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Renaissance Geometry
At the end of the Middle Ages, artists in Italy were rejecting the flat, formulaic figures and spaces that characterize the International Gothic style and were striving for greater realism.1 In particular, the Proto Renaissance artists Giotto and Duccio were painting figures with apparent mass in more spatially convincing settings. A century or so later, twenty-five years into the Quattrocento,2 the discovery of linear perspective would allow a new generation of Italian artists to take the portrayal of space to the next level. Linear perspective introduced a new geometry – a new type of armature upon which two-dimensional works, as well as relief sculptures, could be designed to seem accurately three-dimensional. The epicenter of this revolution in art was Florence.
At the same time in Italy, the popularization of ancient Greek and Roman books was changing the way artists thought about space, proportion, and harmony.3 Euclid’s Elements, on plane and solid geometry, and his Optics, on the geometry of vision, laid the foundation for linear perspective.4 By the middle of the fifteenth century, a mastery of Euclid was considered an essential part of an artist’s training.5 Plato’s Timaeus was another popularized classical book that influenced Italian Renaissance artists. In it, Socrates, the philosopher and scientist Timaeus, and others have a conversation that explains Pythagoras’ theory of the universe – the geometry that created it and the proportional and harmonic relationships among the sun, moon, and planets (“the music of the spheres”).6
For artists as well as architects, the most influential “lost” book was De architectura (published as Ten Books on Architecture), by Vitruvius. In it, the Roman architect and engineer writes, “Proportion is that agreeable harmony between the several parts of a building and the result of a just and regular agreement of them with each other; the height to the width, this to the length, and each of these to the whole.”7 While Vitruvius did not provide specific instructions for designing and constructing proportionally harmonious buildings, there is archeological evidence for the existence of at least one such system.8 Quattrocento artists were especially, even obsessively, interested in Vitruvius' comparison of proportional harmony in architecture to the proportions of a “finely-shaped human figure”9 – his part-to-whole, limb-to-length ratios inspired numerous drawings of “Vitruvian man”, including Leonardo’s.
". . . I contend that the European Renaissance pictorial system, based on the principles of Euclidian solid geometry, attempted at least in the beginning to do no more than replicate what the human eye perceives according to the tenets of Euclidian geometry, which medieval Europeans understood as synonymous with the vision of God. Pictures rendered in perspective permitted human beings to see the world just as God conceived of it at Creation" Edgerton, Giotto’s Geometry, p. 5

1 Christoph J. Scriba and Peter Schreiber, 5000 Years of Geometry, (Basel, Birkhäuser, 2015), 284.
2 “Quattrocento” refers to the art and culture of Italy in the fifteenth century. It is the period of the early Renaissance.
3 Ancient texts that were considered valuable in the Middle Ages survived because they were copied again and again. J. V. Field, The Invention of Infinity: Mathematics and Art in the Renaissance, (Oxford, Oxford University Press, 1999), 178. Some texts, like Euclid's Optics, were popular throughout the Middle Ages. Ibid., 7. Euclid's Elements was first printed in 1482 and immediately became a bestseller. Scriba and Schreiber, 5000 Years of Geometry, 258.
4 Samuel Y. Edgerton, Jr., The Heritage of Giotto's Geometry: Art and Science on the Eve of the Scientific Revolution, (Ithaca, Cornell University Press, 1993), 5. According to Edgerton, Ptolemy understood geometric perspective but only applied it to maps and stage designs.
5 Ibid., 20.
6 Paul Calter, Squaring the Circle: Geometry in Art and Architecture, (Emeryville, CA, Key College, 2008), 21. Also see Rudolf Wittkower, Architectural Principles in the Age of Humanism, (New York, Norton, 1971), 27.
7 Quoted in Roger Burrows, 3D Thinking in Design and Architecture from Antiquity to the Future, (New York, Thames & Hudson, 2018), 207.
8 In the 1980s, a geometric construction called the “sacred cut” was found in the layout of an apartment building in Ostia, a Roman harbor city. The site dates to about 128 AD. Like the quadrature of the Middle Ages, the “sacred cut” produces squares of decreasing or increasing size that are proportionally related. The geometry is shown in Constructions. Quadrature is a topic in Medieval Geometry.
9 About the planning of temples, Vitruvius wrote, “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 figure.” Quoted in Lionel March, Architectonics of Humanism: Essays on Number in Architecture, (Chichester, UK, Academy Editions, 1998) 103. Vitruvius then supplies the ratios of various body parts to one another and to the body as a whole.

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Renaissance Geometry
Renaissance Geometry
Introduction