Anish Kapoor and Cecil Balmond Sculpture Design

The ArcelorMittal Orbit has faced plenty of criticism - but it's a grower

Once people reach the Olympic tower's double-deck viewing gallery and cafe, they may well be persuaded

The Eiffel Tower after a nuclear attack. A catastrophic collision between two cranes. A giant Mr Messy. The Godzilla of public art.

The ArcelorMittal Orbit has been called many things between March 2010, when this extraordinary Olympic eyecatcher was announced, and its completion this week.

"You can like it or loathe it," says Richard Henley, the Arup engineer charged with realising Anish Kapoor and Cecil Balmond's striking design. "But it's a really good example of how we can create, shape and make innovative structural designs in Britain and in a very short space of time. It's been fun getting the Orbit up, but it's also been an intensely busy time and a big emotional effort."

" says Kathryn Findlay, the architect brought on to the team to help make the Orbit a building as well as an artwork. "As you go up in the lifts, it gets lighter and brighter and you see and feel all this amazing red structure whirling - waltzing - around you. And when you get to the top, there's not just a huge vista all around, but a big hole in the centre of the viewing platform so you can look back down on the steelwork. It's quite wild!"

The 115 metre (377ft) sculpture was always meant to walk on art and engineering's wild side. Boris Johnson might have been thinking more along the lines of a latterday Trajan's Column when entries for the Olympic sculpture were judged in 2009, but Kapoor and Balmond were out to push boundaries. The two have collaborated several times, with the help of Arup, on giant artworks, including Marsyas , a shocking red PVC membrane stretched across the Turbine Hall of Tate Modern nine years ago, and Tenemos Britain's biggest sculpture across the docks at Middlesbrough this summer.

The Orbit is in a different league, not simply because of its scale, but because it was planned as a habitable sculpture - the Statue of Liberty is another - and one that would live on for generations after the London 2012 Olympics. Taking their cues from the Eiffel Tower, Tatlin's Tower [Russia's unbuilt monument to the Third International], the Tower of Babel and their own imaginations, Kapoor and Balmond plotted a radical new take.

"Anish and I were thinking how do you beat the Eiffel Tower?" says Balmond. "But then reality hit, with budgets. We didn't exactly come down to Earth, although we reduced the height by 50 metres or so, but we began to explore what could be different in a tower. We came up with the idea of an orbiting structure on the edge of the vulnerable, one where the form looks tenuous. A structure that was not obvious to read."

This way of thinking and visualizing out loud led to the improbable, if perfectly stable, structure you see today at Stratford between the Olympic Stadium and Zaha Hadid's Aquatics Centre. "We knew we were moving away from pure sculpture and from architecture and engineering," says Balmond, "and we knew there'd be a hostile, or confused, reaction at first. That happened to Eiffel, too. It was Hans-Ulrich Obrist [of the Serpentine Gallery] who said, we should think of the Orbit not as art or engineering but as a 'laboratory of space'. We jumped on the phrase!"Balmond joined Arup in 1968. He became the global engineering firm's deputy chairman and formed the Advanced Geometry Unit within it in 2000. This enabled him to explore radical forms of structure including the Pedro y Ines Bridge in Coimbra, the beautiful 2002 Serpentine Gallery pavilion with Toyo Ito and the imposing China Central TV headquarters , in Beijing, with Rem Koolhaas and OMA. Balmond left Arup at the end of 2009 to form his own practice, although he has continued as a consultant on the Orbit.

Although nothing like this orbiting tower had ever been built before, the timetable was about as tight as Eiffel's had been in Paris or Joseph Paxton's when he created the Crystal Palace.Very few drawings and even fewer models were made. "Although we did make a beautiful wax model of the final design; this was done through 3D computer printing. Lakshmi Mittal has it now," says Balmond.

"The Orbit had been well thought through at the concept stage," says Richard Henley, "with an eye for working down from the big picture to the individual components that would make it work structurally and visually. What I like about the finished work is that it's more painterly than you might expect. I was standing under it the other week. It was very sunny and you could look up and see the way that all the different sized parts and tiny changes in details made it look like the way you see an painting close up in an art gallery; all those thousands of brushstrokes."

"Originally, Anish and I had thought of making the structure as smooth as possible," says Balmond. "This wasn't possible because the cost of doing so would have been too high, but we came to like the textured quality you see close up. You can see the effort involved in its construction. It's a bit like carpentry in steel."

Essentially, the Orbit is a swirling tube of red steel tube cut up and bolted together from many sections. The raw material, 65% of it recycled from scrap, came from 38 ArcelorMittal mills around the world.

It has been rolled and cut by computer-guided machinery, at Watson Steel Structures in Bolton, Lancashire. The computer-cut steel components were transported by lorry, to London and re-erected like pieces of a giant Meccano set. "We designed the tower so that it could be assembled in sections each 4.5 metres high," says Balmond. "There's been no scaffolding; each piece has been lifted and connected to the next. It's been a clean process," says Balmond. Construction has been delayed only by weather; the sections could not be lifted in strong winds.

"We used computer programming to test the tower under wind-loading," says Henley. "We knew it would sway - all tall structures do - but we needed to keep this within comfortable limits. We did the testing using Southampton University's 'shaking table'." This might sound like a fairground ride, but the "shaking table" is a piece of equipment that can simulate what it is like to be on a building as it moves in the wind. The decision was made to fit the Orbit with a tuned mass damper, a device installed at the top of tall buildings that swings in the opposite direction from the sway of the wind, stabilizing the structure below. Having found effective ways of shaping and making the Orbit, the big question faced by the artists and engineers was how could it be made to work like a building? With a café in the mix as well as a lobby, lifts, stairs, heating, lighting, service ducts, lavatories, wastepipes and a generator of its own, the Orbit was always going to more complex than Henry Moore's King and Queen or Anthony Gormley's Angel of the North.

Findlay, well known for her adventurous use of geometry in buildings in Japan and Britain, was commissioned to make this "laboratory in space" work as a building. "It's been a case of integrating all those parts - stairs, service ducts - that make the Orbit habitable," she says. "And of seeing it through planning permission. It's been fascinating work, but sometimes you'd look at drawings and think, they're incompatible with what Anish and Cecil mean. So we had our aesthetic 'Clash of the Week' moments, but I think it's all come together and makes visual sense."

Visceral sense, too. When people ride the Orbit's lifts or scamper up and down its 455 stairs, I think they will embrace this challenging structure just as it embraces them in its writhing steel arms. It may even effect buildings of the future just as the Eiffel Tower and the Crystal Palace did.

"I think there's a feedback loop in all our projects", says Philip Dilley, chairman of Arup. "We're quite used to engineering dramatic and unusual structures like the Bird's Nest stadium http://www.guardian.co.uk/artanddesign/2008/feb/11/architecture.chinaarts2008 for the Beijing Olympics [with Swiss architects Herzog de Meuron www.herzogdemeuron.com/ and Chinese artist Ai Weiwei http://www.aiweiwei.com/], and Cecil and Anish's sculptures. There's a pragmatism involved in realising them, but the artistry has been maintained in all them, and we're proud of that. We can do the maths and the testing to make unusual structures work, and safe, but we're getting more interested in art projects of this calibre that can inform engineering and architecture."

Boris Johnson wanted the ArcelorMittal Orbit "to arouse the curiosity and wonder of Londoners". It may well do more than this. Jokes, barbs and insults aside, the Orbit might just push architecture into fresh, and perhaps unsettling, adventures. It may well take some while to adjust to its new ways.

You will have to wait, though, until the Olympic Games themselves before you can experience it for yourself.