To err is human

By on 05/03/2017
Stellated Miller polyhedron

The Italiana journal XlaTangente starts its collaboration with our site. This journal was first published in 2007 thanks to the efforts of a group of academics and to the contribution of the Research Center “matematita”. XlaTangente was initially thought as the Italian edition of the French magazine “Tangente. L’aventure mathématique“, from which it proposed some translated articles, alongside original content. In the last few years, the paper edition evolved in a website,, whose goal is to continue to offer “chewable mathematics”, thanks to the contributions of young mathematicians, increasingly aware of the need to maintain a dialogue with society.In this website, you will find materials (articles, videos, images, links, ideas …) for the popularization of Mathematics. The website is addressed specifically to students of secondary schools and their teachers. It collects materials published in the journal XlaTangente until 2015 and, since 2016, it proposes new online-exclusive content. The site is operated by matematita, an Interuniversity Research Centre for Communication and Informal Learning of Mathematics, University of Milan, Italy. The first contribution to Mathematics in Europe is “To err is human” written and translated by Daniela Della Volpe, with digital images by Alessandro Cattaneo (first published in XlaTangente no. 26, April 2011). 

Leonardo's illustration of the Vigintisex basivm elevatvs vacvvs appeared in De divina proportione (1497) by Luca Pacioli

Leonardo’s illustration of the Vigintisex basivm elevatvs vacvvs appeared in De divina proportione (1497) by Luca Pacioli

We all know it: “to err is human”. However, if Leonardo Da Vinci is the one who made the error, the news cannot but surprise us, especially if it took more than 500 years to discover it! The error appears in one of the numerous illustrations created for the treatise “De Divina Proportione” (1497) by Luca Pacioli (1445-ca 1517) dedicated to the applications of the golden section. (Some of them are proposed on the website “Images for mathematics” searching the word “Leonardo.”)

The fateful illustration represents an “empty” stellated rhombicuboctahedron that Da Vinci called Vigintisex basivm elevatvs vacvvs and that, with the notation which XlaTangente readers are used to, we will call stellated-(3,4,4,4).

This polyhedron is obtained from a uniform polyhedron. In each vertex there is one equilateral triangle and three squares and to it has been applied a process of stellation, adding, in correspondence of each face, a triangular or a squared based pyramid. In Leonardo’s description, the expression Vigintisex basivm means that the uniform polyhedron has a total of 26 faces, while the adjective elevatvs indicates that it is stellated. Finally, the adjective vacvvs tells us that only the skeleton (that is, edges and vertices) has been represented.

Stellated Miller polyhedron

Stellated Miller polyhedron

Here, you can find our reconstruction of Leonardo’s drawing: the fault would be in the lowest pyramid, the one pointing down. It should be triangular based, while in Leonardo’s drawing it has a square base.

The error was discovered by the Dutch mathematician and sculptor Rinus Roelofs who wondered what may have led Leonardo to make this mistake. It is not clear if the Italian scientist had a real-life model, probably he had at his disposal just the instructions given to him by Pacioli.

The hypothesis that we dare to state here is that Leonardo’s drawing corresponds to a stellated Miller polyhedron. Here we refresh the memory of XlaTangente readers. In Miller polyhedron, exactly as it happens in (3,4,4,4), three squares and one equilateral triangle meet at each vertex, but vertices are not indistinguishable (a characteristic that is proper to uniform polyhedra).



The difference between the uniform (3,4,4,4) (see here on the right) and the Miller polyhedron is quite subtle and the two polyhedra are often confused: a clear distinction between them was formalized only (in the) last century thanks to symmetry groups. (It is in fact due to the English mathematician Jeffrey Charles Percy Miller 1906-1981). Our hypothesis would work if, as it would seem, the pyramid that Rinus Roelofs called “the wrong one” is adjacent to two triangular based pyramids: this would correspond to the Miller polyhedron structure.

What is certain is that it won’t be this “mistake” to belittle the genius of Leonardo Da Vinci.

About Roberto Natalini

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