"Lost in Triangulation" ?? 

No,  -- it is not a mistake, -- it is a Pseudo Rhombicuboctahedron!

Scientific American recently posted a brief note based on Rinus Roelofs’ discovery of a mathematical mistake by Leonardo,
which was translated by Dirk Huylebrouck. see:

It discusses a drawing by Leonardo that supposedly shows a "framed" version of an augmented Rhombicuboctahedron,
and it points out several "mistakes" in the geometry in this rendering (below, left).

However, I have now convinced myself that the mistakes may not be as grave as the authors of the above note imply;
because what is rendered may not aspire to depict an augmented rhombicuboctahedron,
but rather the
augmented pseudo-rhombicuboctahedron (right image below),
in which one half of the cuboctahedron is rotated by 45 degrees with respect to the other half.


After constructing the virtual models below, I must conclude that what Leonardo drew is a remarkably good rendition
of an augmented pseudo-rhombicuboctahedron. I think the rendition is simply too close to have happened by mistake.
The left image below is my best shot to find an eye-(camera-)point that makes the projection fit the drawing above.
I believe, Leonardo knew what he was doing!


Yes, there are some subtle geometric mistakes.  The perspective rendering is not perfect.  But more seriously, a slight rotation of the model (right image)
reveals the proper positions of the third struts in the tetrahedra at "5-o'clock" and at "7-o'clock."  At 7-o'clock Leonardo drew one strut in the wrong place:
He drew it  where I show a red arrow in my figure, -- when it should have been drawn in the place of the green arrow.
The fact that he made this "mistake" (?)  only on the left hand side, but not on the right-hand side, makes me wonder, whether this is Leonardo's way
of "watermarking" this drawing with the goal of identifying ripped-off copies of his work.  

There is another glaring breach of symmetry in the bottom left face of the 4-sided pyramid pointing forward and towards the 10-o'clock direction.
I would be rather surprised to learn that Leonardo made such easily avoidable mistakes accidentally in a figure that should show perfect bilateral symmetry.

But overall, to create a drawing a good as he did, without perhaps even having an actual model to look at, must be seen as a remarkable accomplishment.
I am eager to learn more about the background behind this drawing!

(By the way -- even in the pseudo-rhombicuboctahedron every triangle is still surrounded by six squares;
 and in Leonardo's drawing, every triangular pyramid is always surrounded by six quadrangular pyramids.)

Here is a description how to construct cardboard models of the Augmented RCO and Pseudo-RCO.

This whole discussion also prompted me to take a closer look at the rhombicuboctahedron in the Pacioli painting.

Page Editor: Carlo H. Séquin