A Collaboration between an Artist and a Mathematician during the Renaissance.
Leonardo da Vinci: a polymath and a Renaissance Man
Leonardo da Vinci (1452-1519) was a polymath and a true Renaissance man. While he is best known as an artist, his work as a scientist and an inventor sets him apart. He serves as a role model applying the scientific method to every aspect of life, including art and music. His keen eye and quick mind led him to make important scientific discoveries, yet he never published his ideas. He was one of the greatest painters of the Italian Renaissance, yet he left only a handful of completed paintings.
Vinci was a great lover of geometry and devoted much time to it starting in his early forties. He was extremely interested in studying polyhedra (or polyhedrons). Polyhedra are solids in three dimensions with flat polygonal faces, straight edges, and sharp corners or vertices. At the time quite a significant number of people were studying polyhedra, especially the regular ones – sometimes called Platonic Solids (tetrahedron, cube, octahedron, dodecahedron, and tetrahedron). His most outstanding polyhedral accomplishment is considered to be the illustrations for Luca Pacioli‘s 1509 book The Divine Proportion.
Luca Pacioli, the brilliant mathematician. Two great minds meet.
Pacioli taught Leonardo Euclidian geometry and Leonardo impressed the mathematician with his amazing ability to depict intricate geometric shapes as works of art. Pacioli’s text made the golden ratio central to the understanding of painting, sculpture, architecture, music, poetry and philosophy and was illustrated by models drawn by Leonardo, making Divina Proportione, or The Divine Proportion, the only book illustrated by the artist in his lifetime.
Ludovico Sforza, the Duke of Milan between 1494 and 1499, known for inviting the greatest minds at the time – painters, architects and thinkers – to his court in Europe, had invited Pacioli to teach mathematics at his court. It was here that the two great minds of the Renaissance, da Vinci and Pacioli, met. Both became close friends and enriched one another with their individual skills. Pacioli taught Leonardo Euclidian geometry and Leonardo impressed the mathematician with his amazing ability to depict intricate geometric shapes as works of art.
Pacioli spent much of his time in 1497 composing the text of Divina Proportione or The Divine Proportion, which is an in depth study of the golden ratio, or ‘divine proportion’ [a : b = b : (a + b)], in geometry and art and which also detailed Euclid’s regular and semi-regular Platonic solids and concludes with a treatise on perspective and architecture. Pacioli’s text made the golden ratio central to the understanding of painting, sculpture, architecture, music, poetry and philosophy and was illustrated by models drawn by Leonardo, making Divina Proportione the only book illustrated by the artist in his lifetime. Two copies of this work survive, now kept at the Biblioteca Ambrosiana and the University Library of Geneva.
Below is one of the illustrations from that book. The term Ycocedron Abscisus in the title plaque means truncated icosahedron, and the term Vacuus refers to the fact that the faces are hollow. The drawings are beautifully hand-coloured in the Ambrosiana manuscript, reprinted by Officina Bodoni, 1956.
These are the first illustrations of polyhedra ever in the form of “solid edges.” This is a brilliant new form of geometric illustration, one worthy of Leonardo’s genius for insightful graphic display of information.
These are the first illustrations of polyhedra ever in the form of “solid edges.” The solidity of the edges lets one easily see which edges belong to the front and which to the back, unlike simple line drawings where the front and back surfaces may be visually confused. Yet the hollow faces allow one to see through to the structure of the rear surface. This is a brilliant new form of geometric illustration, one worthy of Leonardo’s genius for insightful graphic display of information. However, it is not clear whether Leonardo invented this new form or whether he was simply drawing from “life” a series of wooden models with solid edges that Luca Pacioli designed. There are roughly sixty similar illustrations in the book, mostly in pairs contrasting models with solid faces and models with this solid edge technique, such as these two versions of the dodecahedron seen below.
These printed images of polyhedral models had a significant influence on other scholars and artists. Geometric models of this sort are valuable for an understanding of structure in three-dimensional space. In the Renaissance, they served as mathematical models for students studying Euclid’s Elements—the essential mathematics text for every scholar, including architects.
The widely-reproduced portrait of Luca Pacioli by Jacopo de Barbari (seen below), now in the Capodimonte Museum in Naples, shows him lecturing from Euclid with a dodecahedron on the desk nearby. This gives a good indication of the probable size of the original polyhedron models.
Pacioli is seen standing behind a table and wearing the habit of a member of the Franciscan order. He draws a construction on a board, the edge of which bears the name Euclides. His left hand rests upon a page of an open book. This book may be his Summa de Arithmetica, Geometria, Proportioni et Proportionalità or a copy of Euclid. Upon the table rest the instruments of a mathematician: a sponge, a protractor, a pen, a case, a piece of chalk, and compasses. In the right corner of the table there is a dodecahedron resting upon a book bearing Pacioli’s initials. An rhombicuboctahedron (a convex solid consisting of 18 squares and 8 triangles) suspends at the left of the painting. The identity of the young man at the right is uncertain, but one commentator recognizes the “eternal student” instructed by Pacioli. Some authors have also mentioned the possibility that the student is Dürer.
It seems that da Vinci was more interested in descriptive features of objects (shape, size, perspective) rather than theoretical foundations. Therefore, he illustrated the book, creating around 60 plates for the book.
Applications and importance of Polyhedral Geometry in the 20th century and forward
In the twentieth century, polyhedral geometry has been found to be the basis for a wide range of designs, such as Buckminster Fuller’s geodesic domes, space structures, deployable buildings, and many other types of “nonstandard architecture”. Today, progressive undergraduate architecture programs may include design labs that involve geometric model building, to develop students’ understanding of space. However, the wider culture is not very familiar with polyhedral models, because solid geometry is no longer featured in many high school curricula, and outside of school there are few opportunities to encounter polyhedra.
To increase the general awareness about geometry, it is valuable to build and display such models. Printed figures and computer animations can be widely reproduced, but three-dimensional models, when available, have much more impact because they are real and tangible. Pacioli records that he carried wooden models with him to use as illustrations when he lectured. We know that the value of such models was officially recognized in the Renaissance because there is an entry in the accounts for the building of the Council Hall in Florence indicating that a set of Pacioli’s models was purchased by the City of Florence for public display.
Two traditional model construction materials for polyhedra are paper and wood. Many models required several days to cut, bevel, and assemble the cherry pieces, then sand and finish it, and create the plaque and connecting brass wire.
Currently, 3D technology enables us to digitally produce 3D models of polyhedra. But if you want to understand 3D structures by making them physically with your own hands, watch out for the online and offline workshops on making Platonic Solids that we offer. Find a poster of the same below. You can also try to make them yourself by following these easy steps we shared earlier, here.
Researched by Srilagna Majumdar