York’s Museum of Modern Art, where an enlarged version will open in November, the exhibition includes the full range of the artist’s work. It is something of a corrective, too. Although well known for her paintings, reliefs and sculpture, Taeuber-Arp also worked in textiles, made marionettes, designed interiors, furniture, stained glass and costumes, bags and necklaces. Her work as an applied artist (though she, and we, might balk at the distinction) has often been overlooked in exhibitions of her work. Taeuber-Arp had a rich and complicated life. Born in Switzerland in 1889, she studied textiles and sculpture. She met her future husband, the Alsatian sculptor Jean (Hans) Arp, in 1915. A one-time student of the modern dance pioneer Rudolf von Laban, Taeuber-Arp found herself right at the heart of things: hanging out with Tristan Tzara at Zurich’s Cabaret Voltaire, dancing in a costume of her own devising to Dadaist nonsense poems by Hugo Ball, championed by Marcel Duchamp, friends with Sonia Delaunay. An early boyfriend was Adolf Ziegler, who became Hitler’s favourite painter and organiser of the 1937 Degenerate Art exhibition. She later became friends with Gabrièle Buffet-Picabia, painter Francis Picabia’s first wife and organiser of the resistance cell to which Samuel Beckett belonged. For many years, Taeuber-Arp’s art was overshadowed by her husband’s, in much the same way that Sonia Delaunay’s work was long eclipsed by that of her spouse, Robert. Taeuber-Arp’s early painted geometric abstractions are of a piece with her embroideries, some very small, which spun off into small bags of beaded thread and fabric that continued her abstractions by other means. In 1918, she made a series of marionettes for a play about psychoanalysis. Her clothed and painted jointed cylinders, cones and tubes became its characters: stags, soldiers, a parrot, a figure called Dr Oedipus Complex, and another, a magician, called Freudanalyticus. Their inventiveness is still terrific. Lathe-turned Dadaist wood heads followed. They stare at us, wonky eyed, dumb but knowing. Every part of her work connects. Collages fed into textile works, textiles into paintings or interior designs for a bar in Strasbourg and a house with stained glass windows. The severe design and intense colour of her stained glass and her furniture design almost presages the work of the minimalist, but a kind of precarious, dynamic equilibrium is always in play. Even Taeuber-Arp’s most geometric paintings are on the verge of a kind of figuration (one might say that her painted shapes and lines were an equivalent to her earlier marionettes), and have a dance-like fluidity. Paintings consisting of rows of circles and small rectangles pop and quiver before your eyes. Some look like game consoles or peculiar keyboards. I take the impersonal, exact precision of her painting (sometimes works were completed by assistants) as a kind of deadpan for an art that is, paradoxically, full of life and movement, always alert to the potential of experiment. Circles meet and part on blank grounds that are electrified by the apparent movement of the forms on their surfaces. Her reliefs – with truncated cones and cylinders (certain forms reappear at different stages in her career) – encourage us to take sideways looks as well as approaching them head on. She began to experiment with compound curved shapes and lines, which found their expression in shells and tangles of undulating lines, some illustrating a book of poems by Arp. Words are one thing. Her lines cross and re-cross, stop short, loop away, get complicated, tail off and begin again, like thoughts in motion, poems without words. Hers was an art of endless possibility and unfinished business. Exiled from France in 1942, Taeuber-Arp died one night in Switzerland, in January 1943, poisoned by carbon monoxide from a faulty stove. As it was, her work was generative and rich. Its particular influence (conscious or not) can be felt in postwar Brazilian neo-concretism, in Bridget Riley, in the sculptures of Tai Shani and who knows where else. I wish Taeuber-Arp had lived longer and continued to develop. Now, her art feels like a posthumous conversation with artists who have come after. That’s where her art lives.