The Sovietesque Disappearance of Pacioli’s Rhombicuboctahedron

JF Ptak Science Books LLC  Post 23

What?

I’m lodging a complaint, 16 years after the fact (though new to me), about the Italian commemorative stamp honoring the great Fra Luca Pacioli—a great mathematician, inventor of modern bookkeeping, and friend and dispersal-agent of/for Leonardo da Vinci—it also seems to be the first representation of a living mathematician.  The stamp is based upon and freely rearranges (to the point of obsidian abstraction) elements of the famous painting by (perhaps) Jacopo Barbari (and painted in 1495), which now hangs in the Museo di Capodimonte (Naples). 

The painting has been reproduced in mathematics books and such many times over—with good reason, as it is a gorgeous work and is filled with lots of mathematical bits and pieces. 

The stamp, however, is a curious and annoying work if you are at all familiar with the original.  The most glaring but not most important change is the Soviet-like loss of the figure to the right of Pacioli—although not positively identified, some believe it is Guidobaldo, and others believe it to be the very pretty Albrecht Durer.  He’s gone, replaced by some sort of out-of-century bookcase.  I'm not saying that whoever this happened to be needed Stalinist disappearing  for political reasons, but it upsets the entire composition of the painting, and leads to other more foul errors. 

The closed red book on the table is moved and an hour glass is placed on top of it, taking up the slate that was there in its place in the original, which is now hanging on the wall.  It is highly unlikely that Pacioli would provide a demonstration in this matter. 

Also the mathematical notes here are slightly different from the (correct) references to Euclid's Elements Book XIII, proposition 12 (If an equilateral triangle is inscribed in a circle, then the square on the side of the triangle is triple the square on the radius of the circle.)

An armillary sphere makes a curious appearance, and a few of the mathematical tools on the table top make disappearances.  The sphere may be the artist's attempt to make up for the loss of other symbols, as the sphere represents the Platonic music of the spheres, successive levels of forms being built one on the other and so representing the universe.  (Plato's Republic, where, in the Myth of Er he wrote,". . . Upon each of its circles stood a siren who was carried round with its movements, uttering the concords of a single scale." [Republic p.  354])

The important item that is kept on the stamp from the original (though moved) is the polyhedra form, the dodechahedron, one of the great Platonic solids, and a figure composed of 12 pentagons.  It is a highly symbolic link between ancient thought and the rise of the new mathematics and even newer art. 

The most important missing item, though, is the glass object hanging at the left of the painting—it is the rhombicuboctahedron.  This object is one of the thirteen semi-regular (or Archimedean) solids, and has 26 faces made of 18 squares and 8 equilateral triangles, with 24 vertices and 48 edges.  It is constructed of glass and hangs by a string, and is half-filled with water.  It is spectacularly shown in the painting by Barbari—it is simply a glorious achievement of perspective and light and shadow.

And, as it turns out, reflection.   In a superb 90-page article by Nick Mackinnon: ("The portrait of Fra Luca Pacioli", The Mathematical Gazette, 77 (1993) pp. 130 – 219).  Mackinnon combs through the painting with microscopic precision, and in this case reveals that there is an exterior reflection (of s house) on one of the bottom panels of the figure. (Good work!  He even goes on to try and figure out what it is we are looking at in the reflection.) 
It is annoying for this figure (or a shadow or hint of this figure) not to be on the stamp because of its beauty and symbolism.  It is none other than Pacioli’s friend, Leonardo, who illustrates his book Divina proportione—a masterpiece of thinking and illustrating, introducing the concept of the divine proportion (phi) and its importance in design and in nature in general. Perhaps of greatest significance though are the illustrations of some of the figures in the book, which I believe are the first time that people are able to see geometrical drawings in perspective, which is a very big deal. 

Unfortunately this link is pretty much missing without the Rhombicuboctahedron—you don’t have to know how to say it to appreciate it.  (Or as Bernard Malamud writes in the Natural, when discussing a spaghetti dinner “I don’t know how to spell it, but it sure eats good”.)

All I'm saying is that the designer for the stamp shouldn't been consulted, and the Italian government should just have used the original as the basis for the stamp.  After all, they didn't remove the backgrounds of the Mona Lisa or the Last Super because they were too-conflicty with their subjects in those commemorative issue stamps, did they?

Oh, I forgot--they did. 

Here are a few good books to consult if you’re interested in polyhedra:

H.S.M. Coxeter, Regular Polytopes.  Coxeter was perhaps the most impressive geometer of the 20th century.
Peter R. Cromwell, Polyhedra.
H. Martyn Cundy and A.P. Rollett, Mathematical Models
Alan Holden, Shapes, Spaces and Symmetry. (Very pretty!)
Peter Pearce, Structure in Nature is a Strategy for Design.
Magnus Wenninger, Polyhedron Models.  (Paper construction!)

Comments

[Kaz Maslanka]

Although, I have to plead ignorance to the existence of the stamp, I must agree ... bring back the Rhombicuboctahedron! -- It is a beautiful construction however, not as nice as a regular polytope but a lot better than its replacement on the stamp.

Thanks,
Kaz

[John Ward Logan]

I care not about the stamp, but very much about Pacioli and MacKinnon's article. His speculations: that the student is Duerer, that the painter of the rhombi is none other than L., and that "Il Ritratto" may be the most important painting of the R. in that it shows "the transmission to A.D., and hence to the world north of the Alps, of the geometry of ancient Greece and the basis of the new art in Italy". Recently I have made a discovery concerning "i piccoli numeri" in the numbers F.L. is writing on the lavagna. Also evidence that he may have been in communication with Copernicus. Interested? Specifically, I need help finding Italian or English translations of his works, especially "Schifanoia" which may contain many answers.