Santiago Huerta Oval Domes: Universidad Politécnica de Madrid


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Santiago Huerta
E. T.S. de Arquitectura Universidad Politécnica de Madrid
Avda. Juan de Herrera, 4 28040 Madrid SPAIN
[email protected]
Keywords: oval domes, history of engineering, history of
construction, structural design

Research
Oval Domes: History, Geometry and Mechanics
Abstract. An oval dome may be defined as a dome whose plan or profile (or both) has an oval form. The word “oval” comes from the Latin ovum, egg. The present paper contains an outline of the origin and application of the oval in historical architecture; a discussion of the spatial geometry of oval domes, that is, the different methods employed to lay them out; a brief exposition of the mechanics of oval arches and domes; and a final discussion of the role of geometry in oval arch and dome design.

Introduction
An oval dome may be defined as a dome whose plan or profile (or both) has an oval form. The word [email protected] comes from the Latin ovum, egg. Thus, an oval dome is egg-shaped. The first buildings with oval plans were built without a predetermined form, just trying to enclose a space in the most economical form. Eventually, the geometry was defined by using circular arcs with common tangents at the points of change of curvature. Later the oval acquired a more regular form with two axes of symmetry. Therefore, an “oval” may be defined as an egg-shaped form, doubly symmetric, constructed with circular arcs; an oval needs a minimum of four centres, but it is possible also to build ovals with multiple centres.
The preceding definition corresponds with the origin and the use of oval forms in building and may be applied without problem up to, say, the eighteenth century. From that point on, the study of conics in elementary courses of geometry taught the learned people to consider the oval as an approximation of the ellipse, an “imperfect ellipse”: an oval was, then, a curve formed from circular arcs which approximates the ellipse of the same axes. As we shall see, the ellipse has very rarely been used in building.
Finally, in modern geometrical textbooks an oval is defined as a smooth closed convex curve, a more general definition which embraces the two previous, but which is of no particular use in the study of the employment of oval forms in building.
The present paper contains the following parts: 1) an outline of the origin and application of the oval in historical architecture; 2) a discussion of the spatial geometry of oval domes, i.e., the different methods employed to lay them out; 3) a brief exposition of the mechanics of oval arches and domes; and 4) a final discussion of the role of geometry in oval arch and dome design.
Historical outline of the origin and application of the oval in historical architecture
The first civilizations: Mesopotamia and Egypt
Rounded forms, many times not geometrically defined, were used in building from the most remote antiquity. These rounded forms may be called “oval”. What the ancient builders were looking for was the most simple and economical way to enclose a space. As

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1590-5896/07/020211-38 DOI 10.1007/s00004-007-0040-3

© 2007 Kim Williams Books, Turin

techniques were perfected, some of these plans were geometrically defined using cords and pegs to control their contours, i.e., employing circular arcs or combinations of them.
These enclosures were first covered by masonry in about 4000 B.C., by cantilevering the stones forming successive rings, until the space is closed at the top. This is what we call now a “false dome”. Fig. 1 shows the most ancient remains discovered so far in Asia Minor. Domes were used to form “stone huts” and the technique was developed, no doubt, in the context of permanent settlements associated with agriculture. It is a form of what we today call vernacular construction. The same technique of building has survived in some countries until the present day. (In Spain, for example, the vernacular buildings of piedra seca, dry stone, in Mallorca are similar to those first examples in Asia Minor [Rubió 1914].)
Fig. 1. The first domes covering a closed oval plan in Asia Minor ca. 4,000 B.C. [Baitimova 2002]
The invention of the arch apparently came later than that of the dome. The first arches were built in Mesopotamia or Egypt circa 3500 B.C. to construct the permanent covering of tombs. The books of Besenval [1984] and El-Naggar [1999] contain the most information on arch and vault building in those times.
The first arches were built with crude bricks. It was discovered that if the bricks are disposed in a certain manner in space they remain stable, their weight being transferred from one brick to the next until reaching the earth: the same force which tries to drag the bricks to the earth keeps them in position. It was an amazing invention and an enormous step forward from the more common custom of simply piling the bricks to form walls. 212 SANTIAGO HUERTA – Oval Domes: History, Geometry, and Mechanics

(The practice of brick wall building was itself an invention which evolved very slowly before the bricks and the different bondings were developed; see Sauvage [1998].)
In the first two millennia the builders experimented with several types of arches and vaults and there is no direct line of progress towards the voussoir arch with radial joints, which is our conceptual model. A perusal of the hundreds of surveys contained in the books of Besenval and El-Naggar makes evident a long period of “eclectic” experimentation, in which several forms and types of arches co-existed. Among them appeared the first oval vaults (fig. 2). Some of those vaults were built without centring, by building successive flat slices against a wall where the form of the arch was first drawn. The technique is still used in the north of Africa [Fathy 1976] (fig. 3).
Fig. 2. Oval arches in Asur, Mesopotamia. a) With radial centres; b) Arches built without centring, by constructing successive slices leaned one against the next [Besenval 1984]
Fig. 3. Building oval barrel vaults in North Africa in the 1970’s [Fathy 1976]
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The first vaults were quite small, with spans of only about one or two meters, just enough to cover the tomb. This size favoured experimentation: the vault, if not of an adequate form, will distort and the observation of the movements gave the builders a “feeling” for the more adequate forms. Fig. 4a shows one of the plates of the book of ElNaggar [1999] which explains clearly the kind of forms adopted for the vaults. To an architect or engineer with some experience in masonry structures it will be evident that the vault at the bottom right side is the safest, adopting an oval form which will amply contain the trajectory of compressions (the line of thrust or inverted catenary) within the arch. Choisy [1904] was the first to point this fact as the origin of the oval arches (fig. 5a). (On the design of masonry arches see [Heyman 1995] and [Huerta 2006].)
Fig. 4. a) Different Egyptian oval vaults, showing different degrees of distortion from the oval form [El-Naggar 1999]. The bottom-left vault, which shows no distortion, presents no danger of collapse; b) Construction of the vaults of the Ramesseum [Choisy 1904]; c) Barns of the Ramesseum [El-Naggar 1999]
When vaults grew bigger in the second millennium B.C. – for example, the vaults covering the barns of the Ramesseum (fig. 4b and c) have spans of almost five meters – a good regular building required that the form of the vault be fixed by some construction. The oval forms had to be defined geometrically. The Egyptians were experts in practical geometry using pegs and strings, and a form composed of circular arcs is the most logical. Choisy [1904], observing the form of many vaults, considering the practical geometry of the Egyptians (the 3-4-5 right triangle), and applying the logic of building, proposed a simple oval form and suggested how it might be constructed using a simple system of strings (fig. 5).
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Fig. 5. The geometry and construction of Egyptian ovals, after Choisy [1904]. a) Egyptian oval with the 3-4-5 triangle and comparison with the catenary; b) Possible employment of the oval with leaning brick slice-arches; c) Use of a string method to draw the oval
A geometrical construction is not the only possibility. The mason may sketch the profile of the vault on a wall, perhaps making several corrections until he is satisfied with a certain form. Then, he may fix the form my drawing an horizontal line and measuring the vertical distances to it. Indeed, this was the method followed in a diagram from the Third Dynasty (3000-2700 B.C.) near the Step Pyramid of Saqqara [Gunn 1926] (fig. 6a and b). If the separation between the vertical lines is considered to be uniform, the profile does not correspond to the preceding “typical” oval or a circular arch, and this supports the previous hypothesis. However, Daressy [1927] demonstrated that if the last interval is presumed to be shorter than the others a circular arc may be somewhat adjusted to the curve (fig. 6c). The present author has adjusted an oval, following a simple geometrical construction (fig. 6d). Many other curves may be tried, but any interpretation should be made with caution, considering the historical context and the logic of practical building at that time.
Some scholars claim to have found ellipses and not ovals in the form of the Egyptian arches. In particular, the French archaeologist Daressy [1907] attributed an elliptical form to a drawing of the profile of an arch corresponding to the vaults of one of the tombs of Ramses VI (twelfth century B.C.). This hypothesis has been accepted as true by many scholars who have discussed the laying out of Egyptian arches. (See, for example, [Arnold 1991] and [Rossi 2004]). Heisel [1991] hinted at the possibility of employing a bent wooden strip to lay out the curve. ([Cejka 1978] has explored this method in his research on Islamic arches).
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Fig. 6. a) The oldest diagram of an arch, Third Dynasty, 3000-2700 B.C. (photo and drawings after [Gunn 1926]; b) Drawing by Gunn [1926]; c) Geometrical interpretation as an arc of circle by Daressy [1927]; d) Oval which approximates to Gunn=s drawing (author)
Fig. 7. a) Layout of an Egyptian oval dome. Comparative drawing of an Egyptian oval with an ellipse with foci in D and E ([El-Naggar 1999] after [Daressy 1907]). Choisy=s oval has been superimposed by the author in a thin continuous line; b) Stone vault of an Egyptian tomb of oval profile [El-Naggar 1999]. Choisy=s oval superimposed by the author
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I disagree strongly with the hypothesis that the Egyptians used elliptical forms. The ellipse is a mathematical curve which was defined by Greek mathematicians the fourth century B.C. [Heath 1981]. The supposition that they have discovered by chance the string method for laying out the ellipse (the so-called “gardener=s method”) is also quite difficult to accept, given the long and painful birth of even the most simple and “obvious” inventions. In fact, Choisy=s Egyptian oval adapts itself as well or better than the ellipse to the diagram, as fig. 8a makes evident, and if any geometrical construction was used, this appears a much more probable hypothesis. Finally, a drawing found in a wall of Luxor’s Temple (figs. 8a and b) [Borchardt 1896] provides evidence of the employment of the oval by the Egyptians. In this case it is evident that the form corresponds to an oval, because of the great difference of the two radii. This is a very strong argument, to be added to the others, against the “ellipse hypothesis”.
Fig. 8. Drawing of an oval discovered by Borchartd in 1896 in a wall of the Temple of Luxor. a) Borchardt’s reproduction of the original drawing; b) One of his hypothesis for the geometrical generation of the oval [Borchartd 1896]
Greece and Rome The Greeks knew the arch. Since stone was the usual building material, they employed
the voussoir arch, almost without exception with a semicircular form. The arch was employed in secondary buildings, in sewers or for the gates of the city walls [Boyd 1978, Dornisch 1992]. Some cantilevered domes approached the oval form, but there was no systematic use of the ovals as in the brick architecture of Mesopotamia or Egypt.
The oval form appeared in Europe in Roman times for the design of amphitheatres (see Wilson Jones 1993]) (fig. 9a). Again, some scholars have tried to prove the use of ellipses instead of ovals. However, the oval form is the natural form for laying out the stands of the amphitheatre: it is impossible to construct parallel ellipses, and the only logical method is to use oval forms made of circular arcs. In any case, the differences between ellipses and ovals for the usual proportions are so small that, in fact, one see what one wants to see. Even the most precise mensuration does not serve to settle the matter [Rosin 2005]. It is not a matter of mensuration, but of the history of building traditions.
It appears that the Romans did not built oval domes: the central symmetry was considered a requisite and even in the experimentation of the domes in Hadrian’s Villa all the forms present a centralized character [Rasch 1985]. Some exceptions may be found in the apses of thermae [Lotz 1955], and it is usually presumed that the octagon of the church of St. Gereon in Cologne rests on the oval foundations of a previous Roman building [Götz
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1968] (fig. 9b) and [Krautheimer 1984] (fig. 9c). Choisy [1873] discovered that in the bridge of Narni, the central span of the inclined road was adapted by using an arch formed from two quarter-circles of different radii. Perhaps more examples can be found, but it appears that the Romans used the oval form for arches and domes only in exceptional cases.
Fig. 9. a) One of the hypothesis concerning the oval geometry of the Roman Coliseum [1993]; b) Medieval octagon of the church of St. Gereon in Cologne; it is supposed to be built on the foundations of an oval Roman building [Götz 1968]; c) reconstruction by Krautheimer [1984]
The Middle Ages Most medieval domes have a centralised form, probably due to the Roman influence. In
Spain most Romanesque churches pertain to this type. However, there are also some exceptions, and the church of Santo Tomás de Olla, dated by Gómez Moreno [1919] to the tenth century, presents an octagonal dome on an oval plant, 6 x 5.5 m, surrounded by horseshoe arches (Fig, 10).
Fig. 10. Church of Santo Tomás de Olla, Spain, built in the tenth century [Gómez Moreno 1919]
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In France, Chappuis [1976] has made what appears to be the only exhaustive study of the use of the oval form in the Middle Ages. He has studied some 400 Romanesque churches in the south of France which present some kind of oval arches or domes. Of these, 130 have domes with an oval plan. Fig. 11a shows Chappuis’s classification of Romanesque ovals. Fig. 11b shows clearly the oval plan in two churches. Chappuis has found, then, a precedent in Europe for the Renaissance oval domes, which so far has not been noted by historians of architecture.
Fig. 11. The oval form in French Romanesque churches [Chappuis 1976]. a) Classification of ovals; b) Two examples of churches with oval plans: above, SaintMartin de Gurçon, Dordogne, and below, Balzac, Charente
Another element in which oval arches can be found in Romanesque churches are the groined vaults, resulting from the intersection of two barrel vaults. It may that the medieval masons knew some method to lay out approximately the curve of the groin before construction, and the technique of the “lengthened arch” which will be discussed below, could have been used. But there are other techniques which make possible the construction without the physical building of the diagonal centring, as may be seen in the drawing by Mohrmann [Ungewitter 1890] (fig. 12). (The Romans may have used the technique in building their groined vaults.)
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Fig. 12. Building of a Romanesque groined vault without knowing the form of the groin. The intersection is made “physically” with the wooden planks [Ungewitter 1890]
The Gothic is based on a great simplification of the building procedures. In the cross vaults the ribs, which are always composed of circular arcs, define the geometry of the vault, the masonry shell closing the space between ribs in the last step of the building (fig. 13). In the simple quadripartite vault, the cross ribs are semicircles and the transverse arches and the formerets (wall ribs) are adjusted to the desired height using pointed arches. (The best study on the geometry of the Gothic ribs is still that of Willis [1843].)
Fig. 13. The geometry of the Gothic cross vault is defined by that of the ribs, which are made of circular arcs. a) The technique permits correspondence with the forms of adjacent vaults; b) The ribs are built before and then the shell is closed [Viollet-leDuc 1858]
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Santiago Huerta Oval Domes: Universidad Politécnica de Madrid