When Leonard Rosenberg and Colombe Nicholas’s second child was born, the couple launched a new family tradition: Rather than having traditional family photos taken, they commissioned some of the best-known artists of the 20th and 21st centuries to create portraits of their son, Ian, now 17, and daughter, Morgan, now 21.

“We wanted to instill in the kids a love of art,” Ms. Nicholas said. “You can always hand money down to your children, but we thought this was something really special to give them.”

The first piece they commissioned was a series of photographs by the artist William Wegman, best known for photographing dogs in human poses and dressed in clothes. Mr. Wegman photographed the Rosenberg children on the set of a Cinderella book he was doing, with Morgan dressed as the lead character, a costume later worn by one of Mr. Wegman’s dogs. Other pieces include portraits of the children in chocolate syrup by the artist Vik Muniz, who is known for creating replicas of famous works in unusual mediums such as peanut butter and jelly, and two Julian Schnabel “plate paintings,” with Mr. Schnabel using shards of broken plates as the ground for painterly figures of Ian and Morgan.

Most recently, the couple commissioned videos by Kolkoz, a duo comprising the artists Samuel Boutruche and Benjamin Moreau, which depict day-in-the-life scenes of each of the children, with videogame-style animation. The screens are placed inside cases that look like large cell phones — a pink one displays the video of Morgan and a blue one holds Ian’s. Morgan’s video depicts a scene in which the two French artists travel to upstate New York to visit her in her college dorm; Ian’s video follows him and his friends around their downtown neighborhood.

The Rosenbergs’ 4,500-square-foot, three-floor loft features white walls, high ceilings, and large pieces of modern art on just about every wall, creating a gallery feel.

Along the wall, just below the ceiling on the second floor, is a “living” portrait of the couple that includes a string of events and dates. There is “Bloomingdales 1962,” the date Mr. Rosenberg began a career at the department store; “Felt Factory 1994,” representing the family’s relocation to Ohio for three years to run Ms. Nicholas’s family’s paper-maker felt factory, and “Sarah 2001,” when Mr. Rosenberg’s mother, Sarah, died.

The piece was created by the late artist Felix Gonzalez-Torres, who asked the couple to submit important moments in their lives, and added some of his own, marking important moments in recent history. Some more have been added since the artist’s death in 1996. “His idea was that a portrait is never finished, and we could keep adding to it,” Mr. Rosenberg, who is now a private art dealer, said. “He thought that a portrait was something that was internal, and it was his idea to install it around the room.”

On the same floor, there is a drawing and sculpture by Sol LeWitt, a butterfly painting by Damien Hirst, and dining room chairs by Frank Gehry that are inspired by apple crates. There is a sculpture by John Chamberlain made entirely of automobile parts, which Mr. Rosenberg describes as an important early work of the artist. Originally, the artist had left it untitled, but he later named it “Roseville” in honor of Mr. Rosenberg.

On the wall between the second and third floors hangs a piece the couple commissioned from the artistic team known as Ericson and Ziegler. The artists received tiles from the quarries that supplied some of the city’s landmark buildings. They then inscribed the name of the landmark building and the date it was constructed on each tile, and arranged them north to south, creating a map of New York.

The apartment’s bottom floor serves as the family’s living area. Aside from the bedrooms, there is a family room that is in direct contrast with the rest of the apartment’s sleek, modern feel: a ski lodge-style, wood-paneled living room, complete with fish on the walls, taxidermy, and many rustic pieces the couple found at country auctions during their time in Ohio. The walls of the room are covered with the old-fashioned wood conveyer belts from the Nicholas family’s factory, Orr Felt.

When they first moved to TriBeCa 27 years ago, Mr. Rosenberg and Ms. Nicholas were pioneers. “We used to have to go to New Jersey to do our grocery shopping,” Ms. Nicholas said with a laugh. “The building took a while to take off and people were slow to move in.” When they decided to buy, they bought two apartments, with plans to eventually combine them. Initially, though, they planned to rent out the bottom floor, but had no takers. “Absolutely nobody wanted to live here,” Ms. Nicholas said, pointing to the 1,000-plus-square-foot floor.

The top-floor apartment included a roof, where Mr. Rosenberg and Ms. Nicholas built a deck. They also turned a utility closet under the building’s water tower into Mr. Rosenberg’s office. He still works there.

If ever their modern, gallery-style apartment starts to wear thin, the couple can always retreat to their 1780 farmhouse in Water Mill, N.Y. “It has low ceilings and an antique feel; it’s the exact antithesis of this,” Ms. Nicholas said.