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# Geometric Explorations by Cynthia Phillips Ph.D. and Shana Priwer

In addition to his more practical scientific studies, Leonardo was also interested in more theoretical disciplines such as math, and in particular geometry. Leonardo first became interested in geometry while he was working for Duke Sforza during his Milan period, in the 1480s and 1490s. It seems that Leonardo first became familiar with mathematical constructs such as geometry through his study of architecture and perspective painting. In 1496, the well-known mathematician Luca Pacioli was invited to Sforza's court, nominally to teach mathematics there, and Leonardo may have suggested this invitation.

Leonardo and Pacioli became friends in Milan, apparently spending much time together discussing the overlap between art and mathematics. During his time in Milan, Pacioli was writing a book, later published as the first of a three-volume set in 1509, called Divina Proportione. Leonardo was so interested in this project that he actually drew the figures for this text.

## The Golden Ratio

The book focused on the so-called “Golden Ratio,” which Pacioli called the “Divine Proportion.” It also included a study of polygons, shapes with multiple sides. Leonardo became more and more interested in geometry as he worked on this project, and it appears Leonardo actually ignored his painting because of his obsession with geometry. In more than one instance, Leonardo's work in math and science got in the way of his paying artistic commissions. While most members of society during Leonardo's lifetime could not understand such a choice, we today can of course appreciate Leonardo's achievements in math and science as well as those in the artistic world.

A letter has been found dated 1501 from a member of the court where Leonardo resided to Isabella d'Este, who had commissioned a portrait from Leonardo. The letter states that Leonardo's mathematical experiments were taking up most of his time, and that he could not spare any time to paint. Leonardo's priorities clearly show that he was much more than just an artist.

The golden ratio comes from the mathematical expression a/b=b / (a+b). Its algebraic value is equal to about 1.6180. It is a number that occurs in a variety of places in math and geometry, similar to the number pi (=3.14159).

Euclid studied the golden ratio, which was considered a particularly elegant relationship in terms of both math and art. Little wonder the golden ratio held particular interest for Leonardo, since he immersed himself in the study of both disciplines, as well as his studies in the natural world. As an added bonus, for Leonardo, the golden ratio also turned out to be important to the study of architecture, and the second volume of Pacioli's Divina Proportione actually focused on architectural applications.

Is the golden ratio useful for anything except for math?

The golden ratio actually occurs in nature, as well as in geometry, in places from the layout of seeds or leaves in a plant to the shapes of sea-shells. This fundamental relationship is also found in music and in other places where the natural and mathematical worlds collide.

## Polyhedra

Leonardo's drawings of three-dimensional shapes called polyhedra (one example is a soccer ball) were the highlight of Pacioli's book. Leonardo came up with a new way of drawing these complicated shapes—he showed them with solid edges and hollow faces that let you see right through them to the structure on the other side. For the book, Leonardo drew about sixty pairs of illustrations. Each pair showed a different three-dimensional shape, in both a solid view and a hollow view. Some of the shapes were new—no one had figured out how to draw them before!

The method of drawing shapes was a breakthrough for the day, and it took a visual artist of Leonardo's talents to come up with it. Sometimes art and math are closer together than you think.

Leonardo's obsession with geometry continued even after he finished the illustrations for Pacioli's book. If you look through his notebooks, you'll find sketches of different kinds of geometric shapes in unlikely places, for example, among studies for military fortifications and designs for a fountain.

## Squaring a Circle

In addition to his direct work with Pacioli, Leonardo spent some of his time in Milan conducting his own research into geometry based on that of Euclid and Pacioli. In particular, he was interested in trying to “square a circle,” meaning he wanted to find a way of creating a square with the same area as a particular circle, using only drawing tools such as a ruler and compass.

In another intersection between art and math, Leonardo was particularly interested in drawing rosette patterns, and invented a special compass to help him draw them more accurately. He also came up with a proportional compass that let him scale figures up or down. He was interested in drawing ellipses, and sketched a device called an ellipsograph that would help to create accurate ones. An ellipse is a shape that looks like an oval. It is defined mathematically as the set of points for which the sum of the distances from two points in the middle, called “foci,” is constant.

You could draw an ellipse by placing two pins on a board, attaching the ends of a string to each point, and putting a pencil in the string and drawing a shape, keeping the string taut. The orbits of the planets around the sun are shaped like ellipses, with the sun at one focus.

Leonardo was also interested in knots, and many of his intricate artistic designs are also interesting from the point of view of the modern field of topology. One particular example of this is Leonardo's design for the ceiling of the Sala delle Asse that he created for his patron Sforza's castle. The ceiling includes interwoven trees and branches, and there is one long continuous strand that can be traced over the whole ceiling.

When the French forced Duke Sforza out of power in 1499, Leonardo and Pacioli fled from Milan together, passing through Mantua and Venice before sharing a house in Florence for a while. While there, Pacioli taught geometry at the University of Pisa (located in Florence at that time). Leonardo also remained in Florence on and off until 1506.

Beyond his theoretical work in mathematics, Leonardo was also interested in the mechanical methods of automating mathematical work, and he designed a machine that could have been one of the first calculators. A working replica was built in 1968, but whether or not this replica actually represented Leonardo's intention is another story. The sketches on which the calculating machine replica was based are unclear, and it's possible that the machine was not, in fact, a calculating machine, but a ratio machine instead. In his typical style, though, Leonardo made real contributions tojust about every part of math that he touched. Pretty amazing for someone who's primarily remembered as an artist.

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By Cynthia Phillips Ph.D. and Shana Priwer

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