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1.Mathematical principles of symmetry, overall same proportion(golden ratio) and harmony, just like nature planned for the human body, used in the construction of Greek and Roman theatres and amphitheatres (by Marcus Vitruvius Pollio)     

-From Marcus Vitruvius Pollio (ca. 70-ca. 25 BC), being a Roman architect and encyclopedist, who wrote approximately 10 books discussing such subjects as acoustics and astronomy, we can learn from his 'On Architecture' what mathematical based architectural principles  were the chief reference for the engineers from Greek and Roman theatres.
The importance of proportion (the golden ratio : (1 + sqrt(5))/2, the same proportion nature planned between the parts of the human body)  symmetry and balance and harmony is quoted in many chapters of this book. They were needed if a beautiful, solid and safe constuction had to be built. 

-In the 'On Architecture' we can read in book :
I, ch 1 (8) : " A man must know music that he may have acquired the acoustic and mathematical relations and be able to carry out rightly the adjustments of balistae, catapultae and scorpiones. (9) In theatres, also , are copper vessels and these are placed in chambers under the rows of seats in accordance with mathematical reckoning...The Greeks call them echeia. The differences of the sounds which arise are combined into musical symphonies or concords : the circle of the seats being divided into fourths and fifths and the octave.. Hence , if the delivery of the actor from the stage is adapted to these contrivances, when it reaches them, it becomes fuller, and reaches the audience with a richer and sweeter note."  

III, ch 1(1) "The planning of the temples  depends upon symmetry : and the method of this architects must diligently apprehend. It arises from proportion (named by the Greeks analogia). Proportion consists in taking a fixed module, in each case, both for the parts of a building and for the whole, by which the method of symmetry is put into practice. For without symmetry and proportion no temple can have a regular plan : that is, it must have an exact proportion worked out after the fashion of the members of a finely-shaped human body.... (3) In like fashion the members of temples ought to have dimensions of their several parts answering suitably to the general sum of their whole magnitude. 
***If nature has planned the human body so that the members correspond in their proportions to its complete configuration, the ancients seem to have had reason in determining that in the execution of their works they should observe an exact adjustment of the several members to the general pattern of the plan.*** 
Therefore, since in all their works they handed down orders, they did so especially in building temples, the excellences and the faults of which usually endure for ages." 

V, ch. 3 (4) "The curved level gangways, it seems, should be made proportionately to the height of the theatre; and each of them not higher at the back, than is the breadth of the passage of the gangway For if they are taller, they will check and throw out the voice into the upper part of the theatre. Neither will they allow the endings of words to come with a clear significance to the ears of the people in their seats above the gangways.  In brief the section of the theatre is to be so managed that if a line is drawn touching the lowest and the top rows, it shall also touch the front angles of all the rows. Thus the voice will not be checked. Many and spacious stepped passages must be arranged between the seats...Everywhere, each passage must be continuous and straight without bends; so that when the audience is dismissed from the spectacle, it may not be cramped, but may find everywhere separate and uninterrupted exists." 

V, ch. 4 (2) "By the rules of mathematics and the method of music, they sought to make the voices from the stage rise more clearly and sweetly to the spectators' ears. For just as organs which have bronze plates or horn sounding boards are brought to the clear sound of string instruments, so by the arrangement of theatres in accordance with the science of harmony, the ancients increased the power of the voices."

V, ch.5(1) "Hence in accordance with these enquiries, bronze vases are to  be made in mathematical ratios corresponding with the size of the theatre. They are to be made that when they are touched, they can make a sound from one to another
of a fourth, a fifth and so on to the second octave. (3) Thus by this calculation the voice, spreading from the stage as from the centre and striking by its contact the hollows of the several vases, will arouse an increased clearness of sound, and , by the concord, a consonance harmonising with itself."

2.Greek and Roman theatres : differences in their construction 

-Also in book 5, resp in ch 6 and 7, he describes the differences in the contruction of the Greek and Roman theatres.The Greeks, in choosing the sites of their theatres, almost always availed themselves of some natural hollow on the side of a hill; but the Roman theatres and amphitheatres, with few exceptions, stand upon a plain.

IV, ch.7 (1) In the Greek theatres some things are done differently. Firstly, in the orchestra, the angles of three squares touch the circumference, whereas in the Roman theatre we have the angles of four triangles. In the Greek the line of the proscenium(or stage) is drawn along the side of the square which is nearest to the scenery, where it cuts the circumference. On the same side, parallel to this a line is drawn to touch the outside of the circle, and on this the front of the scenery is marked out.Through the centre of the orchestra, opposite to the proscenium, another parallel line is drawn; where it cuts the circumference  right and left, centres are marked at the ends of the semi-circle. Fixing the centre of the compasses on the right, with a radius equal to the distance of the left point, a circle is drawn to the left side of the proscenium. In the same way, the centre is fixed on the left and with a radius equal to the distance of the right, a circle is drawn to intersect the right side of the proscenium

The Greek theatre in Epidaurus(4th century BC)

-In the Roman theatre the orchestra is a place to sit, instead of a performing area as the Greeks had used it. The stage grows in importance and is brought into direct contact with the audience. The auditorium is a semicircle, often partially supported by a hill underneath as well as concrete vaulting. Corridors under the tiers were used in case of rain. Vitruvius gives several additional measurements for ratio relationships of orchestra radius and diameter to width of passageways, the length of the scene building and height of the podium. In regards to acoustics, he suggests that a roofed colonnade surround the uppermost rim of the cavea and must be level with the top of the stage buildings. All this to insure even distribution of sound thoughout the theatre. From the beginning, seating in the theatres of the Roman empire was designed to seperate the different ranks of society  using parapets. The Greeks, however, had a more democratic approach to seating where each tribe had their own section within the same gallery. The Romans added the scaenea frons, a dividing wall at the back of the stage. The five angle points in the stage area determine the position of the doors and stage entrances. The remaining seven angle points radiate out to divide the seating into sections. The public had to entrer through vaulted passageways to reach orchestra and radiating stairways. The Romans created an unique system of tunnels, stairways, colonnaded galleries. Because they used concrete, their architects were able to design theatres by desiring a place for secular entertainment unrelated to religious practices.

V, ch.6 (1)The plan  of the theatre is to be thus arranged : that the centre is to be taken, of the dimension allotted to the orchestra at the ground level. The circumference is to be drawn; and in it four equilateral triangles are to be described touching the circumference at intervals (just as in the case of the twelve celestial signs, astronomers calculate from the musical division of the constellations).

 Of these triangles, the side of that which is nearest the scene will determine the front of the scene, in the part where it cuts the curve of the circle. Through the  centre of the circle a parallel line is drawn which is to divide the platform(pulpitum) of the proscenium from the orchestra. (2)Thus the stage will be made wider than that of the Greeks because all the actors play their parts on the stage, whereas the orchestra is allotted to the seats of the senators....

The Roman theatre in Orange(50 AD) 

3.Properties of two famous amphitheatres : the Colosseum(Rome) and the Arena(Verona)

-Gladiatorial contests were originally held in large open spaces with temporary seating; but as the games became more frequent and popular, there was need for a larger and more permanent structure. The Romans eventually designed a building specifically for this type of spectacle (called an amphitheatrum (from ajmfiv, on both sides, qevatron, a theatre) because the seating extended all the way around the oval or elliptical performance area(the arena), which was covered with sand for the gladiatorial combats and wild beast fights. The word "arena" derives from "rena", which means in fact sand. One of the oldest stonen amphitheatre was built in Pompei in the first century BC and seating approximately 20,000 spectators. Like Roman theaters, amphitheaters were freestanding. Because they did not require natural hills, as Greek theaters did, they could be built anywhere. 

-All amphitheatres are elliptical because the ellipse is the ideal figure to insure the best distribution and amplification of sound throughout the theatre. The transmission of each sound produces circular waves in the air. If the center of the transmission is lying on the circle circumference we observe that all rays are reflected in different directions. If the sound has its origin in the middle of a semi-circle, each ray is reflected back to the origin. Starting from the focus of a parabola, all waves dissappear parallel to the main axis. But if the sound emerges from one of the foci of an ellipse then from mathematics we know that all waves will  come together in the other focus which creates a second centre of sound that will amplify the original sound. In addition the Romans  kept the golden ratio as the uniform proportion in their construction (main axis/small axis of the basic ellips, width and height of steps, width and height of the rows around the arena, load-bearing walls containing the staircases and surmounted by barrel- vaulting...). In the work "Storia e descrizione de principali teatri antichi e moderni" from Giulio Ferrario we even find a complete mathematical explanation for the dimensions  of the "palco scenico"=platform and the position of the 'proscenico"= proscenium. Confirming the ideas of Vitruvius considering receiving a good view and reception of the sound at each point in the theatre, the maximum angle of inclination shouldn't exceed arctan(1/phi) and the largest distance from the nearest focus (|KF|) should be smaller or equal to |BF|.

The Colosseum in Rome

-The largest amphitheatre built by the Romans is undoubtedly the Colosseum of Rome. Its monumental size and grandeur as well as its practical and efficient organization for producing spectacles and controlling the large crowds make it one of the greatest architectural monuments achieved by the ancient Romans. The Flavian amphitheatre was begun by the founder of the Flavian dybasty, Vespasian. It reached the top of the second level before his death in AD 79 . His son Titus added the third and fourth levels the next year. This amphitheatre was built on the site of Nero's large lake in the gardens of his palace. Drainage of the site had to be made before any construction could begin. The natural flow of water through the excavated hollow was useful for the staging of mock naval battles in the arena.  

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     -The conception, dimensions and arrangements of this vast structure(outer dimensions : 188 m x 156m) were such as to furnish seats for more than 80,000 spectators, round an arena large enough to afford space for the combats of several hundred animals at once with passages and staircases to give ingress and egress, without confusion, to the immense mass of spectators and other attendants.

It was designed to coop with the thousands of victims devoted to destruction; channels for the rapid influx and outlet of water when the arena was used for a naumachia and the means for the removal of the carcasses. The substructure goes down 6.08 m to a brick pavement. The arena is elliptical, 86m x 54m (approx. the golden ratio). For the spectators' protection, a fence surrounded the fighting area. Behind this, 4m above the arena was the podium for distinguished spectators,like city officials. Above this was the cavea, where most of the seating was placed, consisting of two main sections (one of 20 rows of marble seats, the other 16 rows). A wall, pierced by windows and doors, rose up behind the upper section and gave on to a corridor in the back. Behind the collonade on this wall, the wooden seats for women were placed and on the roof of this area was a standing room for thousands of the plebeian  class.


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As a general description of the building the following passage of Gibbon's twelfth chapter is perfect: "It was a building of an elliptic figure, founded on fourscore arches, and rising, with four successive orders of architecture, to the height of almost (150 feet) 50 m. The outside of the edifice was incrusted with marble, and decorated with statues. The slopes of the vast concave, which formed the inside, were filled and surrounded with sixty or eighty rows of seats, of marble likewise, covered with cushions, and capable of receiving with ease about 80,000 spectators. Sixty-four vomitories (for by that name the doors were very aptly distinguished), poured forth the immense multitude; and the entrances, passages, and staircases, were contrived with such exquisite skill, that each person, whether of the senatorial, the equestrian, or the plebeian order, arrived at his destined place without trouble or confusion. Nothing was omitted, which, in any respect, could be subservient to the convenience and pleasure of the spectators. They were protected from the sun and the rain by an ample canopy, occasionally drawn over their heads. The air was continually refreshed by the playing of fountains, and profusely impregnated by the grateful scent of aromatics. In the centre of the edifice, the arena, or stage, was strewed with the finest sand, and successively assumed the most different forms. The subterraneous pipes conveyed an inexhaustible supply of water; and what had just before appeared a level plain, might be suddenly converted into a wide lake, covered with armed vessels, and replenished with the monsters of the deep. In the decoration of these scenes, the Roman emperors displayed their wealth and liberality; and we read on various occasions that the whole furniture of the amphitheatre consisted either of silver, or of gold, or of amber. The poet who describes the games of Carinus, in the character of a shepherd, attracted to the capital by the fame of their magnificence, affirms that the nets designed as a defence against the wild beasts were of gold wire; that the porticoes were gilded; and that the belt or circle which divided the several ranks of spectators from each other, was studded with a precious mosaic of beautiful stones."

-The following ground-plan, external elevation, and section, are from Hirt, and contain of course some conjectural details. The ground plan is so arranged as to exhibit in each of its quarters the plan of each of the stories: thus, the lower right hand quarter shows the true ground-plan, or that of the lowest story; the next on the left shows a plan of the erections on the level of the second row of exterior columns, as well as the seats which sloped down from that level to the lower one; the next quarter shows a similar plan of the third order, and the upper right-hand quarter exhibits a view of the interior as it would appear to an eye looking vertically down upon it. The dotted lines on the arena are the radii, and their points of intersection the centres, of the several arcs which make up the ellipses.

The Arena in Verona

-The arena of Verona was built in the first century A.C. during the last years of the emperor Augustus. The elliptical construction with a central open space called the arena measures externally 152m X123m, including the superior wall. The perimeter of that wall is 391 m.  Of the outer ring, primarily built for decorative reasons, only the famous "ala" (wing) remains and is formed by three arches on three tiers. It is quite possible that the building was not supervised by a single architect but was rather the work of a team of esperienced master-builders. It is not an especially complex building. Once the axial dimensions of the basic ellipse shape of the plan was decided, the same module of load-bearing walls containing the staircases and surmounted by barrel- vaulting had simply to be replicated a certain number of times until they joined. The same unit drawings could therefore be given to as many foremen as there were teams of labourers and craftsmen.

-For the thoroughly practical and realistic Romans the decision to work to  proportional measurements derived less from a desire to achieve aesthetic harmony than from the recognition that it provided an easy way of ensuring effective coordination of all building operations. The system reduced unforseen contingencies to a minimum and enabled all the materials to be prepared beforehand - stone could safely be shaped  at the quarry and brought to the site ready for assembly. This modular construction is particularly solid. The load-bearing structure all dovetails together: the inner three elliptical, barrel-vaulted galleries all fit into th ecross-walls radiating from the circumference of the ellipse: and the external gallery, the one that ran between what is now the outer wall and the 'ala'= wing, was at two superimposed levels. 

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-That permanent element that considerably simplified matters for the builders and at the same time ensured that the modules constructed by the various teams of workmen actually fitted together in the end, was the initial choice of a basic relationship linking the dimensions of every individual part so that every measurement was proportionate to another. Thus the two axes of the inside ellipse forming the Arena measure resp. 73.68 m  and 44.43 m which proportion approximates the golden ratio (phi = 1.618..).

atrapkl.gif (42348 bytes) -We distinguish the same proportion between the width and height of the 45 steps(average with: 72 cm and height :45cm) and also in the angle of inclination overall the steps. Above the three concentric galleries surrrounding the arena ran three corridors dividing the tiered seating into three orders, whereas the position of the staircases that divide the seating into wedge-shaped sectors corresponds to the cross-walls radiating from the circumference of the arena. The three orders of arches comprising the outer wall reflect the three orders of seating inside, the arches themselves correspond to vaulting towards the interior and the walls supporting the vaulting are marked by the pilasters and pilaster strips in the outer wall. Each tier consisted of 72 arches supported by 73 immense pillars built of square stone blocks.

-The magnificent construction features clear connections between its underlying skeleton, the tiered seats around the arena and the exterior of the outer wall, i.e. between the structure, its function and its decorations. The second wall is joined to a third, which is in turn connected to a fourth wall by way of a vaulted ceiling which forms a tunnel 9 m tall and 4 m wide that circles the entire amphitheater. From this tunnel several stairs pass through entrances called "vomitoria" to reach the marble steps of the "cavea". All this gives the monument a severe elegance being surrounded by a thirty metre outside wall which made the arena look taller. This convergence of the ideals of solidity, functional efficiency and formal order ( firmitas, utilitas, venustas) make the Arena one of the finest examples of the superb engineering capacities of the Romans and their conception of public architecture. 

-From the very beginning of its existence, the Arena of Verona has been a powerful and suggestive place of entertainment: gladiatorial fights, jousts, tournemaments and nightly games took place against a rich scenic background. Because of its perfect mathematically balanced proportions it has ideal acoustical properties. During the 18th century, theatrical performances became a constant feature. The solemn Congress of the European Nations, which took place in Verona in 1822, deserves a special mention: on that occasion, Rossini composed and directed in the Arena the cantata "La Santa Alleanza" (The Holy Alliance), for all European sovereigns. Still until today yearly the most famous Opera Festival is organized in this amphitheatre.The first performance of the Festival dates back to August 10th, 1913 when Aida by Giuseppe Verdi was performed. Today the Arena can accommodate a total of 22,000 spectators and during the opera season over half a million people(from both within and outside Europe) flow every year into this amphitheatre : an important place for culture and a harmonious meeting point for society.

Sources 

http://www.greatbuildings.com/buildings/Roman_Colosseum.html
http://www.chch.school.nz/mbc/Colosseu.htm
http://www.ukans.edu/history/index/europe/ancient_rome/E/Roman/Texts/Vitruvius/home.html
http://www.ukans.edu/history/index/europe/ancient_rome/E/Gazetteer/Periods/Roman/Topics/Architecture/Structures/amphitheatres/home.html
http://www.telmec.it/Uk/Arena_frame.htm
http://www.ukans.edu/history/index/europe/ancient_rome/E/Roman/Texts/secondary/SMIGRA*/Amphitheatrum.html
http://wings.buffalo.edu/AandL/Maecenas/France/Orange/ac821204.html
http://www.istos.net.gr/theater/page02.htm
http://www.gardalake.it/arena-verona/
http://www.csuchico.edu/art/contrapposto/contrapposto99/pages/essays/themehfa/keale.html
http://floc99.itc.it/turism/excursion/arena_verona_pr.htm
http://www.emmeti.it/Welcome/Veneto/Verona/Arena/index.it.html
http://www.provincia.asti.it/edu/giobert2/comea/verona/vero.htm

Vitruvius "De Architectura" translated by Frank Granger. Harvard, Cambridge Massachusets Univ. Press
Vitruvio Pollione, "Dell'architecttura", interpretazione di Giovanni Floriani, Giardini editori et Stampatori in Pisa
"Storia e descrizione de principali teatri antichi e moderni" Giulio Ferrario, Arnaldo Forni editore
"La rivoluzione dimenticata" Il pensiero scientifico greco e la scienza moderna, Lucio Russo, Giangiacomo Feltrinelli Editore Milano
"The Verona Arena, the largest opera house in the world",Roberto Pasini, Arsenale editrice, San Giovanni Lupatato
"Vitruvius : handboek bouwkunde", vertaald door Ton Peters, Athenaeum- Polak & Van Gennep
CABRI computer programme

By Kato De Malsche,  Jolien De Boodt,  Eveline Joos,  Ophelia Ongena,  and with the help of 
Mr Pasquale Di Nunno, vice-principal  and  Mr Adriano Collodel,  teacher of latin and history of the classics, and 
their students  from  the Liceo classico of "M. Flaminio" (Vittorio Veneto/Italy)

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