Albert Einstein, when working through the physics that would become the Theory of Special Relativity--and alter the world's understanding of space and time--sat at a desk in a small, conventional, second-floor room in a house at Kramgasse 49 in downtown Bern, Switzerland.

At the same time, during the first decade of the 20th century, adventurous architects and artists were becoming profoundly impatient with inherited notions of space and time in their own disciplines. It is a famous instance of art and science moving along parallel paths.

For architects, ironically, Einstein's room was the problem. Oh, not that particular room, of course, but a whole spatial system of which the young physicist's abode was simply a microscopic part. It was a world made mainly of right-angled rooms and rectangular buildings through which one moved in linear fashion, as if there were a predetermined goal that was the same for everybody, in every circumstance.

To a significant minority of architects, this system of fixed relationships--based on Renaissance perspective and, ultimately, on the classical architecture of Greece and Rome--did not seem fitting for the fast-changing, dynamic world of their times. This disaffection made differing ideas about space a defining characteristic of 20th-century architecture.

Not the defining trait, heavens no. Architecture in this century has been too richly varied to sum up under any single rubric. In no previous century has so much change occurred in architectural materials and technology, for example. Change itself--and the variety of expression that follows from it--has been a significant leitmotif of the century's architecture.

But in many ways it has been a circular kind of change in which the same issues return time and again. Questions about how and why to shape space differently keep coming back. It is an issue as old as architecture itself--all buildings enclose or define space in some way. But in the 20th century, the idea of creating new, more meaningful spaces became a rallying cry for avant-gardes.

During the late 1800s, the way was being prepared for this century's rebels. In Paris and most other European capitals, the plantlike, sinuous lines of art nouveau made striking contrasts to pervasive formality and tired, overwrought ornamentation. Even if the style was more a matter of surfaces than of fundamental change, the implications were strong--architecture was something that could, and should, reflect the natural world.

Meanwhile, in Barcelona, Antonio Gaudi had begun his more forceful, more thoroughgoing mutations of form and space, which continued until his death in 1926. Gaudi's buildings are strong examples of an individualistic, imaginative, curvilinear alternative both to traditional architecture and to much of the modern work that would follow.

Thousands of miles away, Frank Lloyd Wright was perfecting his Prairie houses in the suburbs of Chicago. With their pinwheel floor plans centered on the hearth and heavy, low-slung exteriors, these residences were Wright's first brilliant steps in a lifelong quest for a multifaceted "organic" architecture.

Much later in the century, architect Philip Johnson would refer acidly to the still-living Wright as "the greatest architect of the 19th century"--a dig at Wright's age (he was born in 1867) and his undeniable roots in the Arts and Crafts movement of the late 1800s. In essence, however, Johnson's quip could not have been more wrong.

The spiral Guggenheim Museum on Fifth Avenue in New York (conceived in 1943 and completed in 1959, the year Wright died) is perhaps Wright's most famous building and his most dramatic break with the conventional box. But to Wright, geometry was a malleable marvel--he made dynamic, shifting spaces based on the right angle (perhaps just to prove he could do it) and used circles, half-circles, triangles, hexagons and equilateral parallelograms as the basis of his designs. Wright was, one is forced to conclude, the century's most consistently inventive manipulator of space.

In Europe, the destructive cataclysm of World War I released immense creative forces--the break with the past that prewar architects had envisioned truly occurred during the '20s.

Architects and artists in post-revolutionary Russia together created a vast, varied amount of work collectively known as Constructivism. Radical abstraction, asymmetrical spaces and a fresh approach to construction techniques and materials characterized the architecture, which had worldwide influence. But the movement's greatest spatial concept--Vladimir Tatlin's immense, spiraling Monument to the Third International--was never built.

In Germany, the expressionist visionaries who, during the war, had prophesied new, utopian spaces began to get opportunities to build their dreams. An important example is the Einstein Tower in Potsdam, an astrophysical observatory built in the early '20s and named after Einstein, by then the preeminent German scientist. The architect was Eric Mendelsohn, who believed the white tower with its fluid lines symbolized Einstein's science--"the building's massing has overcome gravity and inertia," he wrote.

The rectangular room with four walls and ordinary windows did not go away, of course, and never will. It serves many purposes well. Even during the heady '20s, it remained the norm because of its age-old structural stability and because people liked it. And even the Bauhaus in Dessau, the famous 1925 building designed by Walter Gropius for the century's most influential art academy, was rectangular in both plan and elevation. It showed its modernity, however, in its abstraction and its skin of transparent glass.

Bauhaus architects and other modernists of the period embraced the right angle as a basis for building because it accorded well with their rationalist beliefs. Theirs was called a "machine aesthetic" because they thought that machines were redefining conceptions of beauty and that the rationality of the assembly line was a model for architecture and planning.

Many a wonderful building was inspired by Bauhaus ideas. With its "open," asymmetrical floor plan, Ludwig Mies van der Rohe's German pavilion for the 1929 Barcelona world exhibition is one of the century's great buildings--elegant, simple and magical. But the functionalist planning produced by this branch of modernism was a worldwide failure when it was put to widespread use after World War II, and the machine aesthetic turned out to be compatible with the most frightfully mediocre, depressingly orthodox architecture. (Washington and its suburbs are littered with examples.)

Perhaps Charles-Edouard Jeanneret, better known as Le Corbusier and one of the century's giants, had this decline in mind in the mid-'50s when he designed the Chapel of Notre Dame du Haut in Ronchamp, France. In contrast to his earlier, rationalist designs, the chapel with its thick concrete walls and "nun's cap" shape is dramatically expressive and its interior space deeply mysterious. The Ronchamp building, as it happened, had a terrible immediate effect--a whole school of self-styled "Brutalists" arose in its wake--but it helped in the long run to keep the spatial debates alive.

History-conscious postmodernists, the avant-garde of the '70s and '80s, rediscovered the virtues of the traditional room along with those of the traditional street. Of the two rediscoveries, the latter is the more important, but the former also had its uses. Many of the spaces created by conventional modern architecture by then had become anonymous and off-putting, and reexamining tradition was a way of reintroducing spatial order and character.

The enfilade of classical galleries in British architect James Stirling's wonderful 1984 Neue Staatsgalerie in Stuttgart, Germany, is an excellent example. Like the big rectangular chambers in the 100-year-old Corcoran Gallery of Art, they are superb, naturally lit containers for paintings.

Still, there was a certain smugness at work in postmodernism, and the '90s have been a wonderful relief. In a sense, the century has saved its best for last, for never have the debates about the meaning and form of architectural space been more vigorous and invigorating.

Admittedly, the old avant-garde optimism is gone, but the possibilities being talked about--and built--are exciting in their variety. Daniel Libeskind's 1999 Jewish Museum in Berlin, embracing the painful paradox of absence at the very heart of its zigzagging spaces, is one example. Frank Gehry's 1997 Guggenheim Museum in Bilbao, Spain, breathing metaphors from every pore of its sculptural, titanium-sheathed skin, is another. Talk about breaking out of the box!

California architect Eric Owen Moss's reaction to the flood of praise the Gehry building received seems just right for a century's end. The building is great and the praise deserved, he noted, "but it is just the beginning of something, not the end."