Every night during Taylor Swift’s 2015 world tour, the runway sticking out from the stage would take off and rotate like a propeller, carrying Swift and her dancers over the fans’ heads. At the 2016 Oscars, five glittering towers of statuettes loomed behind the host, like skittles made of crystals, then turned into video screens. On Broadway and around the world, Aladdin continues to fly on his magic carpet with no strings attached. At the Omnia club in Caesars Palace, Las Vegas, the punters are greeted by what is believed to be the world’s largest chandelier, made up of moving parts that double as LED displays.

On U2’s Innocence + Experience world tour (also in 2015), the video screen was a giant translucent billboard, slicing through the auditorium and incorporating a walkway that allowed Bono to stroll into an animated re-enactment of his teenage bedroom. In 2015, the Pope celebrated mass at Madison Square Garden, New York, beneath a 12-ft sculpture of Christ on the cross. He liked it so much, the Vatican asked if it could be shipped to Rome.

All these creations spring from one place: Lititz, a small town in Pennsylvania’s Amish country, with a population of 9,400. Turning off the freeway on a bright day shortly before the US presidential election, I pass dozens of neat timber houses with posters for Trump, and only one for Clinton. This is a reactionary corner of the US, yet it is also the global hub of a relentlessly innovative industry, one so young that it has no name. In Los Angeles, they make movies; in Nashville, they make country music; in Lititz, they make the things that make the people who go to stadium shows go: “Wow!”

U2 on their Innocence + Experience tour. Photograph: Kevin Mazur/WireImage

Where New York has the garment district, or the meatpacking district, Lilitz has the stadium-show district; or, rather, it is the stadium-show district. There are 12 companies here in the live-event business, and soon, when a new business campus opens, there will be 35. The biggest of the firms is Tait Towers, responsible for Bono’s billboard, Taylor’s runway, Aladdin’s carpet and sets for Beyoncé, Madonna, Kanye, Cirque du Soleil and the London Olympics. Name any big concert tour of modern times and the chances are that Tait made it happen. When Michael Jackson did his first moonwalk in 1983, the floor was built here. As the company puts it: “We are the brand that you have seen a thousand times, but you have no idea about.”

Tait Towers occupies a long, low, brick building, half office and half factory. Inside the door stands a battalion of industry awards. Spread out behind are 30 or 40 engineers and designers, nearly all male, clicking away at double-screen Macs. One man, 60ish, distinguished-looking and trained as an architect, is drawing a giant flower for a casino. Another, the archetypal friendly young geek, is converting mock-ups for the Red Hot Chili Peppers’ tour into virtual reality (“here, have a headset”). In a meeting room, Nick Starr, Nicholas Hytner’s partner in drama, is discussing their new project, the London Theatre Company, over lunch.

In a corner room, sits one of Tait’s joint CEOs, Adam Davis, a 45-year-old bundle of quiet enthusiasm who worked his way up after starting as a stagehand on Broadway. Theatre, he says, was “the epicentre” of staging then, with the mega-musicals; now the cutting edge is in concert tours. Davis is so used to greeting British guests that he has several three-prong sockets built into the table in his office – the ultimate in contemporary hospitality. This says something about the UK’s enduring status as pop’s second superpower. It also says something about Tait, where the staff seem unable to come across a problem, large or small, without wanting to solve it. It’s as if American can-do has been bottled and put in the vending machine.

The firm’s other boss, a little older and louder, is James Fairorth, known as Winky; he worked his way up, too, after going to college nearby. He has now left Lititz to live in LA, and I find him at Desert Trip, California’s brand new festival for very old rockers. The headliners are Roger Waters, for whom Tait has put many a brick in The Wall; Paul McCartney, whom it regularly places on a moving platform; and the Rolling Stones, Tait clients since the 80s, the godfathers of the stadium spectacular. In the list of the 10 highest-grossing tours, the Stones have three entries. Tait has all 10.

Fairorth sees the rise of the gig in terms of the state of the industry. “The tipping point,” he says, “was when Napster changed the music business [in 1999-2001]. The demise of the record label enabled artists to earn much more money touring than they ever earned before. And so a set is now a real extension of the artist, if they are involved in the stage design, which they normally are. It becomes a bit of a competition between [different artists], and a marketing tool. We just happened to be standing there at the right place at the right time.”

The Tait office in Lititz.

For Davis, it’s more about what the fans want. “Repeatedly, there’s this feeling that the future is not going to have public assembly, but every piece of data we have is the opposite. The more digital life gets, the more people need to get together in a public space.”

If it’s the music that makes us choose a particular gig, it’s often the visuals that make us remember it. A star who cares about pleasing the crowd will play the hits, so the music is mostly predictable, and the staging is where the creativity surfaces. It may be just a bright idea, such as when Adele resumed touring after a long gap and her set designer, Es Devlin, found an elegant way to express that sense of returning, putting a black-and-white closeup of Adele’s sleeping face on the big screen as the crowd shuffled in, then having her wake up as the lights went down. Even for a simple stage design, Devlin prefers to work with Tait to put together her designs. “Production managers won’t leave home without them,” she says.

At the other extreme, touring turns into competitive hydraulics. On Justin Timberlake’s 2014 tour, the front of the stage took off vertically and rolled over the stalls to become a moving bridge. Timberlake skipped along it, giving his fans an extra thrill and an unexpected photo opportunity. Musically, his show was mundane, but the engineering made it memorable. And it wasn’t just engineering: the playfulness brought out the small boy inside the slick entertainer.

Looking around Lititz, you are continually struck by the collision of art and science. As Devlin, who lives there when she’s designing a tour, says: “You can’t separate the steel and the mechanics from the poetry.”

Adele plays live in Mexico City. Photograph: Victor Chavez/Getty Images

In the beginning was the sound. In 1966, when hardly any pop stars had a travelling sound man, Frankie Valli and the Four Seasons played Franklin & Marshall College in Lancaster, PA. The sound was provided by Roy and Gene Clair, two brothers from Lititz who used to build speakers in their dad’s garage. They did it so well that Valli took them on the road. One of their firms, Clair Global, is still in Lititz, still making speakers, and still at the top: it supplied the sound at Desert Trip, which was strikingly crystalline – as it needed to be, given that half the punters, and all the performers, were of an age to be hard of hearing.

The other formative moment occurred in London in 1968. A young Australian, travelling in Europe “as you do”, found a job as a barman at the Speakeasy, a musicians’ hang-out in Soho. One night the manager of a new band called Yes was there, saying they needed a driver for their new Transit van. The barman drove Yes to their next gig, at Leeds University. “I don’t think they got through one song without something breaking,” he remembers. “The guitarist would stomp on his volume pedal and break the cable. And I realised that I could make all this stuff work.” He became Yes’s tour manager, sound engineer and lighting designer. “That one gig lasted 15 years.”

The barman was Michael Tait, now 70, silver-haired and an occasional visitor to the building that bears his name. During my visit, he arrives in a yellow jacket and a silver Rolls-Royce. “He’s an icon,” says Tait’s head of marketing, Mia Tinari. “Doesn’t matter how young the artist is, they still know who Michael Tait is.”

Growing up in Melbourne, Tait had played “with batteries and lightbulbs” from the age of four. With Yes, he made common-sense breakthroughs, starting with the stomping guitarist. “He had three pedals, a volume, a wah-wah and a fuzz. I thought, well, if I fix ’em to a board, and put an edge on it, then he can’t stomp on ’em. So I made what was probably the first pedal-board.”

Yes playing in the round at Wembley Arena, 1978. Photograph: Max Redfern/Redferns

His next idea made his name. He went to pick up a music film – “we didn’t call them videos in those days” – in a 16-mm can, which he put on the passenger seat of his car. “I’m driving along, looking at it, and bam! The way Yes were set up in the studio was with Jon Anderson in the centre, and in each corner of a squarish room was one of the other musicians. I thought: the square could be a circle.” He had invented the rock gig in the round. “They said, ‘This can’t possibly work.’ And I said, ‘Look at you now, this is how you’re set up.’ They said, ‘We’ll have our back to the audience.’ And I said, ‘No you won’t, you’ll have your back to these people here, but you’ll be facing all those people over there.’”

It sounds like Spinal Tap – a story in which some of the best gags involve the set going wrong. But Tait’s wheezes worked, and even paid their way. “The rotating stage was a huge money-maker. You double the front row from 60ft to 120ft and everyone in the audience is twice as close, so you scale up the tickets and it pays for itself in, like, four gigs. Once people saw that, Barry Manilow called, Neil Diamond called, and before I knew it I was in the set business.”

Although still based in London then, Tait was storing equipment in Lititz, to be near the Clair brothers. “I built that first rotating stage with a farmer up the road. Then I was having immigration problems in England, so I had to move to America.” He set up in Lititz with a staff of one; now there are 600. In league with Clair Global and others, Tait has recently opened Rock Lititz, a purpose-built rehearsal hall. The size of an arena stage, it has already welcomed Beyoncé and Taylor Swift, possibly the two best-rehearsed acts in pop.

An Amish buggy on the road in Lititz. Photograph: Larry Keller/Getty Images

If the live-event industry started here because of the Clairs, it stayed because of the local culture. “The supply chain allows us to build anything,” Davis says. “When farmers have a problem, they fix it themselves, and eventually it turns into an invention.”

“We still use a company that makes cattle grids,” Fairorth says. “Literally, a Mennonite company that can cut pieces of metal for a rock show.”

As they left the National Theatre in March 2015, Nick Hytner and Nick Starr wanted to build a new company, and new theatres. When they found a space earmarked for culture beneath a new block of flats by Tower Bridge, it was just 12m high. “That meant we could only build in steel,” Starr says, “because you need a material that is strong enough and thin enough to stack the audience on top of each other.” He sees this not as a problem, but as serendipity, since it led them to Tait.

By December 2016, Starr was watching the first of 57 truckloads arrive from Haverhill in Suffolk, where Tait has forged a 900-seat theatre. Starr points delightedly at the back door: “We’ve got 9mm clearance there.” The Bridge will be central London’s first purpose-built commercial theatre in a century. The laws of building state that the client soon falls out of love with the builder, but Starr is still smitten with Tait. “They’re pretty remarkable, aren’t they? They put this huge focus on quality and design. They reckon the money will look after itself if you get that right.”

Not that they’re cheap: budgets range from $100,000 (£80,000) to $20m. “It’s got to be the best shit,” Tait says. “Because we’re a trifle more expensive.”

“The biggest ingredient,” Devlin says, “is time more than money. You’ve got four hours to erect an entire set in an arena and four to five hours to take it down. It’s just the best form of Lego – everything clicks together.”

Madonna live in Tel Aviv in 2012. Photograph: Jack Guez/AFP/Getty Images

Back in Lititz, Davis is peering into the future of live entertainment. “We’ve gone from the performance to the event to the entire experience. We’re working on virtual reality, where you don’t really need a star.” One day, fans will go to a gig and find the air full of drones. “There are four major obstacles to bringing drones inside – safety, swarm algorithms, indoor positioning and indoor communication. We’ve cracked the first three.”

He pauses to let some light in on Aladdin’s magic carpet. “There are strings, but they’re spinning at 1,100rpm. If it moves fast enough, you don’t see it.” But the creation he seems most proud of is something very modest. “You see that purple wheel?” he says on the shopfloor. “That’s our set-cart wheel. It’s fifth-generation, it’s our plastic, our rig, our bearing. It’s not that we set out to make the best wheel. It’s $75, everybody else’s are $35, but it means that two people can push a set cart around rather than four, and it can go over cobblestones.” These artful physicists haven’t just cornered the market: they’ve reinvented the wheel.