Mathematical Excursions to the World's Great Buildings
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From the pyramids and the Parthenon to the Sydney Opera House and the Bilbao Guggenheim, this book takes readers on an eye-opening tour of the mathematics behind some of the world's most spectacular buildings. Beautifully illustrated, the book explores the milestones in elementary mathematics that enliven the understanding of these buildings and combines this with an in-depth look at their aesthetics, history, and structure. Whether using trigonometry and vectors to explain why Gothic arches are structurally superior to Roman arches, or showing how simple ruler and compass constructions can produce sophisticated architectural details, Alexander Hahn describes the points at which elementary mathematics and architecture intersect.
Beginning in prehistoric times, Hahn proceeds to guide readers through the Greek, Roman, Islamic, Romanesque, Gothic, Renaissance, and modern styles. He explores the unique features of the Pantheon, the Hagia Sophia, the Great Mosque of Cordoba, the Duomo in Florence, Palladio's villas, and Saint Peter's Basilica, as well as the U.S. Capitol Building. Hahn celebrates the forms and structures of architecture made possible by mathematical achievements from Greek geometry, the Hindu-Arabic number system, two- and three-dimensional coordinate geometry, and calculus. Along the way, Hahn introduces groundbreaking architects, including Brunelleschi, Alberti, da Vinci, Bramante, Michelangelo, della Porta, Wren, Gaudí, Saarinen, Utzon, and Gehry.
Rich in detail, this book takes readers on an expedition around the globe, providing a deeper understanding of the mathematical forces at play in the world's most elegant buildings.
is the vector Q in Figure 2.22b. The resultant of any number of vectors can also be obtained by placing them end to tip in any order. The perimeter of the diagram of Figure 2.22b shows how the resultant Q of F 1, F 2, P 1, and P 2 is obtained in this way. The important fact is that the magnitude of the resultant is always numerically equal to the length of the vector that represents it. This means that the representation of forces by vectors is much more than a convenient way to think about
rectangular spaces that touched against neighboring ones to form the town’s walled perimeter. Interspersed between the houses were shrines that contained decorative images of animals and statuettes of deities. The settlements that began to develop along the world’s great rivers at around 5000 B.C. profited from the arteries of communication and commerce that connected them. They became economically thriving, literate, urban communities. Those in Mesopotamia (in today’s Iraq) and those on the
feet, respectively. The difference of 2.5 feet is the thickness of the shell. The average weight per cubic foot of the brick and mortar of the shell is about 110 pounds. Chapter 7, the section “Volumes of Spherical Domes,” applies basic calculus to derive the estimate of 27,600 cubic feet for the volume of the shell of the dome above the circular gallery of windows. This implies that the weight of that part of the shell is approximately 27,600 ft 3 # 110 lb/ft 3 . 3,000,000 pounds. Averaging this
2 + y 2 = r 2 in the xy plane. We turn next to the formula for the distance between two points in space. Let the points P 1 = (x 1, y1, z1) and P 2 = (x 2, y 2, z 2) be given. The points (x 1, y1) and (x 2, y 2) in the xy-plane are obtained by pushing P 1 and P 2 into the xy-plane in the direction of the z-axis. Let Q be the point Q = (x 2, y 2, z1). Refer to Figure 4.25. Notice that the distance between P 1 and Q is the same as the distance between (x 1, y1) and (x 2, y 2). By the distance
(h, k, l) and radius r. This is the standard equation of the sphere with center (h, k, l) and radius r. Let’s return to Chapter 3 and the dome of the Hagia Sophia. Refer to Figure 3.3 and set up an xyz-coordinate system in a such way that O = (0, 0, 0) is the common center of the spheres that determine the inner and outer surfaces of the shell and the z-axis is the central vertical axis of the dome. Recall that the radius of the inner surface of the dome is 50 feet, and consider the sphere x