Some see a frozen tear clinging to the “cheek” of the new Salvador Dalí Museum in St. Petersburg, Fla. Others see a melting crystal. Depending on the viewer’s vantage point, still others say they are reminded of a misshapen potato, a nose, an amoeba and a dolphin in a nosedive. The builders of the glazed atrium structure that drapes over the side of the boxy building simply call it the “enigma.” After all, the builders had to solve a mystery of how to shape, engineer and hang a transparent and organic structure—75.5 ft at its tallest, 105 ft at its widest and 27.5 ft at its deepest—and still ensure the delicate-looking structure can withstand hurricane-force winds and windborne debris.

“No one thought this was achievable,” says Mark House, managing director of the Tampa office of The Beck Group, the $29.8-million job’s construction manager at risk.

But even with the complex atrium structure, the Dalí museum is poised to open Jan. 11—on time and $700,000 under budget. “[Based] on cost per sq ft, it is one of the least expensive museums built in the last 10 years,” says House, who notes that Beck and the museum split the savings.

The 68,000-sq-ft museum is a replacement for a facility, also in St. Petersburg, that houses a Dali collection compiled by A. Reynolds Morse and Eleanor Morse. The new building, on a vulnerable site facing Tampa Bay, consists of a cast-in-place, 58-ft-tall reinforced-concrete box with one-way slabs and beams.

Mostly for economy’s sake, the three-story building’s 18-in.-thick perimeter walls are architecturally exposed on the outside, with no stucco or painted surfaces. All the museum’s art, mechanical equipment and computers are housed on the second and third floors to keep them well above the flood line. On the face of it, the building had to be a fortress to resist 165-mph hurricane winds and “the occasional flying yacht during a storm,” says Yann Weymouth, director of design in the Tampa office of architect HOK Florida.

But the architect wanted something provocative inside the mostly rectilinear box as a way to invite the visitor upstairs to the third-floor galleries. A grand spiral stair, crowned by a rooftop skylight, did the trick. The skylight then morphed into the enigma to contrast with the conventional shape of the building, says Weymouth.

Weymouth dubbed the structure the enigma because of its mysterious shape, which wraps around two sides of the building and ends on the roof over the grand spiral. Enigma is also the name of a 1929 painting by Dalí, who was known for surreal artworks with jarring juxtapositions, much like the ordinary box and the free-form enigma and its smaller-entrance sidekick, dubbed the igloo.

The faceted, glazed form, which curves without curves, is “very Buckminster-Fuller-like,” says Weymouth, referring to the inventor of the similarly triangulated Geodesic dome. But “a dome would have been corny,” especially against the Cartesian, Euclidean box, says the architect. The project team might accuse the enigma of being surreal, but never corny. For though they all look alike, no two of the system’s 1,062 triangular glass facets or the 3,000 steel pieces that frame them are dimensionally similar.

It’s as if all they were working on a jigsaw puzzle in which all the pieces were the exact same color, says Ian Collins, CEO of Novum Structures, Menomonee Falls, Wis., which engineered, manufactured and installed the 13,025-sq-ft glazing system under a $2.6-million contract. The entire enterprise, including the geometry and structural and wind analyses, was made possible by computer power. “Bucky couldn’t have done this,” says Weymouth, who selected Novum to supply the system.

Straight, Flat Free Forms

The atrium is the first major installation in the U.S. of Novum’s free-form system. For five years, the firm has been developing its method to modularize free forms and create any shaped surface from straight lines and flat surfaces. Structural analysis to fabricate the pieces has to be automated, including machining the nodes using computer-numerically controlled (CNC) cutting equipment, says Collins. “Otherwise it is a nightmare,” he adds.

HOK used a bevy of digital design tools, including 3D building information modeling, as well as physical models to develop the surface geometry. Then, Novum took HOK’s BIM into its proprietary engineering and fabrication software. The final tessellation, or collection of plane figures that fill a plane with no overlaps or gaps, was determined by Novum’s structural algorithms in order to optimize the glass size for transparency and views. It also helped to minimize the weight of the steel frame. “That would save money, save on joints and save on labor,” says Weymouth.

Novum’s strategy during the two-year preconstruction phase was to normalize the atrium’s grid and optimize the surface geometry while maintaining a shape acceptable to HOK, says Collins. The best-case scenario for optimal cost and fabrication of regular triangular geometries is to use same-size glass shapes and beam lengths, says Collins. For a free-form shape, the goal is to make the glass panels and beams as close in size as possible.

Another goal is to minimize angles under 40 degrees between adjacent components. The rationale is that, as the angle gets smaller, the connection has to get bigger because the members start to collide. Collins says the 40-degree rule was only...