# Paintings with a Mathematical Theme (Part 1)

My main interest in mathematics is geometry and visual mathematics. I can say that I am pretty good at doing various geometrical constructions, but I am not good at doing drawings or paintings. Nonetheless, I can say that I am a very visual person, and my visual proclivities probably sparked in me an interest for the visual arts, especially classical paintings.

In this article I want to discuss a few paintings that have a mathematical theme. By a mathematical theme, I mean that elements of mathematics are part of the subject matter of the painting. For example, a painting can contain geometrical objects, or it can present the subject of mathematics in an allegorical or symbolical form. I chose to treat this subject because most articles on internet discuss the broader connections between art and mathematics. This topic is narrower and should be interesting to people interested in math or paintings. Nonetheless, a good book that discusses many connections between math and art is ‘The Visual Mind: Art and Mathematics” edited by Michele Emmer.

I don’t pretend to be an expert in art or history of art. In my commentary on each painting, I will try to focus on the mathematical elements of the painting. History, including math history, is one of my favorite subjects. So, in my commentaries I will also try to make various historical connections. Besides making interesting historical connections, I hope that my article will help some of my readers discover interesting paintings.

## Allegory of Geometry by Laurent de La Hyre

“Allegory of Geometry” by Laurent de La Hyre is probably my favorite painting that has a mathematical theme. Laurent de La Hyre (sometimes spelled Hire) was a baroque painter from the 17th century. “Allegory of Geometry” belongs to Laurent’s series of paintings on the liberal arts. The classical liberal arts were divided in trivium (grammar, rhetoric and logic) and quadrivium (arithmetic, geometry, music and astronomy). Laurent made an allegorical painting for each of the 7 liberal arts.

In the center of the painting we see geometry embodied by a female figure that leans on a marble block. In her left hand she holds a compass and a right angle, the instruments of geometers. In her right hand she holds a sheet of paper with geometrical diagrams. The diagrams seem to be from Euclid’s Elements. The diagram on top left seems to be from Book 1 Proposition 47, the proposition that deals with the Pythagorean Theorem. The diagram from the middle top seems to be from Book 2 Proposition 9. On top right we see 3 segments that may be related to the golden ratio, but I am not sure. The diagram on the bottom that has a circle, seems to be from Book 3 Proposition 36.

The painting also connects geometry with other sciences or arts. For example, we see a painting on an easel plus a few other instruments used by painters. Laurent probably implies that painting is an application of geometry. The book “Della Pittura”, written in 1435 by Leon Battista Alberti, was probably the first book that showed the geometrical basis of the art of painting. An important note is that Laurent was a friend of Girard Desargues, the founder of projective geometry. Phillipe de La Hyre, the son of Laurent, was a pupil of Desargues. Phillipe wrote a few books on geometry and other areas of math. These biographical details show that Laurent was deeply connected to the field of geometry.

Near the female figure we see the globe of the Earth with a snake on it. This implies that cartography and geography are also applications of geometry. Actually the etymology of the word “geometry’ is derived from ge=”earth” + metria=”a measuring of”. It is also interesting to note that the book that introduced the concept of 7 liberal arts, “The Marriage of Philology and Mercury” by Martianus Capella, treats geometry as being more equivalent to the science of geography. So, the association between geography and geometry exists since the ancient times. The snake that sits the globe may be connected to the goddess Ceres. Ceres was the goddess of agriculture and agriculture is related to the measure of proper seasons or time. However, the measure of time and seasons is more the domain of astronomy since it involves the movement of planets with respect to the Sun and other stars.

Finally, the painting also implies that architecture is also a practical application of geometry. We can see a block of marble, a sphinx, a wall with an image in relief and in the background, we can see many architectural structures. Under the sphinx we can see a libella, the instrument that looks like the letter “A” and has a suspended plumb bob. The libella was a levelling instrument used by architects and stone masons. The image in relief and the sphinx may refer to the art of sculpting as another application of geometry. An interesting historical note is that the geometers Isidore of Miletus and Anthemius of Tralles created the cathedral Hagia Sophia, despite the fact that neither of them were formally educated as architects. Their achievement was incredible since Hagia Sophia is one of the greatest and most beautiful buildings in the history of humanity.

## Allegory of Arithmetic by Laurent de La Hyre

“Allegory of Arithmetic” is another painting made by Laurent de La Hyre. Arithmetic, like geometry, also belongs to the quadrivium and the 7 liberal arts. This painting has very few details when compared to the “Allegory of Geometry” painting. In this painting you don’t see connections with other arts or sciences. The focus is entirely on the female figure, the book that she holds in her hand and the worksheet that is held together with the book.

The most important details are the writings we can see on the book and on the worksheet. On the book we can see the name Pythagoras. On the top of the worksheet we can see the words “Par” and “Impar”. “Par” is the Latin word for even numbers. “Impar” is the Latin word for the odd numbers. Even and odd are one of the most important and basic categories for describing integers. Below the words, we can see the numbers from 1 to 12. Maybe there is a symbolic reason why Laurent chose to list the numbers up to 12, since 12 is important in astronomy and when dealing with time units (12 zodiac signs, 12 months, 2 x 12 hours in a day). Below the numbers we can see the arithmetical operations of addition, subtraction, and multiplication.

It is interesting to see the name “Pythagoras” on the book. Most people associate Pythagoras with the Pythagorean Theorem. Thus, Pythagoras is usually associated with geometry, not arithmetic. However, it is known that mathematicians that belonged to the Pythagorean or the neo-Pythagorean school of thought were interested in the properties of numbers. In ancient times, arithmetic was defined as the study of number, geometry was defined as the study of number in space, music was defined as the study of number in time and astronomy was defined as the study of number in time and space. In the book “Introduction to Arithmetic” by the neo-Pythagorean Nicomachus of Geresa, Nicomachus considers arithmetic the mother of the other 3 quadrivium sciences. Arithmetic deals with numbers and numerical relations fundamental to the other mathematical sciences. Geometry, music and astronomy cannot exist without the concept of number, which is the main topic of arithmetic. On the other hand, arithmetic can exist without the other 3 sciences. The Pythagoreans and neo-Pythagoreans even believed that numbers have mystical properties. The book “Theology of Arithmetic” by Iamblichus uses the writings of Nichomachus and other neo-Pythagoreans and it deals with the mystical properties of the numbers from 1 to 10 (and beyond).