The exhibition is about amazement, marvel and awe. But as droves of Renwick Gallery visitors gape at the large-scale installation art in “Wonder,” the newly reopened museum’s inaugural show, curator-in-charge Nicholas Bell is more amazed by something else: their phones.

Thanks to a few well-placed signs announcing “Photography Encouraged,” smartphones are omnipresent when you walk into the Renwick. It’s no surprise. Who wouldn’t want to photograph this show? Every artwork is an Instagrammer’s dream come true.

There is a giant rainbow made of delicate threads (Gabriel Dawe’s “Plexus A1”), and a magenta room with an elaborate arrangement of exotic insects on the wall (Jennifer Angus’s “In the Midnight Garden”). Your silhouette in front of the stalagmites made of index cards (Tara Donovan’s “Untitled”) makes for a perfect social-media profile picture. Your head poking through what some have described as a giant bird’s nest (Patrick Dougherty’s “Shindig”) is Snapchat gold.

In fact, go to the Instagram page where users with public accounts have tagged the museum in their photos, and you may never stop scrolling. Fashion ’grammers pose in front of the art — or even just in the hallways — some tagging the photos with #ootd, or outfit of the day. The exhibit has been the backdrop for several weddings. And of course, selfiesso many selfies. (But not with selfie sticks — the Smithsonian Institution has outlawed them.)

Every day, Bell scrolls through the more than 20,000 Instagram images with the hashtag ­#RenwickGallery, bewildered and grateful. He has never seen anything like it.

“We’re all flabbergasted, to be frank,” says Bell, who keeps screen shots of his favorite Instagram posts. He flips through a few: a girl in front of “Plexus A1,” eyes wide with amazement. Visitors lying on the floor of the grand salon, photographing Janet Echelman’s “1.8,” a 96-foot installation hung from the grand salon ceiling. A photo of someone taking a photo of the exhibit.

“I wonder, what are they even trying to say? ‘I am here Instagramming’?” Bell says. “It’s like this new first-person narrative of the museum experience. I’m fascinated.”

Since the museum reopened six weeks ago, it has shattered attendance records. As of Sunday, Jan. 3, 176,000 people had visited, spokeswoman Laura Baptiste says. The average yearly visitor count between 2011 and 2013 was approximately 150,000.

Yes, they’re coming to see the art — but more important, they are coming to photograph, and be photographed with, the art. Sometimes it’s hard to tell which they care about more. “Wonder” is an exhibit that “invites the visitor to treat it superficially,” Washington Post art critic Philip Kennicott wrote in his review, foreshadowing some visitors’ current obsession: snapping a perfect skinny-arm-posed photo in front of the art.

Visitors surround Gabriel Dawe’s “Plexus A1.” (Alex Fogg)

A visitor poses inside Patrick Dougherty’s “Shindig.” (Lisa Peterson)

Many museums prohibit photography in temporary exhibits because of copyright concerns. Conservation is a worry, too: Camera flashes can damage delicate works, and crowds elbowing their way through to take photos may inadvertently jostle nearby pieces. And of course, there is the visitor experience to consider: At the ­Louvre in Paris, you can barely see the Mona Lisa through the forest of arms holding up cellphones in front of it.

In an attempt to combat the throngs of cameras, Amsterdam’s Rijksmuseum announced a program in October that encourages guests to sketch works of art instead.

But there is another reason museums may want guests to put away their phones. According to a 2013 study, people who take photos of a work of art don’t connect as deeply with it and are less likely to remember it.

“You’re not looking at this giant, rich, textured, nuanced thing,” says Linda Henkel, a psychology professor at Fairfield University in Connecticut who wrote the study. “You’re already reducing it by looking at it on the screen.”

But not everyone comes to a museum for a memorable experience.

For many folks, “part of their purpose is they want to show other people what they’re doing,” Henkel says. If that’s what they want, “the photo is a trophy. Their experience is not a very rich experience then.”

This doesn’t worry Bell. Far from finding the picture-taking disruptive, he says, “I think that for different visitors, they’ll find their own ways to engage most intimately with the exhibition. For some people, that will be to not take photographs, and for some people, that will be to take photographs. I don’t think that we should judge.”

Dawe, whose 48-foot rainbow-threaded installation is among the most popular with photographers, agrees.

“In the exhibition, you’re immersed in the work, and there’s no way of not being immersed in the work,” says Dawe, who likes it when people tag him in photos of his art on Instagram. “I don’t think trying to capture that moment with your phone takes away from the experience.”

Although, he adds, “selfies are kind of obnoxious.”

Visitors observing Janet Echelman’s “1.8.” (Ishani Sharma)

But those selfies have brought a whole new audience to the Renwick, including many people who didn’t know that the museum of contemporary crafts existed before it started showing up in Snapchat. (Blame The Post, too: The Weekend section called the Renwick the “Most Instagrammable Location” last month.)

“Everyone was posting about it on Instagram and Snapchat, so it looked really cool,” said Samantha Quintanilla, a Towson University student who was aiming her phone earlier this week at Leo Villareal’s “Volume (Renwick),” a digitally controlled twinkling chandelier in the grand staircase.

“It’s super trendy right now,” said Meredith Hester, a physician assistant who was snapping pictures of John Grade’s “Middle Fork,” a massive tree trunk constructed from reclaimed cedar.

“Wonder” has been so thoroughly photographed that visual cliches have emerged: people popping through the holes in Dougherty’s nest, or silhouetted in front of or behind Dawe’s rainbow.

As photographers try to distinguish themselves, Bell has observed micro-trends. People are taking photos of his “Photography Encouraged” signs. During finals week, students posted jokes about studying while taking photos with the index cards. A few people have been spotted doing yoga poses inside the art. (Not a good idea, Bell says: “If you’re going to do a headstand, I’m a little concerned that you’re going to fall into the artwork.”) And more people have been focusing on detail shots of the pieces.

Interestingly, Henkel found that participants in her study who were instructed to take detail photos actually remembered more about the art than those who took zoomed-out shots.

Tara Donovan’s “Untitled.” (Rachel Barron)

Jennifer Angus’s “In the Midnight Garden.” (Rachel Barron)

There are still people who like to see art the old-fashioned way.

“I tend to want to absorb and want to experience it more than living through the camera,” said Suzanne Gesin, one of the few people who walked into the grand salon and didn’t immediately take out her phone. She wondered if all the photographers were “just cataloguing it, like checking a box.”

“People who take a picture with that rainbow thing are like, ‘Look at me, I have great taste,’ ” said visitor Petro Nungovitch, who added that he is “not huge on posting pictures.”

To achieve a sense of wonder, Henkel suggests that Renwick visitors take photos at the end of their time in each room, rather than at the beginning.

“Have the experience, see it, and then try to capture it,” she says. “With art, it’s that initial impact, that initial moment,” that packs an emotional punch. “In a distracted moment, you might lose some of the message.”

But Bell isn’t concerned about people losing the message. He is basking in the museum’s newfound popularity.

“I’d be lying if I told you that we don’t kill ourselves to produce exhibitions here that don’t get very much attention,” he says. “The show that we did when we closed the building — I did a show about baskets — I loved that show.” But it was “like a tree in the forest that falls and nobody hears it.”

The exhibition’s reach will affect programming for years to come. Although Bell says that the gallery can’t do large-scale installation art all the time — “It’s not really what this museum is about” — the Renwick and the Smithsonian American Art Museum are discussing ways to make it more prevalent. And with only a few exceptions, the “Photography Encouraged” policy is here to stay.

In fact, it has become integral to the museum’s new self — or selfie? — image. “This is really a part of who we are now,” Bell says.

A scene from Janet Echelman’s “1.8.” (B.J. Adams)

Correction: A previous version of this story misspelled Gabriel Dawe’s name.