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Katleen first traveled to the Hamptons nearly a decade ago while working and living in New York City, and she was immediately entranced. "It was the topology of the place," she says. "You have countryside next to the beach. You can bike-ride to the ocean. Nothing like it exists in Europe."

A few years later, she and her husband fell in love with a home they were renting for the summer. "It had a saltbox feel," she says, comparing it to the first colonial-era houses built in the area hundreds of years ago. To her eye, the nonlinear rambling flow of the architecture—comprised of small structures linked with sunlit passages—made the result resemble a group of farm buildings. Without the hay. Or the smell. "But at the same time, it was very modern," Katleen adds, like some geometric abstraction of an all-American country house. By the end of that season, she and her husband had purchased the place from the architect who had just built it for himself.

Sebastian and his cousin, Maxime, 11, play at a chalkboard-topped table from Offi.