“Starting with the Universe,” an exhibit on the life and works of R. Buckminster Fuller that opens today at the Whitney Museum of American Art, presents the unique opportunity to step into one of the masterpieces of Brutalism to see a show on uninhibited idealism.

The Whitney twice dodged a bullet, first in 1985, when the Michael Graves plan to drown it in a Postmodern pastiche was scrapped, then again in 2003, when the insolent Rem Koolhaas proposal for a slouching tower was shelved. By 2005, when Renzo Piano drew up plans that were ultimately and unfairly rejected by the community, the museum was finally seen for what it is, a jewel box of a building, as much a work of art as anything hanging on its walls. Marcel Breuer, its architect, was a craftsman, and even the rarely used stairwell has exquisite details.

This was not a kind of architecture practiced by Bucky, as Fuller (1895-1983) is universally known. The Whitney was finished in 1966, one year before the opening of Fuller’s greatest architectural achievement, the United States Pavilion for Expo 67 in Monteal. Where Breuer’s building is a good host, and an inviting place to linger over Fuller’s creative output, the 200-foot transparent geodesic dome was the headliner at Expo 67, a soaring form that outperformed the show on display inside.

Now called the Biosphère, the dome is still eye-catching, and seems like something from the future. That is, if you free yourself from all the associations that such domes now carry: sweat lodges and moldering plywood houses, military warehouses and temporary science stations in the Antarctic. Once at the cutting edge, the dome has become a shabby artifact of hippies and the government.

How strange that one of the most remarkable feats in the history of engineering could become mundane.

For thousands of years, buildings followed a simple rule: Walls hold everything up. Then the industrial revolution introduced a kind of construction that we still use today, in which iron columns provide the structure and the walls are pinned to a frame. Now walls are a skin, a way to keep out the elements and, one hopes, add some ornamentation.

But in a geodesic dome, the skin is the structure. Even more revolutionary, it’s a structure that is far lighter, stronger, and easier to construct than any other similar building type. Bucky lore is filled with stories about unskilled laborers assembling a monumental dome in a couple of days.

More than 300,000 have been built since the geodesic dome was patented in 1954, a staggering number, and I doubt that includes the playground equipment that’s a part of the childhood of everybody born after the mid-’60s.

The Whitney show saves the geodesic dome for the end. There’s only so much dome you can get into the Breuer building (though the curators try their best, and re-create a 1954 dome out of cardboard), and the most beautiful objects are his teaching models, complex geometric forms made out of painted dowels and colored paper, most the size of a basketball. They look like Montessori toys, which in a sense they were, simple visual tools that show how nature is a puzzle to be solved.

It’s this utopianism that gets Fuller into trouble, or at least tarnishes his legacy. Born in 1895 into one of New England’s oldest families — only a blueblood could get away with a name like Bucky, be kicked out of Harvard twice, and spend World War I patrolling the Maine coast with his mother’s yacht — he was a hereditary freethinker.

He could famously lecture on engineering or Neolithic culture or sailing for eight, 12, 15 hours straight, using coinages such as “livingry” (meaning the opposite of weaponry), popularizing words such as “synergy,” and calling this planet “Spaceship Earth.”

He stopped drinking, according to a 1966 profile Calvin Tomkins wrote for the New Yorker, republished in the exhibition catalog, because he didn’t want people to think his were the rantings of a drunk.

But there is a loopiness to Fuller’s ideas, and the perception isn’t helped by a lifetime of near-successes. A prototype of his Dymaxion House, a mass-manufactured “machine for living,” was included in the inaugural 1939 show for the Museum of Modern Art’s new home, but none was ever sold. The Dymaxion 4D Transport, a three-wheeled car from the early 1930s faster and more nimble than any automobile then in existence, was in a traffic accident and never saw production. (A stunning 1934 model is on display at the Whitney.) In 1955, Walter O’Malley wanted a geodesic dome over a new stadium for the Dodgers, but Robert Moses blocked the plan, and two years later the baseball team left for Los Angeles.

Had any of these projects made it through, he would have been celebrated as one of the most successful engineers in the history of this country. Instead, he’s merely one of the most visionary.

It took a half a century for the world to catch up to Fuller, and by the 1960s he had become a counterculture hero. The very last item in the exhibit is a video loop from 1967 of Fuller, dressed in a natty three-piece suit, talking to a group of shaggy if earnest men and women in their 20s. The video lasts about an hour, just a taste of how he could hold forth, alternately losing his audience and blowing their minds.

By then he had been blowing his own mind for more than 70 years. In 1985, two years after his death, an element was discovered, C60, that had the same structure as the geodesic dome. It was named buckminsterfullerene, and its discovery led to a Nobel Prize. Now an entire class of carbon molecules are known as “buckyballs.”

It turns out that Fuller didn’t invent the dome so much as he divined it. The universe had engineered it long ago.

Until September 21 (945 Madison Ave. at 75th Street, 212-570-3600).