Mercat de la Boqueira, Barcelona. Photograph by Filipe Varela

### by Alexander J. Hahn

Let’s start with a piece of fiction. The setting is the crowded Mercat de la Boqueira in Barcelona, by reputation one of the best fresh food markets in Europe. Scores of fruit vendors display varieties of apples, pears, peaches, plums, apricots, cherries and oranges in but the few square meters of area that each has available. In competition with each other, they vie for the attention of housewives and tourists who press ahead from stall to stall. One of the merchants has arranged her Valencia oranges in an eye-catching display. Her fruit is large, round, polished, beautiful, and the tall pyramid of their arrangement glows rich in color, light and shape. It is striking and draws much attention. The child of a customer is about to reach for an orange near the bottom of the stack… but our story turns to a close look at the pyramid.

The merchant’s attractive display had been constructed very carefully. The pyramid has a triangular base of 12 oranges to a side. From each side of this triangle rises one of the sloping triangular faces of the pyramid. Each triangular face starts with its row of 12 and continues with rows of 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1 oranges as it ascends. An addition of these numbers tells us that each of the three faces of the pyramid is a triangle with 78 oranges. Another simple addition informs us that there are 78 + 78 + 78 = 234 oranges visible on the rising surface of the pyramid.

But this count is too simple. Counted once for each triangular face, the orange that presides over the pyramid from its perch at the very top is included three times, and each of the 11 oranges positioned below it along the three slanting edges of the pyramid is each taken twice. To correct for the multiple count, 2 has to be subtracted for the orange at the top and 11 for each of the three edges. It follows that a total of 234 – (2 + 3 · 11) = 234 – 35 = 199 oranges are visible on the three ascending triangular sides of the pyramid. A similar count for each of the 12 horizontal triangular sections of the pyramid, would have informed us that our merchant’s pyramid consists of 364 oranges in all. In addition to the 199 oranges visible on the pyramid’s surface, there are another 165 oranges that form the smaller interior pyramid that holds up the visible array from below. It might come as a surprise that so many, namely 199 of the 364, or about 55%, of all the oranges of the tower are visible on its surface.

The next issue is the stability of the pyramid. How did it remain stable during its construction? Wouldn’t the oranges on the three sides of the triangular base have been forced outward by the weight of the oranges above them… leading to the collapse of the growing stack? Could billiard balls have been arranged in the same way? Our merchant had been careful to always place a flatter part of an orange at the bottom. With this strategy and the force of friction between adjacent fruit, the lower sections of oranges remained in place as the higher tiers were added. This would not have been so for billiard balls. Very round and very smooth, they would not have provided sufficient resistance to these outward thrusts. Only a triangular rack of the sort that is standard equipment at every pool table could have kept the lowest triangle of balls (and hence the entire scheme) tightly in position. As a precaution, our fruit merchant had placed a similar barrier around the perimeter of her display.

The story of the oranges provides a sense of what *Mathematical Excursions to the World’s Great Buildings *is all about. Of course, the setting of its narrative is not a fruit market and the subject is not a pyramid of oranges. Rather, it is a carefully selected flow of some of the great buildings of the world and the subject is the analysis of their most prominent features: their beams, arches, vaults and domes. By analogy with the example of the pyramid, the numerical, geometric and analytic information that the mathematics adds is essential to the understanding of the architecture of these structures. It provides insights that verbal descriptions or even sets of images cannot convey. The *Mathematical Excursions* discusses the pyramids of Egypt, the Parthenon in Athens, the Colosseum and Pantheon in Rome, the Hagia Sophia, historic mosques, great Romanesque, Gothic, and Renaissance cathedrals, some of Palladio’s villas, the US Capitol, and three icons of the 20th century: the Sydney Opera House, Gateway Arch in St. Louis and the Guggenheim Museum in Bilbao.

It is not the aim of this narrative to provide a comprehensive study of these famous structures but only to describe them sufficiently so that their geometric features, such as symmetry and proportion as well as structural features, such as thrusts, loads, tensions, compressions and reactions can be analyzed mathematically.

The lithograph *Principal High Buildings of the Old World* by George F. Cram provides a snapshot of the *Excursions*. It depicts many of the buildings of the narrative and their most prominent aspects.

The mathematics includes basic aspects of Euclidean geometry, trigonometry, properties of vectors, coordinate geometry in two and three dimensions, and (at the very end) basic calculus. These mathematical topics comprise a second narrative, one that develops much current elementary mathematics from a historical perspective. The book* *intertwines the two stories, one architectural, one mathematical, into an interlaced fabric. On the one hand mathematics enhances the understanding of the architecture; on the other, the architecture is an attractive stage that makes basic abstract mathematics visible.

The Sydney Opera House provides an illuminating example. In January 1957, the young Danish architect Jørn Utzon won a competition to design an opera and concert hall complex on a dramatic piece of land that juts out into Sydney Harbor. Utzon submitted a design featuring an arrangement of high vaulted roofs that looked like a cluster of sailboats under full sail. See the first page above. The Sydney Opera House would become one of the world’s great buildings. But its realization was a monumental challenge. The plans called for two large auditoria, a restaurant and the necessary support facilities. There was to be a major hall for musical events of large scale such as grand opera and orchestral concerts, and a smaller minor hall for theatrical performances and recitals. The firm of Ove Arup, a Dane with British citizenship, was called on to provide the structural engineering expertise for the complicated project. The analogy to sailing ships extended to the building phases of the complex, both architecturally and functionally. There is the substructure “below the deck,” the essential “mast and sail structure,” and, finally, there is everything else “above the deck.” Accordingly, the project split into three basic stages: the podium, the curving shells of the roof vaults, and the rest, including the acoustics and seating schemes of the interiors of the auditoria.

The podium was to extend over a large rectangular area covering almost the entire site. The image on the first page depicts the podium in brownish tones. The podium was to include some performing spaces, a kitchen and cafeteria, storage and dressing rooms, administrative areas, paint and carpentry shops, and electrical and telecommunications facilities. Critically, the podium needed to be built in such a way that each of the performance venues, especially the two principal auditoria, was isolated from the sounds and vibrations generated elsewhere in the complex. The podium was to include a large roofed circulation space for vehicular traffic. Known as the concourse, it was the approach to which limousines whisked divas and tenors, buses dropped off theatergoers, and delivery vehicles brought in all manner of supplies. The entrance of the concourse can be seen on the first page just below the curving concrete roof at the very right.

The flat roof of the concourse is very interesting from the perspective of our discussion. The top of the roof was to be a large open platform, reached by a wide and grand staircase, that served as primary access to the performance venues. Given the large expanse of this roof, Utzon’s original plans called for a central array of columns to support it. But Arup’s team was able to do away with the columns to create a covered concourse area with a clear space of about 160 feet by 312 feet, a clear space the size of a football field! How was Arup able to achieve such a large clear space without interior supports?

It is a fact that unreinforced concrete is strong in resisting compression, but weak in resisting tension or pulling. The ratio between the compressive strength and tensile strength of concrete is roughly 12 to 1. Reinforcing concrete beams by embedding iron or steel rods and bars strengthens them, but not enough for very demanding applications. The process of pre-stressing concrete relies on the ability of concrete to resist compression.

Consider a concrete beam that is cast from a form or mold. Include as part of the mold a carefully, lengthwise positioned set of metal ducts. After the concrete has been poured and has hardened, steel cables with great tensile strength are threaded through the ducts. These cables are pulled tight (with hydraulic jacks) and anchored––under great tension––firmly into place at the ends of the beam. The finished beam is under great compression from the pull of the cables. The remaining space in the ducts is pressure grouted with a special mortar. This fixes the cables into place within the beam and prevents corrosion. If this is done correctly––and here is the point––then the beam will always be under compression, no matter what loads and tensile forces it is subjected to! Pre-stressed concrete, now common in structures that are subject to large loads, such as bridges and viaducts, was just coming into wider use at the time of the Sydney opera project.

Arup’s engineers created the large clear space of the concourse by building its horizontal roof with a parallel arrangement of 52 pre-stressed concrete beams, each 6 feet wide and about 170 feet long. The figure above shows the cross section of a concourse beam along its length. The slanted part of the beam on the right supports the grand staircase. The diagram shows the pre-stressing cables in dashed lines running from the top of the beam on the left, to the bottom at mid-span, to the top again near the transition to the slanting part. The location of these cables is a response to the particulars of tensile forces on the beam. The pre-stressed tie-beam (also shown in dashed lines) in the foundation is designed to counteract the outward thrust that each beam generates.

It is common knowledge that cardboard is strengthened when it is made with corrugated paper, paper with parallel ridges and grooves. Arup’s engineers applied this idea in the design of the beam. So that the beam can sustain the pre-stressing adequately and be structurally most effective, the corrugation is designed to vary along the length of the beam. The corrugation features a “folded-slab” design with a cross section that varies from one support to mid-span to the other support. The cross section changes from U-shaped to V-shaped to T-shaped, and back again to V and U. The sequence of Us, Vs, and T that runs across the top of the diagram illustrates the pattern.

But how should a continuous beam with such a variable pattern of cross sections be designed? How should its cross section transition from one shape to the next? An ingenious geometric construction combined with coordinate geometry and trigonometry provided Arup with the answers. This construction is derived from the figure in the upper corner of the diagram. This solved one problem.

The path from Utzon’s imaginative design of the roof vaults with its cluster of billowing, sail-like shapes to their realization turned out to be much more of a challenge. What should be the explicit geometric definition of these freely flowing sculptural forms? With what combination of materials and methods of construction should these completely new vaulted roofs be built? It would take from 1957 to 1963––years of exploration, analysis, disagreements, and hard work––to answer these questions.

Roof structures of the required size and complexity cannot be built without an explicit geometry that can be expressed mathematically. Without such a mathematical model it is not possible to calculate the loads, stresses, and rotational forces that the vaults would be subjected to, and to estimate the impact of wind and temperature changes on their stability. Without an explicitly defined geometry, the necessary computations and computer analyses cannot be undertaken, and the construction of the unprecedented structure could not proceed. Parabolas and ellipses were Utzon’s first choices for the profiles of the vaults. For reasons that we will be apparent shortly, neither provided a buildable option. Another problem concerned the construction of such curving vaults. It had been Utzon’s initial thought that they would be built as thin, concrete eggshell-like structures. Such roofs were coming into use at the time. In fact, Arup’s firm had executed such roofs with concrete shells only 3 inches thick. However, the pointed, steeply rising vaults of Utzon’s design were not compatible with such a design. Since a departure from his architectural concept was not an option, this idea was abandoned.

Original site plan, including report, of the Sydney Opera House. *Via*

Given the various parameters, Arup became convinced that each of the sail like roof structures could only be built as a sequence of curving ribs––narrow at the bottom and increasingly wide as they rise––that would spring from a common point at the podium and fan outward and upward from there. See the geometric patterns above the depiction of the opera complex of the first page. Each roof vault would consists of two such curving fan-like structures––one the mirror image of the other––rising from opposite sides to meet at a circular ridge at the top. Utzon endorsed this concept enthusiastically, “I don’t care how much it costs, I don’t care how long it takes to build or what scandals it causes, but this is what I want.” This solved one problem, but the problem with the geometry remained. The large size of the shells meant that the ribs would have to be constructed in segments. The demands of economy and time meant in turn that these rib segments would have to be mass-produced. Now we understand why the parabolic or elliptical geometries would not do. In either case, the ribs would curve differently. Was there a geometry that would make it possible to build the curving sail-like structures with standardized, repetitive components? If the answer was no, then it would be impossible to execute Utzon’s design and the project would collapse.

Suddenly Utzon had a flash of an idea. The only surface that curves in the same way in all directions is a sphere of a given radius. Utzon’s flash was the realization that a limitless variety of curving triangles could be drawn on a sphere and that all the shells for his roofs could be designed as curving triangles from the same sphere! This was the idea that saved the project. Both Utzon and Arup finally saw light at the end of their tunnel.

To be able to visualize what triangles Utzon had in mind, let’s return to our Barcelona fruit merchant. Suppose that an eager customer wishes to sample one of the Valencias. The merchant takes a beautifully spherical orange and slices it exactly in half. He puts one of the halves on a flat horizontal cutting board and cuts through it at a two different angles to obtain a perfect wedge. This wedge is the sample for the customer. To understand the shape of Utzon’s triangle, another cut is required. The merchant leaves the wedge in its position and slices through the half of the orange with a cut perpendicular to the cutting board and through the wedge (at an angle with the wedge) in such a way that the cut separates the wedge into a larger and smaller segment. The merchant cleans all the flesh from the smaller segment very carefully until only the peel remains. This curving triangular piece––its shape unaltered by the cutting and cleaning––is a model of Utzon’s vault geometry. Two mirror images of such curving triangles, joined along the cut described last, shape his vaults. Cut in different ways, they provide the shapes for all of his vaults. Weighing matters of both aesthetics and structure, Utzon and Arup decided on 246 feet as the radius of the sphere from which the design of all the matching pairs of spherical triangles would be taken.

The image above depicts the four triangular shells for the vault sequence of the major hall. The figure shows the spherical triangles for shells A1 to A4 as well as the cross-sections of the spheres that give rise to them. The triangle labeled A1 models the vault of the entrance foyer. Triangle A2 is the vault over the stage, backstage, and wings. With its height of about 220 feet, it is the highest vault of the complex. Triangle A3 is the roof vault over the seating area and A4 is the vault over the spacious lounge. It is closed off by a wall of glass through which opera and concertgoers could experience a dramatic panoramic view of Sydney Harbor Bridge and the activity in the harbor.

By conceiving the spherical triangular shell for each roof vault to consist of a fan-shaped arrangement of ribs (as illustrated in the diagram on the first page) and each rib as a configuration of segments, each shell could be constructed rib by rib as a sequence of concrete rib segments. In this way it was possible to construct the vaults from mass-produced repeating components. The teams of Utzon and Arup could now complete the particulars of the design. The segments of the ribs are built with pre-stressed concrete. Their cross sections are designed to vary from a narrow T near the podium, to a narrow solid Y, to a wider, open Y higher up. This pattern of segments flows upward in exactly the same way for each of the shells. The heights of the shells determine the extent of this flow. At the same height the rib segments of any two shells are identical. Only if one shell is higher than the other is there a difference in the upward flow of rib segments.

The building of the vaults could begin. The various concrete rib segments were prefabricated at the site. A total of 1498 standard and another 280 nonstandard rib segments, each 15 feet long, were cast. There are twelve different types. Of the seven types in the lowest parts of the shells, respectively 280, 280, 260, 196, 174, 110, and 82 were made. Each rib of a triangular shell has a matching rib in the mirror image of the shell on the opposite side. By deploying heavy construction cranes and the use of an erection arch made of curving triangular steel trusses, the shells rose to completion, segment by segment, a matching pair of ribs at a time.

Utzon and Arup knew that the idea of the single geometry had been a critical advance. When Arup recalled later that, “we did not want to pull the architect down to hell, but we wanted him to pull us up to heaven,” it was this breakthrough that he had in mind. Utzon was certainly influenced by Arup’s single-minded focus on “how do we build it?” However, the spherical solution had been his.

The remaining challenge was to cover and seal the outer surfaces of the shells. This was done with a sophisticated tile system. The tiles configured in V shaped formations, or lids, were locked into place with brackets and bolts that are adjusted to give it the precise orientation it needed to have on the spherical surface. It goes without saying that much mathematics and computer analysis went into the complex design of these tiled surfaces. The spherical geometry made it possible to standardize the manufacture of the more than 4000 lids that were required. When the last lid was lowered into position in January 1967, the roof vaults of the opera complex were finally complete.

With the completion of the building of the vaults, the complex could now sail to a quick and successful conclusion. But problems over seating capacity, acoustical properties, cost overruns and the timely completion of the project had been brewing. The technical challenges had resulted in delays in the construction. The cost estimates that Utzon provided soared from about US $8.5 million in March of 1958 to US $45 million in July 1965. The lotteries that financed the building were becoming more and more frequent. (The cost of the project ultimately exceeded US $90 million.) Not surprisingly, the Sydney Opera House became a political football, and when the opposition party won the Australian election in 1966, Utzon was forced to resign. A panel of architects was appointed to replace him and finished the structure with Arup’s firm by its side. On October 20, 1973, sixteen years after Utzon had won the competition, Australia celebrated the opening of its performance hall complex with the Queen of England in attendance. The Sydney Opera House is a large, white sculpture that catches and mirrors the sky of its harbor setting with all its varied lights from dawn to dusk and day to day. It has captured the imagination of people the world over and has become a symbol of the city of Sydney, indeed of Australia.

The Sydney Opera House confirms that great architecture needs to make the most of the possibilities of the site, its response to light, the creation of space, scale and proportion, the use of appropriate materials, and mathematical and computational analyses. These are the crucial considerations that a great building needs to respond to.

**About the author:**

Alexander J. Hahn is professor of mathematics at the University of Notre Dame in Indiana, USA. His research over the years has had a focus on algebraic concerns, in particular on classical matrix groups over fields and number theoretic domains, and on related structures such as quadratic forms and Clifford algebras. At the same time he has been exploring the history of mathematics and science and is the author of *Basic Calculus:* *From Archimedes to Newton to its Role in Science,* *The Pendulum Swings again: A Mathematical Reassessment of Galileo’s Experiments with Inclined Planes*, and *Mathematical Excursions to the World’s Great Building*s. Hahn’s interest in architecture has been informed and inspired over the years by his friends and colleagues of Notre Dame’s School of Architecture, especially Michael Lykoudis, its Dean.