Mathematics on the "Ritratto di Frà Luca Pacioli"

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In the *Museo e Gallerie di Capodimonte* in *Napoli (Italy)* one can admire this *"Ritratto
di Frà Luca Pacioli"*.

The only thing that is certain is that the central person
is Frà Luca Pacioli, one of the most famous mathematicians
from the Renaissance period. The painter is unknown, although some people are convinced the painter is
*Jacopo de' Barbari*. Other people think that the painting isn't the work of one artist. As to the other person on the
painting, some people suppose it's *Guidobaldo, Duke of Urbino*, other people are convinced it's the famous painter
*Albrecht Dürer*.

A very interesting paper about the historical and mathematical elements was published by Nick Mackinnon: "The portrait of Fra
Luca Pacioli", *The Mathematical Gazette, 77 (1993) pp. 130 - 219.*

The closed book on the table is supposed to be the *"Summa de arithmetica geometria proportioni et proportionalita"*,
written by Pacioli. A regular dodecahedron is placed onto this book.
The open book is book XIII from Euclid's Elements. Pacioli clearly is exposing a theorem to one of his pupils.
Book XIII treats the Platonic solids or regular polyhedra.

The drawing on the slate occurs in proposition XIII.12, although Mackinnon mentions this drawing is also linked to other propositions concerning the regular dodecahedron and icosahedron.

The rhombicuboctahedron on the "Ritratto di Frà Luca Pacioli"

The object suspended in the top left is a

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However, the RCO on the painting has a different orientation. This orientation easily can be derived by viewing the object from the right side. The diagonals of the square in the centre are horizontally en vertically oriented as can be seen from the level of the water. This is a valuable help to determine the point on the upper triangle where the suspension string has been attached. |

We rotate the RCO as seen on the ritratto over -90 degrees around the vertical axis and we suppose the length of the edges of the RCO equals 1.

As to the material construction of a RCO as seen on the Ritratto it's worth to notice that, if the length of the edges of the RCO is 10 cm, then the volume is

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