Leonardo was not a genius in mathematics

Dirk Huylebrouck KULeuven Faculty of Architecture Brussels - Gent, Belgium Leonardo’s mathematics Dirk Huylebrouck

(Pseudo-scientists pretend Leonardo used f.) f=1.618… Amboise Eger

1 Vinci

Leonardo: born in Vinci, Italy, 1452 A more serious link  Died in Amboise, France, in 1519. with Hungary (Nagy D.) Leonardo’s mathematics Dirk Huylebrouck

inventions mathematics

Wikipedia: Leonardo = painter, sculptor, architect, musician, mathematician, engineer, inventor, anatomist, geologist, cartographer, botanist, philosopher, and writer. Leonardo’s mathematics Dirk Huylebrouck Not about errors as a painter Nor about errors as an engineer (He only finished four pain- tings) Anti-torque tail rotor? Hole on top? Cannot go forward

The Last Supper, 1495-1498, 20th century fraud, vanished in 1600 now in museums Leonardo’s mathematics Dirk Huylebrouck Nor about alleged golden Nor about vain attempts on section research circle squaring or cube doubling Leonardo’s mathematics Dirk Huylebrouck Leonardo: became interested in mathematics (in 1497, age 45) after meeting Franciscan friar - mathematician Luca Pacioli (1445–1517). He illustrated Pacioli’s ‘De Divine Proportione’. We will talk about this work in particular… Leonardo’s math

A first error, in geometry Leonardo’s mathematics: a first mistake Find the diffe- rences

Leonardo Rinus Roelofs (The Netherlands) Leonardo’s mathematics: a first mistake

Drawing by Leonardo Drawing by Rinus Roelofs Leonardo’s mathematics: a first mistake Drawing Rinus Roelofs

Rhombic cuboctahedron Adding pyramids transforms it drawn by Leonardo. into the stellar polyhedron. Leonardo’s mathematics: a first mistake Why discovered only now? After over 500 years? Also, Roelofs made the discovery about 10 years ago. Roelofs mentioned it in a talk, for people who politely smiled. Except … (Rinus’ conscience) It is Leonardo! Go for it!

(me)

 (Rinus) meeting 2 complementary minds… Leonardo’s mathematics: some reactions Admirers of Leonardo:

it was not a mistake:

- The error was a mysterious riddle - The error was a mysterious code - Roelofs and Huylebrouck may know everything about mathematics, for sure they don’t know anything about mysticism.

OK, agreed Where is the best- seller and the money? Leonardo’s mathematics: some reactions Others said Leonardo drew a pseudo rhombic cuboctahedron. Today, this solid is well-known, cf. Wikipedia:

Pseudo- Rhombi- rhombi- Cubocta- Cubocta- hedron hedron

Yet, the first mention of the pseudo-rhombicuboctahedron dates from 1905. Not truly an Archimedean solid (less symmetry). It is similar to rhombicuboctahedron and thus sometimes called 14th Archimedean solid... Leonardo’s mathematics: some reactions Roelofs: No pseudo rhombic Leonardo structured his cuboctahedron! drawings well…

Leonardo’s drawings closely follow Luca Paccioli’s book De Leonardo’s mathematics: some reactions The illustrations in De Divina Proportione: (1) polyhedron; (2) hollow version; (3) stellar version; (4) hollow stellar version Leonardo’s mathematics: some reactions The illustrations in De Divina Proportione: (1) polyhedron; (2) hollow version; (3) stellar version; (4) hollow stellar version Leonardo’s mathematics: some reactions The illustrations in De Divina Proportione: (1) polyhedron; (2) hollow version; (3) stellar version; (4) hollow stellar version ??? Leonardo’s mathematics: some reactions

Roelofs: the drawings have labels Leonardo says so!

planus solidus planus vacuus

elevatus solidus elevatus vacuus Leonardo’s mathematics: some reactions

The drawings have labels Leonardo says so!

planus solidus planus vacuus

elevatus solidus ??? elevatus vacuus Leonardo’s mathematics: some reactions Pacioli described in words how the solid had to be formed: CHAPTER LIII. ABOUT THE BODY WITH 26 BASES AND ITS ORIGIN, THE FLAT VERSION, PLANAR OR HOLLOW, AND THE STELLAR VERSION, PLANAR OR HOLLOW. THE ORIGIN OF THIS BODY IS THE HEXAHEDRON CUT EVENLY ON ALL SIDES, AS ITS PHYSICAL FORM CLEARLY SHOWS TO OUR EYE. Wikipedia animation Leonardo’s mathematics: some reactions

Summary: (1) Leonardo intended to draw a stellar version of the rhombic cuboctahedron but drew a stellar version of the pseudo rhombic cuboctahedron; this happens easily, it is a common error. (2) Pacioli and Leonardo wrote what their intention was, in the labels and in the text of the book. Or was it a conspiracy by these two authors? (3) Leonardo and Pacioli enumerated all the polyhedra they knew, so why would they have ‘forgotten’ the pseudo rhombic cuboctahedron?  Good excuses, but … no partial credit! … Leonardo’s mathematics: observations

Did Luca Pacioli and Leonardo have real 3D-models? Portrait of Luca Pacioli (by Jacopo de Barbari?) A mysterious rhombic cuboctahedron in glass: was it real? Leonardo’s math

Errors in analytical mechanics and arithmetic Leonardo’s math: error in an. mechanics? People wrote me they discovered other mathematical errors. Engineers/mathematicians Tibor Tarnai and András Lengyel (Budapest University for Technology and Economics): discovery of an error in analytical mechanics. Leonardo’s math: error in an. mechanics? What kind of error was this?

3 × 3 = 9 9 × 3 = 27 27 × 3 = 80

The ‘simple error’ was Tarnai and Lengyel tried other hard to explanations, on computer, acknowledge but in vain… Leonardo’s math: error in arithmetics? There was a more obvious arithmetic error about 4096 × 2… … on the same page …

Though in the codex Arundel, Leonardo could do the powers of 4 correctly… Leonardo’s math: error in arithmetics? Some calculation errors were noticed before, in birth dates, and simple errors like 25 + 2 + 16 + 6 + 1 = 48 or 6 × 34 = 104. Aaron, Phillips, Larsen: Leonardo:  ‘developmental dyslexia’ Authors = ‘psychohistorians’.

Or was it an error in copying notes? (Leonardo often copied notes…) Leonardo’s math: other geometric errors?

Intarsia by Fra’ Giovanni in Verona (photos Kim Williams). Leonardo’s math: other geometric errors?

Intarsia; below: soccer ball. Paper T. Hartwig, K. Williams (2000), note by Benno Artmann: an error, copied from an error by Leonardo. Leonardo’s math: other geometric errors?

3 representations of the = polyhedron, attributed to Leonardo. Milan version Geneva version Printed version I found the remark by Artmann while cleaning my attic. In the year 2000, when the paper appeared, I did not pay much attention to this remark, because it seemed insignificant… Yet, things changed since then, due to all the criticism Leonardo could not have made errors … Leonardo’s math: other geometric errors?  I inspected other polyhedra drawn by Leonardo. Take, for instance, the rhombic cuboctahedron

Leonardo’s drawing. Drawing by Rinus Roelofs. Leonardo’s math

Quiz Leonardo’s math: other geometric errors?  Can you make some observations your self? A QUIZ!  An easier case to start: please compare, and make an observation that waited 500 years… Leonardo’s math: other geometric errors? QUIZ  Some easier ones. What it wrong here?

Correct  Upside down? Leonardo’s math: other geometric errors? QUIZ  Some easier ones. What it wrong here?

Inside out?

Leonardo does this many times! Leonardo’s math: other geometric errors? QUIZ  What it wrong here?

Correct Wrong Leonardo’s math: other geometric errors? QUIZ  Try these…

Rotated + mirrored Leonardo’s math: other geometric errors? QUIZ  Anything special here…? Leonardo’s math: other geometric errors?  Which edges are missing? Leonardo’s math: other geometric errors?  A more difficult one…

Edges from behind come in front… Leonardo’s math: other geometric errors?  And 2 more quick ones… Leonardo’s math

More than ‘inaccuracies’ Leonardo’s math: more than inaccuracies - Some of the previous observations: with Jos Janssens (The Netherlands), so: not just an individual opinion. - Yet ‘scientific’ referees: not true, ‘we don’t see the error’. - Many journals refuse to publish it … - Comments: It are not errors but ‘inaccuracies’. However… - Leonardo’s math: more than inaccuracies Pacioli writes in his book: Projected onto a plane, this body will always rest on 6 tops of pyramids, one of them being a pentagonal pyramid, the other five triangular. Pacioli gives no proof, but, he says, it can be seen “… through ‘great abstractness’, ‘deep science’, ‘algebra’ and ‘almucabala’ ”. Yet, there can be no proof, because the statement is wrong. Leonardo’s math: more than inaccuracies The middle point comes out of the plane. Suggested by Jos Janssen, confirmed by Rinus Roelofs (both from The Netherlands) Janssen noticed more errors:

Yet, Leonardo specialists say they don’t even see this error… Leonardo’s math

The specialists’ opinion Leonardo’s math: the specialists’ opinion So, what do these Leonardo specialists do? They organize exhibitions, for instance, to illustrate the bridge above… Leonardo’s math: the specialists’ opinion

Leonardo’s model

? ? ? ? ? Museum models: useless pillars; noticed by L. Luyten (Belgium) Leonardo’s math: the specialists’ opinion Amboise, France († Leonardo). The model for the bridge collapses; somebody put a pillar

The model outside has 4 pillars, connected above. Leonardo’s math: the specialists’ opinion

Traveling exhibition: ‘Leonardo the genius’, Australia. My publication was removed from University web site, ‘because they have a lot of money’  to Dean’s office  Question in the Brussels’ City Council about threats… Leonardo’s math: the specialists’ opinion Vinci, Italy (°Leonardo). A museum dedicated to da Vinci; mathematical models in the tower. Leonardo’s math: the specialists’ opinion

Can this form an octahedron?

This example again was a discovery by Rinus Roelofs. However, half of the museum posters: wrong ... Leonardo’s math: the specialists’ opinion

 I proposed the museum director to correct the posters for free (I found a sponsor).  The director accepted! …but… he was very ill and so I had to wait, over a year… Leonardo’s math: the specialists’ opinion The new director accepted too and Rinus brought the new posters. Since Sept 2015: correct posters. Viva l'Italia… Leonardo’s math… I asked and for her opinion, in private…

She smiled at me. Thank you for your attention