LLEONARDO AND THE PSEUDO-RCO
J ¨ORG M. WILLS
Abstract.
There are two hypotheses on Leonardo’s polyhedron basedon the Pseudo-RCO and drawn for Luca Pacioli’s book: Leonardo madean error, or: Leonardo draw it with intention, as it is. We give argu-ments, which support the Intention-hypothesis.
In 2011 the Dutch artist and mathematician Rinus Roelofs discovered an“error” in one of the famous polyhedra paintings for Luca Pacioli’s book“De Divina proportione” [2]. Roelofs’ discovery appeared in the ScientificAmerican [1] and various other scientific journals. The polyhedron is rathercomplicate (see the figure; the best figures are in Carlo H.Sequin’s contri-bution [3]).
Leonardo’s polyhedron based on the Pseudo-RCO
The “error” is not easy to find, which explains its discovery after morethan 500 years. But the crucial point is, that the polyhedron is cor-rectly drawn: It does not contain any false line or vertex. It is only anunexpected polyhedron, based on the Pseudo-RCO rather than the RCO(rhombi-cuboctahedron), one of the 13 Archimedean solids. Leonardo’s jobwas to draw the 5 Platonic solids and 13 Archimedean solids for Pacioli’sbook and for each one four variations: The basic solid, the correspondingstar polyhedron (pyramids on the faces) and for both the edge-skeleton,where the edges are replaced by thin struts. In each case the basic solid a r X i v : . [ m a t h . HO ] J u l J.M. WILLS is the simplest and the star polyhedron with struts the most complicatedpolyhedron. For shortness we call it the final polyhedron. If one looks atthe complicated final polyhedron (see the figure), one might get the im-pression, that Leonardo lost the track. But: The construction of the finalpolyhedron had to start with the basic polyhedron, and then it was builtup step by step. During this process the basis could not change (like thefoundation of a house). So Leonardo made his decision (or error) at thebeginning with the basic polyhedron. It is rather difficult to mix up theRCO and the Pseudo RCO at this step: If he started with a model (of18 squares and 8 regular triangles) or if he started with a drawing (of theedge-skeleton of a cube and successive edge-cuts and vertex-cuts). In bothcases it is nearly impossible to ignore the symmetric RCO and to choosethe less symmetric Pseudo-RCO. This supports the “Intention-Hypothesis”,or as Carlo H.Sequin formulated [3]: “Leonardo knew, what he was doing”.Finally one may ask for Leonardo’s motivation: Why did he draw the un-expected polyhedron? Here we leave science and come to psychology andspeculation: Perhaps he wanted to irritate the viewer. Or he wanted tobreak the routine, because he had to draw roughly 60 polyhedra, perhapstoo much routine for a genius. We do not know the motivation.
References [1] Dirk Huylebrouck,
Een fout van Leonardo da Vinci , EOS-magazine, 2011; shortenedEnglish translation in:
Lost in Triangulation: Leonardo da Vincis Mathematical Slip-Up , Scientific American, 2011, .[2] Luca Pacioli,
De Divina Proportione , Milano, 1497.[3] Carlo H. Squin, ”Lost in Triangulation”?? No, – it is not a mistake, – it isa Pseudo Rhombicuboctahedron! , 2011, . University of Siegen, Fachbereich Mathematik, Emmy-Noether-Campus,Walter-Flex-Str. 3, 57068 Siegen
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