In the heart of St Petersburg, on the north bank of the Neva, the spire of the Cathedral of SS Peter and Paul lances the sky like a baroque syringe. But the spire has been surpassed, for height and force, by a structure that soars slantwise some 400 metres into the air, and prods the clouds like an accusing finger.

The building - if that is what it is - spans the river and presents to a viewer of its eastern aspect a flat, grey lattice that might be a crane or a vast gun emplacement levelled at the heavens. It soon resolves itself, however, into discrete but unified parts. The iron spine of the thing speeds out of the earth at an angle of 60 degrees. Encircling it, a double helix of the same metal, supported by vertical and diagonal struts, tapers to an oddly indeterminate apex. Inside this narrowing cage, an arrangement of three (or is it four?) huge glass geometric shapes is visible. Perhaps, as one strains now to take in the whole from a nearby bridge, there is a grinding of gears far above, and something revolves at the upper reaches of the tower, half lost in a flurry of snow.

This view, of course, is a fiction. It derives from a short film made in 1999 by the architectural historian Takehiko Nagakura, a specialist in unbuilt monuments. The tower in question - Vladimir Tatlin's Monument to the Third International - was designed and modelled between 1919 and 1921 but never erected. It survives as a monument of the mind: half ruin and half construction site, the receiver and transmitter of confused messages regarding modernity, communism and the utopian dreams of the century gone by. At one time the retrospective emblem of a recharged western Marxism - in the wake of 1968, its silhouette provided the logo for New Left Books - it has more recently seemed a staple of modernist kitsch: an architectural device that summons too easily, for contemporary architects and artists, the mystique of the might-have-been.

Yet, despite its familiar antique futurism, Tatlin's tower retains something of its original strangeness, its potential, its revolutionary promise. Far from being a melancholy unmade memorial to the decay of architectural vision or the hubris of utopian thinking, the tower is an object lesson in how to make a monument that is not at all monumental.

It had its official origin, however, in a strict and specific edict of the new regime, with which Tatlin was then not exactly at odds. (He would later, in the Stalinist era, find himself deprecated for his "formalist errors".) In April 1918, Lenin announced a programme of "monumental propaganda"; venerable tsarist statuary and inscriptions were to be spirited away and replaced by properly Bolshevik monuments to a revolution then only months old. It seems that Lenin had a limited sense of what such a renovated public art might look like. A photograph from that year shows him orating in front of massive half-length sculptures of Marx and Engels - Muscovites quipped that they seemed to be sharing a bath - that are utterly conventional in style. Tatlin's, on which he began work the following spring, was by contrast (in the words of Vladimir Mayakovsky) "the first Russian monument without a beard".

It is unclear exactly when or why Tatlin determined that his monument ought to take the form of a tower. We know that he had in mind a riposte to Gustave Eiffel's tower of 1889, from which his monument takes some of its structural cues, though his would have been 100m taller. We know too, thanks to TM Shapiro, who worked on the first model of the tower at Tatlin's studio, that he had found in the cranes and girders along the Neva "an inexhaustible source of inspiration ... a poetry of metal". He was building a solidly materialist affront to Eiffel's airy commercial fantasia; rather than tapering elegantly to nothing from its vast supporting arches, Tatlin's tower seemed to be labouring like an iron Atlas to heft the planet into a communist future.

The tower, according to what evidence remains of Tatlin's intentions, was to have served as a propaganda hub for the city, the state and the world beyond. The mysterious glass volumes inside the structure might have recalled the segmented design of a Russian church, but they were primarily administrative spaces. A vast cube (in Tatlin's first model, it has become a cylinder), rotating once a year, was to host conferences and congresses. Above it a pyramid, revolving once a month, was given over to office space. Above that a cylinder, disseminating propaganda to the global proletariat, would spin once a day like an ideological dynamo, charging the air with information. In at least one rendering of the tower, the cylinder was topped with a glazed hemisphere that housed radio studios and transmitters, while according to some accounts the lower reaches of the monument were to be clad with cinema screens. At night the tower itself might vanish, but still project the viewer outside himself, beyond the city, into a new and machine-made future.

The imagined tower may have bristled with new technology, but the material expression of Tatlin's vision was only ever a matter of the handmade and the human scale. Two drawings remain, showing the structure from the "side" (slanted) and the "front" (vertical). But the most resonant documents are the few photographs of Tatlin's original 5m-high model, executed in wood, tin, paper, nails and glue. Flanked by the artist and his young studio collective, it is a curiously moving object, both rigorous and frail. A second model was made, and exhibited at the International Exhibition of Decorative and Industrial Arts in Paris in 1925. A simplified version was revealed at a May Day parade the following year. Neither of these replicas, nor the several assiduous late-20th-century copies of Tatlin's first model that reside in museums today, can quite capture the urgency and aspiration of the original.

Lynton concludes that Tatlin, commissioned in the aftermath of the first world war, must have known from the outset that the monument in these photographs could never be built. The real fulfilment - itself by turns brash and complex, polemical and poetic - of his architectural ambition is to be found instead in the written responses to the scheme. Official reactions were guarded; Trotsky applauded the artist's rejection of traditional forms, but (rather stating the obvious) wondered if the tower didn't look like so much "unremoved scaffolding". The novelist and journalist Ilya Ehrenburg approved of the design, but wrote that most Bolsheviks still preferred the old plaster heads. Tatlin, however, had his champions, notably the critic Nikolai Punin, who hymned the tower as "a synthesis of the different types of art" and welcomed the aesthetic cleansing of old forms: "the charred ruins of Europe are now being cleared."

In fact, in at least one case the tower was welcomed as the ultimate emblem of 20th-century aesthetics. The literary critic Viktor Shklovsky, whose concept of poetic estrangement (ostranenie) was at the heart of the new formalist criticism of the teens and 20s, merely aired a conventional modernism when he wrote that "the monument is made of iron, glass and revolution ... Here for the first time iron is standing on its hind legs and seeking its artistic formula." But he said something more profound when he claimed that the monument was analogous to literary language: it drew meanings and associations to itself in the way that poetic words did, and they hung about it like snowflakes. The tower was an iron stanza scrawled across the frozen cityscape.

In other words, it was a complexly readable object in a way that advanced writers of the era hoped their works might become. It referred back to numerous precursors, and forward to several possible futures. It conjured architectural wonders both ancient and modern, real and imagined. It resembled the Tower of Babel, the Colossus of Rhodes, the Pharos lighthouse at Alexandria, the emblematic landmarks of Pisa and Paris. It could even be viewed as a diagram of the thrusting gesture of the Statue of Liberty. At the same time, as the architectural critic Owen Hatherley has pointed out recently in his book Militant Modernism, the Russian avant garde was transfixed by the mythology of the red planet - the tower is also a Martian invention, bestriding St Petersburg like a tripod from The War of the Worlds

Tatlin had invented, or perfected, an extraterrestrial or airborne architecture; despite its great mass, his tower looked as though it might take to the air's uncharted ways. (Tatlin, it transpires, subsequently spent several years trying to design the perfect Soviet flying machine.) It is of a piece, too, with the visions of an evanescent or immaterial architecture that had exercised artists and engineers since the middle of the 19th century. Sergei Eisenstein, who greatly admired the tower's transparent innards, made the link to Joseph Paxton's Crystal Palace, and dreamed of a film shot entirely inside a huge glass house. Surprisingly, however, Walter Benjamin, with his taste for 19th-century iron arcades and 20th-century media, never mentions the tower, though it is in some sense a sci-fi expression of his insights into the modern city: an arcade aimed at the sky.

It is the tower's combination of futurism and pathos that has made it such an inspiring subject for so many later artists. Between 1964 and 1990, Dan Flavin created 39 "monuments" to Tatlin, rendering the ziggurat profile of the tower in white neon. In the 70s and 80s, the monument became, for a new generation of Russian artists, an emblem of an avantgardism that might have been; Ilya Kabakov's installation The Palace of Projects (1995 - 2001) reimagined the structure as a museum for utopian schemes, both personal and political. Even when they do not refer to it directly, something of the tower's retro-futurism ghosts the films of Tacita Dean and Jane and Louise Wilson, while its spectral profile has been a faint presence in the paintings of Julie Mehretu, the Ethiopian-American artist whose work is highly attuned to the signals still sent by obsolete modernism.

The art historian Stephen Bann once wrote that the mystique of the tower for later generations - especially in the west - had a good deal to do with how little people knew about it and its inventor. The building was a "vacuum" to be filled retrospectively with utopian intent. For sure, Tatlin's vision seduces us now with its unbuildable scale, its technological overreaching, the pathos of its doomed collectivity. But the more we learn about the tower - and Lynton's book is an invaluable addition to the literature on Tatlin - the more it seems that it was not a "monument" at all. It was instead a constellation of inspiring fragments, dispersed across the century by an artist who dared the future to build something out of the ruins of his dream.