No, it's not in his bed, but it has been in his head, and now it slices through his bedroom at eye-level, disappearing seamlessly into a shelf of books.
In his fifth-floor flat, an early post-war brick council slab in Fulham, West London, the wing continues into the next room, fatly spanning a spacious sitting room. All is still - the wing speed is zero.
It is an installation called Phlight,
built with a little help from the Arts Council. 

Tyszko has a history of startling works. In 2002, he created a stencil kit to form the words Absolut Hypocrisy from 14g of pure cocaine. In the same year he received global attention with Suicide Bomber Barbie, a doll with a take on the glamorization of suicide bombers for Hamas, al-Qaeda and the like.Phlight's original inspiration came from the shock of 9/11. Tyszko considers the moment just before the second impact, when the plane seemed frozen, as 'the apex of modernism' politics, technology, commerce and religion all about to collide. 'The weight of the metaphor literally brought the building down,' he says. Phlight, too, is a frozen moment, but has evolved from 9/11 to a more elusive spectacle.
This is art colliding with architecture, and the domestic setting makes it a personal experience. Tyszko's Polish father flew with the wartime RAF, and Tyszko used to tell the kids on the common-access balconies that he was building an escape machine, like the glider in Colditz.

The wing is faithful to original Douglas blueprints (below), although aluminium. Its construction in a workshop in Kew by ETS Design (which recently engineered a revolving caravan for a Richard Wilson work) has been archived by David Ellis for a future documentary, and video clips are on the Phlight website. The alignment of the sections between the bedroom and lounge is millimetre-perfect. There are no markings, just rivets and edges. Much is left to the imagination.
The choice of a Dakota is significant. From the Forties, California churned out thousands to transport troops and cargo to war, and later give the Berlin Airlift its take-off. Over the jungles of Papua, it enabled the Allies to keep ahead of the nimble Japanese Army. There, Papuan tribes who had never seen an aeroplane before thought it was a god - perhaps the first case where technology has been deified.
The Dakota is a heroic machine, reflecting the optimism of modernism in which technology promised to deliver Utopia. It even became divine. Now, its wing is a modernist icon slicing through a personal space - in this case, a flat built with the modernist idealism of social housing. Whether Phlight is a postmodernist statement on modernism, a heroic romance in metal or just a flight of fancy, the ideas it invokes are numerous.

Phlight will also be the venue for a series of events with the likes of Will Self, lain Sinclair and others. Tyszko will literally be artist-in-residence for at least a year, and from this month will issue boarding passes for public viewing (see www.phlight.org). Even more than the Sixties' council flat in Manchester that hosts the Apartment Gallery, the venue is part of the statement. A collector is already interested, but he would have to buy the entire flat, Whatever happens, this Dakota wing remains in the fabric of Fulham, and of time.