Abstract. Answers to a reader's query why Roman amphitheatres are elliptical while their derivatives, the Iberian bullring and the modern circus, are circular, in the Nexus Network Journal.

Query: Why are Roman amphitheatres elliptical?

ORIGINAL QUERY:
Date: Tuesday 20 May 2003
From: Dag Nilsen <dag.nilsen@ark.ntnu.no>
Norwegian University of Technology and Science (NTNU)

In taking over some lectures for our retired professor, I had to illuminate our first year students on the, to them, strange subject of of classical antiquity. The notion of architectural history is still so foreign to them that they don't know what to ask about. Or, maybe I overload them with information and slides - it's not that easy to squeeze it all into just three lectures.

Standing there, speaking about Roman theatres and amphi-theatres, I started wondering. I am embarrassed to admit that it was the first time it struck me that there must be a reason why the Colosseum and other amphitheatres are elliptical in plan, while their derivations -- the Iberian bull-fighting arenas, and the modern circus -- are circular, and the Roman theatres are semi-circular. I can't remember having seen any explanation in the literature. And -- are they true ellipses, or made up from circle arcs of different radii?

From: Luigi Pepe <pep@unife.it>

Non so se è la risposta esatta, ma ho sempre pensato che la forma ellittica sia dovuta alla proprietà dell'ellisse che la somma delle distanze da due luoghi privilegiati (i fuochi) sia costante.

(I don't know if this is the exact response, but I have always thought that the elliptical shape was due to the property of the ellipse that the sum of the distances from two privileged loci (the focuses) is constant.)

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From: Maurizio Lorber <mauriziolorber@yahoo.it>

I'm without an answer too about the ellipse-cirle problem in architecture (by the way, as you know, the colosseo is not an ellipse) but I think that you can find an interesting geometrical construction in the first book of Sebastiano Serlio and probably in the books of architecture of Vitruvius (especially in the illustrated edition of Daniele Barbaro and Palladio).

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From: Paul Rosin <Paul.Rosin@cs.cardiff.ac.uk>

Jean-Claude Golvin in his book, L'Amphitheatre romain: essai sur la theorisation de sa forme et de ses fonctions, discusses these issues in great detail. First, he argues (as have many others before and after) that ampitheatres are oval rather than elliptical.

As for their overall shape, he says that a square or rectangular ampitheatre would result in combatants getting stuck in the corners. Circles/ovals make better use of the space. And finally, ovals/ellipses are better than circular ampitheatres since they have a dominant direction, giving a structure to the fight, whereas a circle would lead to an impression of confusion.

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From: Mario Docci <mario.docci@uniroma1.it>

Potrà trovare la risposta ai suoi quesiti sulla rivista Disegnare, idee, immagini, n°18-19 Editore Gangemi-Roma. Comunque gli anfiteatri romani sotto tutti degli ovali a quattro o più centri.

(You can find the answers to your questions in the periodical Disegnare, idee, immagini, no. 18-19, Editore Gangemi, Roma. In any case, Roman amphitheatres are all ovals with four or more centers.)

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From: Brigitte Van Tiggelen <vantiggelen@memosciences.be>

While honeymooning in Greece some years ago, I remember visiting an elliptical theater in Thorikos. It seems that early stone theater were not circular... and that more theater of this kind have been excavated in the last decade.

Have a look at the following web sites (some in German) :

About the reason for elliptical, one should also question the probable need for a rectangularlike stage instaed of a circular one...

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From: João Pedro Xavier <jpx@arq.up.pt>

It's almost impossible to distinguish an ellipse from an oval when the major and minor axes are not too different. Anyway, there is the question of the "parallelism": it is not possible to draw "parallel" ellipses while it's easy to do that with ovals (I'm remembering Serlio's ovals and the correspondent instructions to trace concentric ovals). I mean, if we have an ellipse and we intend to trace a line with each point equidistant to that ellipse we do not obtain another ellipse. So it seems reasonable for builders to adopt the oval form which facilitates the construction of the rows of benches and do not corrupt the whole form. But we have to wait for Sylvie's commentaries...

(Editor's note: Sylvie Duvernoy addressed the ellipse-oval issues in her Nexus 2002 presentation, "Architecture and Mathematics in Roman Amphitheaters", published in Nexus IV: Architecture and Mathematics.)

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From: Roger Herz Fischler <rhfischl@math.carleton.ca>

I discuss this in Shape, Form and Space (editor's note: this is the text the Herz Fischler prepared for his own course on architecture and mathematics, a chapter of which is excerpted in the NNJ as "Didactics: Proportions in the Architecture Curriculum.")

The references that I give are:

Thoenes, C. 1963. "Studien zur Geschichte des peeteresplatzes", Zeitschrift fuer Kunstgeschichte, 26 (1963), 97-145 [this is for St. Peter's, suggests that a quadarc was used].

Gridgeman, N.T. 1970. "Quadarcs, St. Peter's and the Coloseum", The Mathematics Teacher, 63 (1970), 209-215. Suggests quadarcs in all of these.

The important thing to remember is that conic sections in Roman times were all defined as cuts of cones; equation forms did not exist! So laying out an elliptical field is something that may not have even entered someone's mind.

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From: John de Pillis <jdp@math.ucr.edu>

REFLECTING PROPERTY of ELLIPSES:
================================
A property of truly elliptical amphitheaters is that sound emitted from one of its foci is concentrated onto the other focus. That is, although sound may travel in all directions from one of the foci, each "sound wave" is reflected off the wall at just the correct angle so as to arrive at the second focus.

APPLICATION:
============
This means that a person can whisper while standing at one focus, and be heard clearly by someone else, standing at the second focus. (This feature is shown to visitors in several government buildings and museums.) People not located at either focus can not hear the whispers.

=======
Whether the Romans used this "secret channel" to communicate, I do not know. Could be.

ATTACHMENT:
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The (original) graphic,"Elliptical Arena," will illustrate more exactly, the ideas presented in my verbal response to why amphitheaters may be elliptical.

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From: Graham Pont <pont@tpg.com.au>

I seem to recall reading (many years ago) an explanation of the theatre form in Francesco Milizia, Trattato completo formale e materiale del teatro (1794).

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From: Rudi Penne <rudi.penne@pandora.be>

I am not sure whether my contribution is useful, because I haven't taken a close look at the shape of the amphitheatres yet. However, I guess they are not elliptic, but rather "super-elliptic" (at least they should be), with equation

(|x|/a)^p + (|y|/b)^p = 1

with p a real number, typically between 2 and 3, but it might be larger. If p equals 2, we get an ellips, and if p tends to infinity, we get a rectangle. Superellipses (or Lamé ovals) are a "compromise" between ellipses and rectangles.

These super shapes often appear in nature since they perform better than circles (ellipses) and squares (rectangles) in matters like optimal fluid transport (e.g., a plane section of a bamboo stem). I 've been told that Piet Hein (mathematician? town architect?) designed a traffic square in Stockholm using a super-ellipse as model (I should look up again about the exponent p here), optimizing the traffic around the square.

I don't think that the ancient designers of the Colosseum were aware of the existence of super-ellipses, but maybe they had the right intuition?

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Relating to the form of Amphi-theatres, I suggest to having a look at:

Vladimiro Valerio, "Sul disegno e sulla forma degli anfiteatri," Disegnare, 6 (1993): 25-34 (English summary).

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Regarding the setting out of amphitheaters, I'd say they are generally made of circular arcs to simplify the builder's work. While there are means of drawing ellipses on a drawing board or even on a builder's lofting floor, these techniques are difficult to transfer to a large scale building project. This is especially true in the case of an amphitheater, because as the "ellipse" expands with each additional row of seats its aspect ratio (or angle of projection) changes, and it become gradually more circular. Determining the foci, etc., of each row would become quite a chore. On the other hand, four circular arcs can be combined into quite a presentable pseudoellipse. (They seem to look best when they are tangent to a circumscribed rhombus, touching at at the center of each side) The task of creating equally spaced "ellipses" on this type of plan is reduced to increasing the radii of the component cirular arcs by equal increments.

Regarding the shape of the arena floor, it would appear that the ellipse has several descendants, including the circus ring, the square "ring" of boxing and wrestling, and the rectangular court. Perhaps one should say that the ellipse achieves a balance of the central and bilateral geometries. The former seems to be suited for staging direct combat between men and/or beasts, whereas the latter lends itself to territorial contests*. I think the Roman amphitheater was used for both forms of competition. Additionally, an elongated plan establishes seating closer to and farther from the center of action, suitable for accommodating a socially stratified audience.

*Of modern playing areas, one might wonder if the more rounded ones (ellipses in the case of Australian Rules Football, and radiused rectangles in the case of ice hockey) don't promote a higher proportion of physical combat relative to territorial objectives. And perhaps tennis, played in virtually square arenas, should be seen as an indirect form of combat rather than an intimate game of territory.

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From: Pietro Totaro <pietro.totaro@fastwebnet.it>

The most recent (as far as I know) book about the amphitheatres shape and history is:

D. L Bomgardner, The Story of the Roman Amphitheatre. London/New York: Routledge, 2000. ISBN 0-415-16593-8.

You can find an article [by Camillo Trevisan] "Sullo schema geometrico costruttivo degli anfiteatri romani: gli esempi del Colosseo e dell'Arena di Verona" excerpted from Disegnare, idee, immagini, no. 18-19 at this web address: http://www.iuav.it/dpa/ricerche/trevisan/anfite/anfite1.htm. Another article of the same author published on the web: http://www.iuav.it/dpa/ricerche/trevisan/anet/trait_en.htm is related to the oval problem and it could be useful too. Moreover it has a complete English version, unfortunately this is not available for the former.

The Coliseum plan is a true oval (polycentric construction) but it closely approximates the ellipse (see the remarks of Mario Docci above), the same property is relevant, e.g., to Verona Arena. So, just considerations of cosmogonical nature or the one of João Pedro Xavier could discriminate between the two possibilities.

It is true that ancient Greeks called the conics stereoi topoi (solid loci). However, they knew well the plane properties of these curves ( boast of Apollonius of Perga, 3rd-2th century B.C., but already partially known to Menecmo) . They did not have the mathematical tools to calculating the perimeter of the ellipse exactly (i.e., the elliptic integrals introduced in the works of Euler and Legendre ) but they could calculate the area of the ellipse (Archimedes, On Conoids and Spheroids, prop. 6, its approximation depending only on p ) . Apollonius's work was known during the Roman imperial age. The discovery of the gardener's method, namely a method to trace an ellipse by means of a rope string and two pivots, is attributed to Anthemius of Tralles (mathematician and one of the architects of Sancta Sophia). One can always suppose that this method is far older: according to some researchers it dates back to Neolithic age.

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From: Laurence Kasparowitz <PLN795@co.santa-cruz.ca.us>

seems to me quite simple...at least for the oval...it is a true test of charioteers to drive straight for awile ...and then have to make a half circle...and then keep doing it !!!

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From: Biagio Di Carlo <mail@biagiodicarlo.com>

This is to continue the discussion of superellipses begun by Rudi Penne:

Under the pseudonym "Kumbel," Piet Hein(1905-1996), Danish scientist, designer, inventor and poet, wrote some 7000 "Grooks" (short poems), later published by MIT Press. He was the inventor of the SOMA cube (1936) and developed the studies of ellipses begun by French mathematician Lamé. The scientific constributions of Piet Hein can be compared to those of Einstein and Bohr in the field of physics.

The SUPERELLIPSE was obtained by modifying the equation for the ellipse. The form of the superellipse is similar to a rectangle with its angles rounded, and was used by Hein to design a large piazza in the center of Stockholm, as well as for designs of tables and decorative objects (1950-1960). By rotating the superellipse on an axis one obtains a three-dimensional figure, the SUPERELLIPSOID, also called a Super Egg.

In 1973 Lloyd Kahn published in the magazine Shelter the information necessary for the construction of a superellipse, that is, the chord factors relative to a 4v icosahedron, expanded to the form of a superellipse.

The ellipsoidal form was successfully applied in the field of geodesic domes, particularly by John Rich, Ernie Aiken and Carey Smooth (links at http://www.biagiodicarlo.com).

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From: Taro Nagazumi <tnagazumi@cc.e-mansion.com>

In the book Earth from Above by Arthus Bertrand there is an aerial view of a horse like figure in Oxfordshire. If you compare the form of it to the African Animal trap relic also in the same book you can see the movement through space as in hunting or horse riding as a giver of forms. If you look at Sterlin's book of Architecture you can see Precidents as Tarxien Temple in Malta etc. Then if you look at the main purpose of the Collosium as Chariot Racing, the purpose of enlonging one axis becomes so clear. Besides the Centriato( if I spell it correctly) the unit that Romans used to calculate farm land is very linear. Which makes the most minimum turns for the cattle driven plow to turn more efficient. Even in 18th-19th century Edo (Tokyo) which had a million population, you can see a similar form for horse training(with bow & arrow shooting range perpendicular to it).

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a mathematical point of view it is practically impossible to differentiate among an ellipse or an oval. Ellipses vary from the circumference, when both axes are equal to the ellongated ellipse, where the axes are very different, passing through the golden ellipse whose axes are in a golden relation (See Stonehenge Temple in my book, From the Golden Mean to Chaos (Buenos Aires: Vera W. de Spinadel, 1998). previously mentioned). An oval is any closed curve that encloses a convex region. Among them, the most well known are Casssini´s ovals, which are defined as the set of points in the plane, whose product of distances to two fixed points is constant (remember that the ellipse is defined as the set of points in the plane whose sum of distances to two fixed points is constant). With this definition of Cassini´s ovals, you get an enormous variety of forms!

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From: Mark Wilson Jones <M.W.Jones@bath.ac.uk>

There have been quite a few responses already, so there is not call for much more comment. However, as the inquirer notes, not many of the responses address the problem of why amphitheatres should be the sahape they are (whether elliptical or oval). Golvin's work (cited in one reply) is the most extensive to tackle this theme, and I have also written on the subject in an article: "Designing amphitheatres," Romische Mitteilungen 100, 1993, 391-441.

The following is lifted (more or less) from pp. 391-2:

"The origin of the architectural form of the amphitheatre is somewhat obscure. ... No doubt it actually evolved by a process of adapting rectangular spaces, often in the context of civic fora, the setting for early gladiatorial combats. In a rectangular arena the action could get "trapped", as it were, in a corner; and every corner is a relatively long way from spectators at the opposing one. Cutting off or rounding off corners naturally leads towards the smoother shape of the ellipse/oval, which may be likened to a stretched circle, or a circle with a tendency towards linearity. This form has further advantages as a compromise between those suited both to spectacles (centric) and processions (linear).

It is generally assumed that this evolution took place in Campania ... However, influential prototypes may (also) have been the wooden structures erected for occasional gladiatorial shows in the Forum Romanum. ... At some stage the elliptical/oval arena layout took hold, a novelty which, quite apart from the functional justification mentioned above, may have been suggested by the splayed or oblique shape of the Forum. ... (a parallel sided arena would have implied uncomfortable tapering spaces on the outside.) Did someone reason as did Michealangelo, who in the 16th century chose an elliptical pavement for the similarly shaped Campidoglio piazza, precisely so as to avoid this sort of formal conflict? Otherwise the ellipse may have recommended itself for the simple reason that it represented a departure from established building types such as the theatre and stadium, besides making a decisive break with forms which ultimately descended from Greek usage.

In contrast to relatively static rectilinear or centralised groundplans, the formal qualities of the ellipse are quite literally dynamic. Successive rings change in their proportions,the perimeter being much rounder than the arena [Plates iii and iv]. The varying curvature and the lack of a single focus naturally generate a sense of movement, and, without a good view of a large portion of the structure,it can be quite difficult to percieve exactly where one stands in relation to the whole. All this was quite revolutionary for its time...."

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From: Dag Nilsen <dag.nilsen@ark.ntnu.no>

I got enough references for "how" to keep me busy for the summer, though my initial wonder was mainly "why?", which one of the responders picked up: sounds quite plausible that an oval form by being directional gives structure to fighting, while still being round, i.e. with no corners.

I realized almost immediately after having sent the query that the amphitheatre could not be a true ellipse, at least not both the arena and the seating perimeter, and moreover, that it would be impractical in layout on site -- but thanks to the answers, I noted the difference between "elliptical" and "oval", which I was not aware of before.

I will be better equipped if awkward questions pop up in next year's lectures.

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From: Chris Lynn <chris.lynn@doeni.gov.uk>

Coming from a rather different angle I too wondered why amphitheatres were elliptical or oval? Earlier respondents have given good practical explanations of 'how' in terms of geometry and why- lack of awkward corners, the opportunity for linear progression. I also thought that the emperor, seated in the middle of one of the long sides would also get an optimum view in and oval- everything visible, nothing too far away. But a circle or a rectangle with hemicyclical ends could have achieved the same result and would have been easier to lay out, to cut stone for and to build. I wonder if the ellipse or oval form had any antecedents in the world of sacred architecture, thinking of the shows in the amphitheatre as originally funeral games? The design of the amphitheatre could have had a cosmological inspiration, the building representing a massive convocation of the community, seated internally in reverse of the normal spatial order of their ranks and assembled for a purpose which may originally have been (quasi) religious.

In the normal sense society would have been imagined with the emperor, magistrates and vestals at the top, then the 'knights' and then the mass of the free populace at the bottom of the (tripartite) social pyramid (with slaves below that again). To pursue this reverse analogy, it might have been imagined that the contests and shows were taking place in another region either above in the 'sky' or below in an infernal region. The idea that the amphitheatre, or rather its arena, was a portal to the infernal regions is reflected in the dress of the slaves who removed the bodies of fallen gladiators and the name of the gate from which they were removed.

Back to the initial question again, but slightly re-stated, is it possible that defining the arena in an oval or ellipse also symbolised its 'otherness' as well has having practical advantages for the organisation of the dreadful spectacles?

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From: Per Arnt Carlsen <pacarlsen@sensewave.com>

As a simple architect I am inclined to suggest that building types with an audience, where two parts meet, had two focuses, en elliptic plan. The whole scheme would have been made for the two individuals, such as the fight betweeen the two gladiators, or the dialog of the actors in the theater.

Building types for the audience having one focus, could have a circle, such as in an arena for the bullfights where there was one focus, the bull, and the man circled around him before the audience. In a circus, the artist, or his performance, would be the focus.

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