ROAMING IN ROME:
   THE PANTHEON

Rome

Rome, the ‘Eternal City’, is the capital of the Republic of Italy. It is situated on the banks of the Tiber River about 27 km from its mouth in the Tyrrhenian Sea, and is built on seven hills of the region known as the Roman Campagna. It contains within it The Vatican City State, the centre of the Roman Catholic Church. With a population of nearly 3 000 000, Rome is a busy, even chaotic, place to visit, but if you are fit, it is easy enough to explore on foot now that some traffic is barred from the central area during the day. What captivates tourists is its 2 500 year heritage which is in evidence all over the city, although buildings of the Renaissance period seem to dominate the huge range of architectural styles. The remnants of Ancient Rome, including the Forum, the Colosseum (pictured), the Appian Way and the Catacombs, compete for the visitor’s attention with the baroque splendour of the many churches (including St Peter’s Basilica), piazzas, gardens, museums and fountains.

      
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The Pantheon

The best preserved of the ancient monuments of Rome is the Pantheon, a temple dedicated to all the gods. Situated in the Piazza della Rotunda in the centre of Rome, it was built during the reign of the Emperor Hadrian (117–138 AD). It was built as a radical reconstruction of a temple which had been built in 27 BC on the same site by Augustus Caesar’s General Agrippa but later destroyed by fire. The inscription on the front of the porch reads: M[arcus] Agrippa L[uci] f[ilius] Co[n]s[ul] tertium fecit. Constructed of concrete and pumice, the dome of the Pantheon was the largest circular structure ever built until modern times, eclipsing even that of the dome of St Peter’s Basilica. The dome stands on a base which is 6.15 metres thick, and as the dome rises, the concrete envelope diminishes its thickness until it is only 1.50 metres thick at the opening. As well, the materials were carefully graded so that heavier materials were used at the base of the dome, lighter materials near the top. The building is truly one of the most remarkable buildings in the history of architecture. It contains the tombs of the painter Raphael and other notable people.































The proportions of the Pantheon  

The main body of the building essentially consists of a hemispherical dome of diameter 43.2 metres on top of a cylinder of the same diameter and height 21.6 metres. Thus a sphere determined by the dome would fit exactly inside the cylindrical walls, and rest on the floor.

 Investigate If a cylinder of radius r and height l has volume V = r2l and a sphere of radius r has volume V'= 4/3.r3, find the interior volume of the Pantheon.

Natural light enters the Pantheon only through a circular hole of diameter 8.5 metres in the top of the dome. To find the surface area of the portion of the dome remaining after the hole was cut, we first find the surface area of the spherical zone bounded by the circular hole, i.e., the area of the removed piece of the dome.

 Investigate It is known that the surface area of a spherical zone is A = 2rh, where r is the sphere radius and h the altitude of the zone. Use Pythagoras’ theorem to prove that the area of a spherical zone is equal to that of a circle whose radius R is the chord of the arc which generates the zone (see the figure).

 Investigate You should have shown above that A = R2. Now find the surface area of the hemispherical dome remaining after the hole is cut in it.    

























The sphere and the cylinder

The Roman orator Cicero claimed that he had sighted in Sicily the tomb of Archimedes on which was engraved a sphere inscribed in a cylinder. This was said to be a tribute to Archimedes’ favourite book On the sphere and cylinder (perhaps inspiring the design of the Pantheon).

 Investigate   Using the standard formulae, verify that
(a) the volume of a sphere is 2/3 the volume of the circumscribed cylinder.
(b) the surface area of a sphere is 2/3 the total surface area of the circumscribed cylinder.
(c) the surface area of a sphere is equal to the curved surface area of the circumscribed cylinder.

A puzzling calculation

Some mathematical puzzles are intriguing because of the apparent lack of information provided. Consider the following.

 Investigate Suppose that a cylindrical hole of length 4 cm is cut through the centre of a sphere (see figure). What is the volume of the remaining ring?
(Yes, there is sufficient information!)


























A curious curve

Let a sphere be intersected by a cylinder, where the radius of the cylinder is half the radius of the sphere, and the side of the cylinder (the generatrix) passes through the centre of the sphere. Then the line of their intersection forms a curve known as the Viviani curve, named after the Italian mathematician Vincenzo Viviani (1622–1703).

If the radius of the sphere is unity ( = 1), then its equation is x2 + y2 + z2 = 1, and the equation of the curved surface of the cylinder is (x – 1/2)2 + y2 = 1/4.

Let P be a point on the surface of a unit sphere centred at the origin O. If r denotes the radius of the circle of latitude through the point P, the coordinates of P are
x = r cos = cos cos, y = r sin = cos sin, z = sin , where is the latitude and the longitude of the point. (See diagram.)

 Investigate  Substitute these values for x and y into the equation for the cylinder given above to find the coordinates of the points on the Viviani curve. Show that = ±, and deduce that the points on the Viviani curve are given by

                          x = 1/2(1 + cos 2),  y = ± 1/2 sin 2,  z = sin .   

Further reading ...