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.
structure are called reactions. The bottom row of blocks of a pyramid puts the underlying ground under pressure. Any shifting in the structure that results is called settlement. Uneven settlement can lead to critical dislocations and failure of the structure. This did not occur for the pyramids of Figure 1.6 because they were built on a foundation of natural limestone. Such rock can support loads of 100 tons per square foot. However, other Egyptian pyramids settled unevenly during and after
built in the sixteenth century. It is in French high Gothic style. The cathedral is famous for its stained glass windows, many in beautiful blue hues dating from the thirteenth century. Plate 14 depicts the rose window of Figure 3.24 viewed from the interior. In the same way that colored tiles with intricate designs are central to the artistry of Islam and delicate golden icons and mosaics are the hallmark of Byzantine art, stained glass windows in splendid hues exemplify the art of the Gothic
Pisa was planned in 1063 and largely completed by 1118. Laid out in the basic basilica plan, it is one of the best examples of Romanesque architecture in Italy. Byzantine influences are evident in its interior. Architecture Inspired by Faith 85 Figure 3.35. The facade of the Palace of the Doges, Venice. Photo by Benjamin Sattin A Byzantine mosaic depicting Christ the Savior is in a dominant position behind the main altar. It is similar to that of Plate 6 but of lesser artistic value. In
they were not incorporated into their number system. Neither their numerical notation nor their arithmetic extended to them. So at the end of the day, the number system of the Greeks was not large enough to capture their geometry. What was needed was a number system with which all lengths could be recorded. Only such a system could establish a link between geometry on the one hand and arithmetic and algebra on the other. Such a system needed to come with a notation for numbers so devised that
configurations of heavy stone slabs. From 600 B.C. until about A.D. 200, Greek geniuses, working in Greece and its colonies along the coast of the Mediterranean, laid the foundations of mathematics and science. Many of their answers “all matter is made up of the four basic constituents earth, air, water and fire” were wrong or incomplete, but the important fact is that when they asked “are there basic elements that combine to make up all matter?” they posed the right questions. Mathematics was