An Exchange

Abstract

D irk Huylebrouck’s article on Leonardo’s geometric ‘‘slip-up’’ is interesting (The Mathematical Intelligencer 34, No. 4 (2012), 15–20). In fact the discovery of Leonardo’s ‘‘false’’ polyhedron by Rinus Roelofs is a little sensation. But the conclusion, that this polyhedron is false and that Leonardo made a mistake, is doubtful, and probably false itself. The polyhedron is only false if we, the viewers, insist that thebasepolyhedron is the rhombi-cuboctahedron, that is, one of the 13 Archimedean solids. If the ‘‘pseudorhombi-cuboctahedron’’ (the so-called 14th Archimedean solid) is taken as base, then Leonardo’s polyhedron is correct and it corresponds to his own label. Any undergraduate student with some geometric intuition can deduce this solid from the rhombi-cuboctahedron. So it is very likely that Leonardo (or Pacioli or some other Renaissance artist) knew this ‘‘14th solid,’’ and that Leonardo constructed his (correct) polyhedron on purpose from this base. Why might Leonardo have done this? The answer is quite simple: Riddles and puzzles in paintings were common in the Renaissance and in later times. (Another famous Renaissance example is the enigmatic polyhedron in Albrecht Dürer’s ‘‘Melencolia’’ with the hidden golden section.) Leonardo may have constructed his correct polyhedron to confuse the viewer, and to make her or him think about geometry. He could not know that it would take 500 years until Rinus Roelofs ‘‘rediscovered’’ this polyhedron.