Lost in Triangulation: Leonardo da Vinci’s Mathematical Slip-Up

A geometric drawing by da Vinci contains an error, as revealed by Dutch mathematician and sculptor RinusRoelofs

By Dirk Huylebrouck  | March 29, 2011 | 29

Image: Courtesy of Eos

Artist, inventor and philosopher Leonardo da Vinci (1452–1519) was without a doubt a genius. Yet, there is some criticism. In his book 1434: The Year a Magnificent Chinese Fleet Sailed to Italy and Ignited the Renaissance (William Morrow, 2008) British author and retired submarine commander Gavin Menzies claims that da Vinci swiped most of his ideas from the Chinese. Menzies’s theory was poorly received by the world of science. Besides, isn’t da Vinci’s brilliance beyond question? Definitely, but the Dutch mathematician and artist Rinus Roelofs did find an error in one of the Renaissance man‘s drawings (at right).

A clue can be found in a portrait (below) of Luca Pacioli, a mathematician who, like his contemporary da Vinci, worked at the court of the Duke of Milan. The polyhedron in the left upper corner, hanging from a string, is called a rhombicuboctahedron: a polyhedron with an equilateral triangle that is always surrounded by squares. Leonardo illustrated it separately in Pacioli’s book.

In this portrait of Luca Pacioli, from 1495, you can see a rhombicuboctahedron (half filled with <a href='/topic.cfm?id=water' >water</a>)

Error in pyramid

The rhombicuboctahedron can also be found in the star-shaped figures (pictured below), at least if we put a pointed protrusion on each side surface—that is, a pyramid with a triangular or quadrangular base.

A rhombicuboctahedron within a star-shaped figure

In this picture it clearly shows that the red pyramid has a triangular base. It goes without saying: a triangular pyramid is always surrounded by six quadrangular pyramids. But in da Vinci’s drawing (again reproduced, directly below) this isn’t the case: The pyramid at the bottom of his rendering has four upright ribs, although it should have three. And the pyramids adjacent to it, at the bottom, along with the two above them that point directly left and right (at nine and three o’clock), are also doubtful: They appear to be triangular whereas they should in fact be quadrangular. The apexes of two triangular pyramids at the bottom left and right (eight and four o’clock) seem to be missing as well. The latter omission can be explained as a matter of interpretation from the observer’s point of view. But the drawing of the pyramid pointing down is clearly wrong.

Can you see the differences between Leonardo da Vinci's drawing and Rinus Roelofs's correct drawing?

Model mystery
When you look at a rhombicuboctahedron and put a pyramid on each side surface, it seems obvious (a corrected version of da Vinci’s drawing is below). Well, this is how it always goes in mathematics when the right insight occurs. But da Vinci was the first to draw the rhombicuboctahedron for print. So it’s very unlikely that anyone else had already drawn pyramids on it to portray the star-shaped bodies in print.

Can you see the differences between Leonardo da Vinci's drawing and Rinus Roelofs's correct drawing?

Da Vinci only had Pacioli’s instructions (whether or not with a model) and he had to gain insight in hundreds of illustrations, imagine them, draw them, and prepare them for print. The artist probably did not have any real models. Maybe he made simpler examples by using twigs and little wooden sticks. But big, solid models of polyhedrons, made by da Vinci, have never been found. It isn’t even certain whether he had a model for the rhombicuboctahedron: the figure on Pacioli’s portrait seems to be made out of glass and therefore had to be very heavy.  Moreover, it was half filled with water. How could such a heavy model hang from a small string? And how could the glass side surfaces be attached to each other in a waterproof way, in a time when superglues didn’t exist yet?