Each semester, I begin my course on Structures in Architecture at Columbia University with an exploration of structural forms through history. I do this to reinforce how social, economic, and technological conditions shape our design decisions. History shows that structural forms weren't arbitrarily used—they emerged in response to specific challenges and constraints. As designers, we should strive to do the same in our work today.

Ancient structures

The concept of structures is as old as our species, and for many thousands of years our nomadic ancestors built temporary dwellings from whatever they had laying around. None of these structures were built to last. In contrast, our relationship with “permanent” structures is on the order of 15,000 years old, and it was catalyzed by some of the first major human settlements in the Fertile Crescent region (now Palestine, Jordan and Israel).

Our ancestors’ approach to innovation was pretty simplistic: intuition and the concept of trial-and-error drove the evolution of ancient structures for millennia. When something fell down, builders learned what not to do. When a new idea succeeded, the breadth of humans’ realm of structural capability got incrementally bigger.

Trial-and-error continued to drive our development for a long time, through the first millennium of the Common Era and beyond. Throughout this time, the design of a building - architecture, structure, airflow, light, and construction means and methods - were collectively the remit of a singular designer: the master builder.

The use of stone as a primary building material was ubiquitous throughout this ancient era - it was an ideal material for resisting weathering and rot. If you were a king, shah, basileus, pharoah, tlatoani or devaraja, the places where you slept, worked, entertained and worshipped were constructed from stone. The structures required immense cost and effort, and in return they’d be around for a long time.

The stone used for these structures varied in strength, but all stone was strong relative to the alternatives. When the Romans built the Temple of Jupiter at Baalbek, for example, they used granite imported from Egypt.

*Temple of Jupiter at Baalbek (image courtesy of Paul Saad).*

Being granite, the strength of the material in these impressive columns is likely on the order of 30,000 pounds per square inch, meaning that the material would fail by rupturing if 15 tons of force were applied to each square inch of the column’s cross section. Given that the granite weighs about 170 pounds per cubic foot, the Romans would have been able to stack the granite to a height of about 25,000 feet before the sheer weight of the column would crush the stones at the base, higher than any mountain outside of the Himalayas.

Stability, not strength

Obviously, the Romans never got that high. Long before a stone column would fail by crushing, it would simply tip over. No matter how exacting one’s methods might be, a stacked stone column will inevitably succumb to this kind of instability from a cocktail of real-world conditions that include dimensional imperfections, slight eccentricities in stacking, and the chaos of the environment around it. In the world of ancient structures, stability was job number one.

Through trial and error over centuries, skilled builders discovered key structural forms that kept stacked stone buildings stable. Columns and walls provided vertical support, while arches, vaults, and domes made horizontal spans possible.

Both of these construction vectors were put to the test in the first millennium of the Common Era, when the size of religious structures - notably cathedrals for Christianity and mosques for Islam - grew to accommodate the ballooning size of their congregations. The Hagia Sophia, an iconic structure from the Byzantine era, was completed around the year 537 with sufficient capacity for thousands of worshippers. To do this, designers used a series of cascading domes and half-domes to span the stone roof from its peak down to the surrounding perimeter walls.

*Sectional view of the Hagia Sophia (drawing courtesy of Wilhelm Lübke).*

Evident in this section are the semi-circular, constant-radii curved forms intrinsic to the Late Roman and Pre-Romanesque eras. Builders in this time found that such forms could effectively span large spaces, but required significant thickness to do so, meaning that roof structures were very heavy. Correspondingly, the walls they landed on needed to be immensely strong to resist the lateral thrust of these domes, which pushes sideways at the top of the perimeter walls. To resist this thrust, builders added massive piers for fortification, which can be seen very well in the aerial image of the building below.

*Aerial view of the Hagia Sophia, showing large piers to buttress the perimeter wall against lateral thrust (image source unknown).*

The weight of these massive forms had a significant unintended consequence: as buildings of this era got larger, they also got darker. The four piers on the western wall of the Hagia Sophia (seen in the above image), dramatically reduce the amount of natural light captured by the perimeter windows. The obtrusive forms also forced builders to reduce the size of windows in the perimeter of such large structures.

By the end of the millennium, much of the Pre-Romanesque architecture in Europe was characterized by interior spaces that were dark, with stagnant air and little ventilation or passive climate control. (I need to note here that some scholars contend that dark interiors may have been intentional at times, to foment a sense of “mysticism” within the space. That perspective notwithstanding, it’s clear that the architectural style precluded large windows and our contemporary understanding of human wellness characteristics paint this as a clearly problematic thing).

Comfort wasn’t the only problem plaguing Pre-Romanesque designers. The sheer volume of stone material needed for these structures was massive. All of that stone had to be quarried, transported and prepared on site. Formwork had to be large enough to hold the growing weight. Cost and labor requirements ballooned. Construction schedules grew more and more protracted. Inefficiency reigned. A revolution in design thinking was needed.

A revolution in France (no, not that one)

The solution to these very big problems became evident in the suburbs of Paris during the 12th Century, when Abbott Suger of the Abbey of Saint-Denis was presented with a novel logistical challenge. Suger was tasked with the design and construction of a new cathedral for Saint-Denis. He had to build the new cathedral on the same site as the older, Romanesque-style church. And he had to preserve the functional capacity of the existing church until enough of the new cathedral was built to enable a transfer of liturgical practices.

Such constraints pressed the need for lightness in both the new cathedral’s structure and its corresponding formwork. Suger brought to the challenge an awareness of then-niche design practices that deviated from the semicircular forms found in the Hagia Sophia and elsewhere. Instead, Suger adapted a catenary form for his arches and roof vaults, which had been proven in Islamic and Byzantine use cases to remain stable when built out of much thinner material.

To save on this stone material and reduce the size of the structure, Suger had to contend with more complex geometric forms. The result however was exceptionally compelling: when the nave first opened in 1144, the consecration ceremony was attended by King Louis VII, who was introduced to a cathedral that exemplified the Gothic era - a step change in topological complexity that yielded a space bathed in light and openness, a dramatic departure from its Romanesque predecessor.

*Apse of the Hagia Sophia, left, versus the apse of the Basilica of Saint Denis, right (images courtesy of Tugba and Wikimedia Commons).*

Suger had trialled the Gothic Style and proven that it yielded a better end product and a far more efficient process in getting there. As the style spread, it unlocked such efficiency that a cathedral building boom followed: more than 80 cathedrals were built on the Île-de-France over the next 300 years. To deliver them, builders quarried more stone than was needed for all of the Pyramids of Giza. Central to this catalyst was a toolbox of new forms: the pointed arch and the gentle curves of the slender flying buttress.

*Flying buttresses beside the nave of Notre Dame de Paris (image courtesy of Getty Images).*

The science behind it

The Abbott Suger pioneered large-scale use of niche forms not widely used across Europe at the time. But his faith in these new forms didn’t stem from a clear understanding of why the pointed Gothic forms performed better than their semi-circular Romanesque counterparts. His motivations came not from deep theoretical understanding, but rather from practical intuition and centuries of accumulated trial-and-error knowledge in design and construction.

A broad understanding of the principles behind the efficiency of catenary forms would come in the 17th century, through early work by Galileo Galilei and formalized by mathematicians and physicists including Christiaan Huygens, Gottfried Leibniz and Johann Bernoulli. These individuals demonstrated that stability - not strength - was of utmost importance in determining the fidelity of a stacked stone structure. They asserted that the most materially efficient structural form - the form that required the least amount of material - carried the applied loads to the point of support using pure compression.

What is the ideal compressive form then? The great minds of the 17th century formalized the notion that the most efficient compressive form for a stacked stone structure was the exact opposite of the most efficient tensile form, or the shape a string or chain would take under the same loading pattern.

Left image: The shape of a string subjected to three different loading patterns (consisting of one, two and three intermediate weights suspended from it), and the corresponding “purely compressive” form - an inverse of the purely tensile. Right image: Jasper Johns’ The Lagoon, comprised of a series of perfect mathematical catenaries.

The principle was summed up eloquently by Robert Hooke, a 17th century English polymath: “As hangs the flexible line, so but inverted will stand the rigid arch”.

The mathematical shape of a string or chain suspended from both ends is a catenary. It is the natural shape a purely tensile spanning element takes. If any other weights are hung from that tensile element, its shape will naturally adapt to the ideal shape to carry the new loading pattern. In essence, the ideal form depends heavily on how it is loaded. Once reversed, as shown in the sketches above, the ideal compressive form is found. The ideal form, whether tensile or compressive, is referred to as the funicular form.

The pointed arch used in the Gothic era was much closer to the true funicular form for cathedral arches and vaults, and it enabled builders like Suger to achieve stability with much less material. After all, as we have seen, a square inch of limestone or granite can carry a lot of force, if you can get it to stay where it is.

*Comparison of sections between the Hagia Sophia, left, and Notre Dame de Paris, right. The comparative lightness of Notre Dame’s structure is apparent.*

A new analytical era

The work of 17th century minds like Galileo, Huygens, Leibniz and Bernoulli led to a new foundational understanding about how structures work. In addition to understanding the role of funicular forms in structural stability, scholars like Thomas Young established principles on the strength of materials, and Leonhard Euler developed formulas to predict the specific point of loading at which slender members would buckle. Together, these 17th century minds bequeathed society with a robust set of analytical tools that, for the first time, enabled designers and builders to accurately predict how new forms would behave before they were built. We were no longer limited by the trial-and-error approach to structural iteration that had been the cornerstone of advancement for thousands of years.

This wealth of new understanding enabled builders in the subsequent centuries to vastly broaden their experimentation with form. Henri Labrouste, exposed to these ideas at the École des Beaux-Arts, put the principles to work in the complex iron forms in the Bibliothèque Sainte-Geneviève and Bibliothèque Nationale. Rafael Guastavino advanced thin-shell masonry forms in Europe and the Americas. Their new ideas could be tested on paper, enabling them to channel more innovation into each iterative design.

Antoni Gaudi was perhaps one of the designers most influenced by this new paradigm. Inspired by physical forms found in nature, Gaudi sought to realize the optimal structural form on several of his projects, most notably Sagrada Familia. Leveraging the work of 17th century scholars, Gaudi developed sophisticated methods for determining the funicular form for complex structures, including the use of physical modeling using twine and miniature sandbags.

Starting from the ceiling, Gaudi would model the lowest walls and columns of the building using suspended twine. He would then create catenary curves between these elements, gradually building up the configuration level by level. As he added each new layer, the weight would cause the components below to shift into new optimal positions, naturally adapting to create the most efficient overall form. The final inverted twine model revealed the ideal structural shape for the cathedral—one that would require minimal material while maintaining stability.

*Catenary model for Sagrada Familia, crafted by Antoni Gaudi. The model itself is now an exhibit piece in Barcelona.*

Gaudi’s catenary models enabled him to anticipate the way a new structure would perform before building it. It reflects the sea change that was occurring at the time, as designers leveraged new physical principles to more dramatically iterate on past forms and construction techniques. In contrast with the generic pointed arches used by Suger and others during the Gothic era, Gaudi’s arches were unique in their shape, driven by the loading patterns of the structure above. This had significant implications for construction, as Gaudi could not simply build one timber form for all of his arches. Many more timber forms were needed, and the effort required to establish the structure’s geometry on site became immensely cumbersome. So much so, that the construction of Sagrada Familia would become historically protracted: started in 1882, construction of the cathedral continues to this day.

Labor dominates

Gaudi’s work could be seen to cap a thousand-year-long evolution from the simple curves of the Pre-Romanesque to the analytically-driven purity of true funicular forms. Early in that journey, advancements toward Gothic architecture led to incontrovertible improvements in the efficiency and quality of structural design: Gothic cathedrals were easier to build, and more inviting to occupants than their predecessors. Gaudi’s work achieved even more material efficiency, but that came with a significant premium on the job site: more laborers - and more time - were needed.

As ground broke on Sagrada Familia, the western world was entering into the industrial revolution. Labor requirements and working conditions became paramount considerations in factory production and construction. The economics of modern construction offered a counter-narrative to geometric purity: saving a bit of material isn’t worth it if it takes more laborers more time to build.

The quest for structural purity was also challenged by the emergence of a dominant new material: structural steel. For thousands of years, builders of permanent structures had to contend with stone as their base material, which could only be relied upon in compression. For those thousands of years, carrying load in tension wasn’t really an option. That all changed with steel. Now, compression and bearing weren’t the only ways to make structures stand up. This shift undermined the fundamental constraints upon which the evolution of structural form had been based.

This dramatic paradigm shift created immediate opportunities for designers with a more nuanced vision of structural efficiency, one that brought harmony between material and labor usage. In the Americas, in the 20th century, Felix Candela demonstrated this through his application of single-ruled surfaces, a set of geometric shapes that enabled laborers to construct using simple formwork made from straight timbers.

*The straight-line timber formwork used to construct Chapel Lomas de Cuernavaca, Mexico, and the resulting final form.*

Candela’s forms are not funicular forms. Were they to be made from stacked stone in the same manner as historic cathedrals, they would not be stable and would not remain standing. Instead, these structures utilized steel-reinforced concrete, with its inherent capacity to carry loads in tension and bending, to allow these geometric deviations. The forms he chose were materially efficient, though not strictly optimal. But they could be built quickly, with a modest number of laborers skilled only in conventional construction techniques. His approach reflected a brilliant compromise between material efficiency and labor efficiency. It enabled Candela to be a prolific builder, realizing about 300 significant works in his lifetime, compared to about 15 by Gaudi.

Other builders of the 20th century adopted similar nuanced approaches to their work, embracing material efficiency and leveraging mass-production and industrial technologies to streamline construction. In Switzerland, Heinz Isler adopted analytical techniques similar to Gaudi to design thin-shell reinforced concrete structures that could be constructed with reusable formwork. His modular approach enabled him to realize significant labor efficiencies without compromising on form optimization.

*Sportcenter* *Paradies, Kanton Basel-Land, Switzerland, showing modular reuse of formwork for thin-shell concrete structures.*

A vastly different approach to modular construction was demonstrated by Pier Luigi Nervi, who used precast concrete techniques to develop more precise components that could be transported to site and assembled with significantly less on-site formwork.

Other builders carried the spoils of the industrial revolution into an exploration of tensile structures. Vladimir Shukhov was an early pioneer in tensile forms, using cable nets to create doubly curved roof structures supported by masts and megacolumns in the late 19th century. 50 years later, European designers Frei Otto and Jörg Schlaich used cables, rods and a range of prefabricated connections to create a range of celebrated architectural works.

By the middle of the 20th century, these advancements in structural materials, analytical techniques and construction methods had fully transformed the way builders approached the design of permanent buildings. Buildings had also become far more complex, integrating modern systems such as electricity, mechanical and plumbing that required the designer to become a coordinated design team. The buildings built today by these teams are rarely castles for kings or temples for thousands of worshippers. They are more often commercial office towers, hospitals, or multifamily residential buildings. In most cases, considerations of structural efficiency are contextualized alongside energy performance, simplicity in building systems design, facility operations and long-term maintenance. Structure is regularly concealed behind cladding, and is in support of the building’s architecture instead of being intrinsic to it.

Conclusions

The evolution of structural form throughout history shows how our approach to design is shaped through our theoretical understanding, material capabilities, and practical methods of construction.

The progression from intuitive to analytical understanding fundamentally changed how we approach structural design, moving from trial-and-error to predictive methods
While pure structural efficiency was once the primary goal, modern design must balance multiple competing factors including labor costs, construction complexity, and integration with building systems
The introduction of new materials, particularly steel and reinforced concrete, dramatically expanded the possibilities for structural form beyond traditional compression-only systems
Success in modern structural design often comes from finding an optimal balance between material efficiency, constructability, and practical constraints rather than pursuing theoretical perfection

The journey from ancient builders to modern design teams shows how structural form advances in tandem with our technological capabilities, analytical understanding, and society's changing needs. Today's definition of efficiency reflects current conditions—the site context, client goals, and building use patterns. While design technologies, building materials, and construction methods will continue to diversify, the pursuit of efficiency remains fundamental, even as the path to achieve it evolves.

From Stone to Steel: The Ever-Evolving Quest for Structural Efficiency