Mathematical Excursions to the World's Great Buildings

Author(s): 
Alexander J. Hahn
Publisher: 
Princeton University Press
Year: 
2012
ISBN: 
978-069-1145-20-4
Price (tentative): 
£34.95 (net)
Short description: 

Two intertwined stories are told. On one hand the evolution of mathematics, in particular of geometry and statics, and on the other hand architecture and this involves the form given to the building and ornaments (geometry), and how people have dealt with all the forces, weights, and stresses (statics) that determine the stability of what they constructed. Starting from Stonehenge and the Egyptian pyramids till the Guggenheim museum in Bilbao and the opera in Sydney. The mathematics that are needed are elementary, and the last chapter is added to illustrate the application of calculus in the analysis of structures. Each chapter has a number of problems and discussions, mostly mathematical exercises but also other ones.

URL for publisher, author, or book: 
press.princeton.edu/titles/9693.html
MSC main category: 
00 General
MSC category: 
00A67
Other MSC categories: 
97G40, 97G70, 01A07
Review: 

With the format of this book (8,5 x 9,5 inch) with wide margins often containing images, and with numerous larger illustrations (including inserted color plates) one immediately realizes that this is not just an ordinary book about mathematics or a dull enumeration of data about historical buildings and architects. This may well be considered an art book. Two narratives are marvelously intertwined over a time span from the cave dwellings to the modern architecture of the 20th century. Architecture is of course one of the narratives and the other one is about the evolution of mathematics, both placed in the appropriate social and cultural environment. The two stories are connected in a very natural way. It would be difficult to read only the part about architecture and skipping the mathematics or vice versa. Of course geometry and symmetry play an important role, but also the forces in the structure that need to be diverted to keep everything stably in place is easily explained using modern vector calculus. The mathematics is often the mathematics needed for describing the statics of the structure. It is amazing to realize that many of the historic buildings were constructed before the mathematics was available to do all the appropriate computations and yet survived for many centuries.

The mathematics are however kept very elementary. Some basic calculus is amply sufficient. In fact the last chapter, which is the most advanced on a mathematical level is an introduction to the necessary calculus to compute for example the length of an arc, the center of mass or the moments. On the other hand, also architecture has its own vocabulary that not every reader may be familiar with. The author has solved this by including a glossary of architectural terms. Every chapter is followed by a number of diverse problems and discussions. These can be mathematical (a proof or a computation) but can as well be a question about the forces in an arch by looking at a picture of a cathedral, or open questions such as commenting on a shape appearing in a figure

In the first 6 chapters successive historical periods and some of their characteristic architectural monuments are discussed. Sometimes the mathematics are a pretext to discuss properties of the buildings, sometimes it is the other way around. Never is it a dull enumeration of historical or architectonic facts, and neither is it mathematics, just for the sake of mathematics. This is not an encyclopedia and the author has definitely made a choice in which buildings to include and which not. Most of the obvious ones, i.e., the ones known by the majority of the readers find some place. The choice is however mostly confined to the ancient cultures of the Mediterranean area and later Europe with few exceptions as may be clear from the enumeration below.

To give an idea about the contents, I include a quick summary of the first six chapters (without being exhaustive).

  1. Human awakening (Stonehenge, papyrus, Egypt, theorem of Pythagoras)
  2. Greek geometry and Roman engineering (Parthenon, round arches and forces that keep them in place, Colosseum, Pantheon)
  3. Architecture inspired by faith (Hagia Sophia, Dome of the Rock in Jerusalem, Mosque of Cordoba, Gothic churches, Milan, Venice, Pisa, and the ways of deviating the forces from the dome or the roof to the walls, and of course the symmetry of Islamic ornaments)
  4. Transition in mathematics and architecture (conic sections, coordinate systems and algebraic geometry, the dome of Florence)
  5. The renaissance (music, harmonics, Venice, Brunelleschi and the Basilica di Santa Maria del Fiore in Florence, Da Vinci, Michelangelo and Bernini with St. Peter's in Rome)
  6. A new architecture (domes of St. Paul's in London and the Capitol in Washington, analysis of structures, the Sydney opera house, Guggenheim museum in Bilbao, Sagrada Familia in Barcelona)

It is not only a picture book but also a book that is a pleasure to read from cover to cover and I can imagine that after reading it, after a while one will pick it up again and again to just enjoy the illustrations or reread sections and chapters.

Reviewer: 
A. Bultheel
Affiliation: 
KU Leuven

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