by Unknown
XII. Virtual visualization of the Pantheon’s geometry. (Conception Mark Wilson Jones, realization Robert Grover)
The geometry of the Pantheon calls to mind the title of an important work of ancient science, as would have been evident to any ancient mathematician standing in the center of the Rotunda. On the Sphere and Cylinder is a fundamental work of Archimedes. In this his longest treatise, he established the formula we learn at school for calculating the volume of a sphere, V = 4/3 π r3.7 The subject is the same as Book XII of Euclid’s Elements, written over half a century earlier, but which gave no rules for calculations.8 Archimedes’ findings on the sphere were totally new for the third century BC and are still definitive today. His procedures came from examining a sphere and a cylinder of equal diameter, just as in the Pantheon.9 His breakthrough was linked to the concept of symmetria, or mathematical harmony (literally the coming together of measures), an ideal that was intrinsic to ancient architectural design.10 In the introductory letter, after stating the main relationships between a cylinder and sphere of the same width and height – that the volume and the area of surface of the former are both 3/2 as great as those of the latter – he went on to observe: “Now these properties were all along naturally inherent in the figures referred to, but remained unknown to those who were before my time engaged in the study of geometry, because none of them realized that there exists symmetria between these figures.”11 Here, “symmetria between these figures” means that they are commensurable and expressible through the relationship of small whole numbers. Attilio Frajese, who published the first complete Italian edition of the work, says, “Archimedes senses that lying beneath complex geometrical facts there must be corresponding simple arithmetical facts.”12 Indeed, apart from the relationships already mentioned, Archimedes proved that the surface of the sphere and the curving surface of a circumscribed cylinder must be equal. Thus in the Pantheon interior, the surface area of the drum is equal to that of the dome it carries. The harmony between these two figures is expressed by the simplest possible ratio of 1:1, both for the radii and the surfaces.
Archimedes also wrote of conoids (i.e., paraboloids and hyperboloids) and spheroids (i.e., ellipsoids), but it was the sphere and the cylinder that he loved best, perhaps because of this elemental symmetria. Cicero found proof of this, it seems, when he was quaestor of Marsala, in Sicily. In 75 BC, he went to Syracuse to find Archimedes’ tomb outside the walls: “I remembered certain doggerel lines inscribed, as I had heard, upon his tomb, which stated that a sphere along with a cylinder had been set up on the top of his grave.”13
The connection between abstract mathematics and physical spatial forms was certainly perceived by Archimedes. His Method of Mechanical Theorems relates how he applied the notion of the center of gravity and the lever to the investigation of geometrical figures by dividing solids into straight strips and then “weighing” them on a notional balance, as in the science of mechanics. This approach was as innovative as it was typical of Archimedes. Areas acquire a virtual weight and are balanced against each other, by which means the relative surface areas could be gauged. As he noted in a letter to Eratosthenes, the mathematician and librarian at the Museum of Alexandria in Egypt: “[I]t is easier to supply the proof when we have previously acquired ... some knowledge of the questions than it is to find it without any previous knowledge.”14
The theorems on the sphere and the cylinder, too, were conceived as problems of mechanics. Thus, the concepts of geometry, symmetria, and balance were related to one another. Until Archimedes’ Method of Mechanical Theorems was rediscovered at the beginning of the twentieth century by Johan Ludwig Heiberg,15 his reasoning was only known through the quotations of Hero of Alexandria.16 Hero, a mathematician from the time of the emperor Nero, also wrote a treatise for architects on the lifting of weights.17 The central importance of this way of thinking in the creation of the Pantheon seems to be confirmed by the simple dimension, 150 feet (or 100 cubits), that defines the diameter of the ring of its interior columns. What is more, a square inscribed in this circle can be “flipped” over to produce another square that locates the columns of the portico (see Fig. 1.5 and Plate XII).18
The coffers of the dome of the Pantheon are divided into five rows of 28, a number that expresses an idea of perfection.19 The number 28 is in fact a “perfect number,” one that is equal to the sum of its factors (28 equals 1 + 2 + 4 + 7 + 14, each of which divides into 28). Perfect numbers are rare; units, tens, hundreds and thousands have one each: 6, 28, 496, and 8128, respectively. Following a tradition going back to the Pythagoreans, it was in Hadrian’s time that Nichomachus of Gerasa included in the first book of his influential Introduction to Arithmetic a discussion of perfect numbers.20 For Nichomachus, such numbers are associated with virtue, moderation, and beauty; arithmetic, music, geometry, and astronomy are like “bridges” and “stairways” to knowledge.21
There are other interpretations of the intentions behind the choice of 28 for the numbers of lines of coffers. Mark Wilson Jones has explained that this is a key ingredient of the interplay of rhythms and alignments – and selective lack of alignment – orchestrated between the pattern of the floor, the articulation of the wall, and the coffering of the dome (see Plate X).22 As in so many other Roman buildings, a series of subordinate proportions entered into the composition and deployment of smaller units, including the exedras, columns, aedicules and moldings (see Chapter Five).23 There is complexity, but never does it banish the underlying geometrical simplicity; the two poles of design are kept in balance.
This concept of balance, neither too much nor too little, is central to the aesthetics of architecture. At the end of the classical era of great Western domes, around AD 560, Procopius of Caesarea described the dome of St. Sophia in Constantinople in these terms: 24 “[I]t proudly reveals its mass and the harmony of its proportions, having neither any excess nor deficiency.”25
In classical architecture, geometry is like one of Nichomachus’s stairways, leading to higher realms of both aesthetic achievement and knowledge. The interior of the Pantheon arouses sentiments on the part of many a visitor similar to those expressed by Procopius, without necessarily knowing the ideas of Archimedes or Nichomachus. Yet knowledge of them gives access to further intellectual pleasures.
Description of the Structure
How did the architect of the Pantheon turn the elemental concept of cylinder and hemisphere into reality on such a scale and build the largest dome that had ever been built? To understand this, we must first understand the structure of the cylinder-drum and the hemisphere-dome, both of which are neither immediately visible nor comprehensible in their three-dimensional entirety.
It would be vain to make this attempt except on the basis of a thorough account of the physical fabric. In Hadrian’s time, Lucian of Samosata, a Syrian orator, marked the beginnings of art literature by popularizing the literary genre called ekphrasis, which means “description” in Greek. An ekphrasis recreates a work of art in words, stirring the imagination and arousing emotions in the reader; it communicates the idea and the effect of the work to someone far away. To help us understand the structure of the Pantheon, there follows a selection of some of the most concise modern ekphraseis, presented not in chronological order but, rather, moving upward from the bottom to the top. These passages, by Adam Ziolkowski, Luca Beltrami, and William MacDonald, respectively, may be further appreciated by viewing the color drawings of the elite nineteenth-century French architects Achille Leclère and Chedanne (see Plate XI). Ziolkowski, author of the entry on the Pantheon in the authoritative Lexicon Topographicum Urbis Romae, describes the drum seen from the inside:
The drum rests on a ring of concrete 7.3 m wide and 4.5 m deep. ... Its wall, notionally 6.2 m thick, made of concrete faced with brick, contains cavities arranged on three levels, marked by the three cornices on the outer face of the drum. On the lowest level there are eight large apertures, the entrance and seven exedrae opening to the inside on the
rotunda’s main and diagonal axes. The four diagonal exedrae are trapezoidal, the other three apsidal. In front of each side exedra there is a pair of columns set in line with the wall; the architraves superincumbent on these columns are continuations of a cornice running round the interior and marking the top of the lower zone. ... All these apertures are two storeys high, each of the six side exedrae being topped above the architrave by a sort of unfloored chamber.26
On the third story there is another set of large chambers (see Plate IV and Figs. 5.1b, 6.2), this time of uniform configuration, whether they align with the cross axes or the diagonal axes. Beltrami, who directed important investigative campaigns in 1892–1893, explains their geometrical division:
The exedrae have chambers that are divided into three sections by two radial walls. Vertically these divisions fall over the axes of the Corinthian columns (of the lower level). ... These 1.2 metre thick walls act as buttresses and connect the masonry at the springing of the dome to that of the external drum. ... The 6-metre perimeter thickness [of the drum] is divided into three: a 1.9 metre thick inside wall, another 1.9 metre thick outside wall and 2.07 metre wide ring chambers.27
In alternation with this system of voids is another family of smaller semicircular chambers that occur on all three levels. On the ground floor and also on the top level, they are reached from the outside via small openings shaped like doorways. On account of the great number of all these different types of voids, MacDonald, author of an inspirational introduction to the Pantheon, likens the structure of the drum to a honeycomb and describes its external configuration:
On the outside the rotunda reads as an almost solid cylindrical wall of brick. There are openings in it here and there, at various levels, that give on to some of the many different chambers that honeycomb the rotunda structure, a honeycombing that is an integral part of a sophisticated engineering solution to the problem of supporting the huge dome.28
The exterior of the Rotunda is subdivided by cornices into three parts, or stories. The first cornice lies at the height of the frieze over the Corinthian columns inside the building, the second lies at the springing of the dome, and the third registers the top of the drum (see Figs. 1.12, 1.13, 6.3). Moving on to the dome, MacDonald notes:
Rather more than half of the exterior rise of the dome is defined by a series of concentric step-like rings that are actually buttresses, masses of masonry placed over the dome’s lower part where they are most needed structurally. ... Partly because of these ring buttresses, the exterior silhouette of the dome is not hemispherical but bowl-shaped; inside, the hemispherical surface of the dome rises from a level well below that of the outer high terrace. The upper part of the cylindrical wall of the rotunda is built up high, also as a shoulder-like buttress, reducing the prominence of the exterior of the dome. The only exterior spherical portion rises above the highest of the step-ring buttresses, extending upward and inward to culminate in a horizontal circular opening, an oculus, more than thirty feet (9.45 m) in diameter, which is centered over the paving a hundred and fifty feet below.29
The above extracts give us a clear picture of the structure of the dome and the distribution of spaces within it. We have now inspected the structure of the cylinder like a bee in the honeycomb described by MacDonald. We have a clear picture of the structure of the building and the spaces within it. We can easily make out eight piers and exedrae in the plan of the building (see Plate IV and Fig. 5.1b). The spaces are made up of exedrae and chambers up to the third story, arranged along the eight axes of the circumference. The vaulted chambers are enclosed by the internal and external walls of the drum and by radial walls. The exedrae look onto the rotunda and reach up to the second story. The third-story chambers are floored at the springing of the dome and open out onto the outer face of the rotunda. The cupola in its hemispherical purity is visible only in the interior, while on the exterior it is partly concealed by the step-rings.
The Drum
The drum of the Pantheon is an immense structure, roughly 108 feet (32.2 m) tall and 21 feet (6.2 m) in thickness at the base, reducing to 20 feet (5.9 m) at the top. The ratio of the drum to the dome (44.08 m) is about 1 to 7.3.30 Apart from its huge scale, what is most striking is the presence of the numerous voids that MacDonald likened to the cells of a honeycomb. Giuliano da Sangallo drew attention to them by using a dark tint on his plan in the Codex Barberinianus (Fig. 4.1),31 perhaps to represent the darkness of the empty spaces. Some decades later, Sebastiano Serlio remarked that “I think the spaces are there to avoid using too much material. In any case, being circular they are very strong.”32 The sections of the wall between the apertures (i.e., the entrance and seven exedrae) act as eight huge piers onto which stress is directed by the vaulting over the apertures. The drum can thus be described both as a series of piers connected by walls or as two concentric walls connected by transverse walls.33 The drum is what in modern terms we call a “diaphragm structure”; this structure is comparatively light and incredibly strong.
4.1. Plan of Pantheon by Giuliano da Sangallo, after 1465. (Biblioteca Apostolica Vaticana, Vat. Barb. lat 4424, f. 13 recto)
In Roman architecture, a beautiful example of a diaphragm structure in the form of a hollow pillar is Trajan’s Column, inaugurated in AD 113 (Fig. 4.2). Its shaft comprises 19 hollowed-out monolithic marble drums, with a helical staircase running through them. This hollowing produces a structure that weighs a third less than a similar full column but has virtually the same stiffness.34
4.2. Exploded perspective of Trajan’s Column. (Wilson Jones 2000, Fig. 8.8)
In the Pantheon the semicircular chambers inside the drum (24 in number, 8 for each tier) are oriented so that they act like arches braced against the outward thrust of the rotunda. The Romans used this arrangement in retaining structures, as for example at the Mausoleum of Augustus (see Fig. 5.1a).35 In the third story of the Pantheon (Fig. 4.3, and see Figs. 5.1b and 6.2), the niches are each divided by a radial wall, an arrangement that had also been adopted in the Mausoleum to counter the lateral pressure from the huge core. In Nero’s Nymphaeum under the Temple of Divus Claudius, on the Celio in Rome, there are chambers with semidomes that lie on a structure consisting of two walls separated by a semicircular corridor.36 In Hadrian’s Villa, the pumpkinsemidome of the Serapeum has chambers at the springing, with a system of pillars and niches-openings-windows underneath (see Figs. 5.9 and 5.10).37 This diaphragm strategy finds its most complete manifestation in the Pantheon.
4.3. West elevation of Pantheon; engraving by Francesco Piranesi, Pantheon. (Seconda parte de’ tempij antichi che contiene il celebre Pantheon, Rome 1790, Plate VII, Istituto Nazionale per la Grafica, Roma)
In the three storys of the Pantheon, the distribution of the masonry and its voids changes subtly. In the lower levels, semicircular voids alternate with the exedrae, while by the third story, above the springing of the dome, the voids are distributed more uniformly along the circumference. The dome therefore discharges its weight relatively evenly, while the drum then concentrates the load on the eight “piers.” Piers and interlocking walls work together to support the dome.
The general idea behind this system for stiffening a structure while lightening it operates in Roman bridges, too. Piers are sometimes hollowed out by a smaller arch in order to prevent the pressure of a river in flood from bringing down the abutments. An example is the Pons Fabricius on the Tiber, built in 62 BC, which joins the Isola Tiberina to the Campus Martius. The two segmental arches have a span of 24.5 meters. The road on top is 5.5 meters wide, almost the same thickness as the Pantheon’s drum. In the Pantheon, the system of piers and barrel vaults within the drum can be likened to a circular bridge, or rather, a circular aqueduct with three rows of arches, as in the Pont du Gard near Nîmes.
Let us now focus on the relationship between the spatial articulation of the rotunda and its fabric. The great mass of the drum is concrete encased in brickwork that acted both as formwork and facing (Figs. 4.3 and 4.
4, and see Fig. 3.4). The wall of the drum is built in opus testaceum,38 involving bessales (bricks about 2/3 ft or 19.7 cm square), sesquipedales (bricks about 1 1/2 ft or 44.4 cm square), and bipedales (bricks or tiles 2 ft or 59.2 cm square). The latter have a thickness greater than the other two, typically in the range 4 to 4.5 centimeters (whereas bessales and sesquipedales range between 2.5 cm and 4 cm thick). After having been baked in these sizes, bricks were often cut into smaller units. The bessales and sesquipedales were generally cut in half on the diagonal to make semilateres that were embedded in the concrete like the teeth of a saw, with the hypotenuse of the triangle on the surface (see Fig. 5.8). When used for the arches of the rotunda, however, the sesquipedales and bipedales were usually employed either whole or as rectangular halves or smaller portions. In this context, individual bricks do not fit a perfectly radial pattern, but tapered bipedales usually alternate with ordinary ones for the sake of economy. The concrete consisted of mortar made of lime and pozzolana into which were laid, not poured, pieces of aggregate, often as large as a fist, made of stone (often tufa) and, to a lesser extent, pieces of broken brick. At intervals, the concrete is divided into horizontal sections by “through” or “bonding” courses made up of a single stratum of tegulae bipedales.
4.4. Study of structure in upper part of drum by Josef Durm. (Durm 1905, Fig. 641)
The outer face of the drum of the Pantheon has interlocking arches of two kinds, discharging and relieving. Discharging arches and relieving arches differ in that the first have an opening underneath, whereas the second have no opening or none visible at the surface. In their disposition, these arches may be likened to the wicker arches of a basket. The dome is like the upside-down basket seen on the top of the crane of the Haterii,39 or like the baskets seen on the frieze of Trajan’s Column used by Roman soldiers to transport earth, mortar, and caementa.