Da Vinci's Three Elevated Archimedean Solids - Cuboctahedron, small rhombicuboctahedron, and icosidodecahedron

Da Vinci's elevated polyhedra

Leonardo da Vinci (1452-1519) made outstanding illustrations for Luca Pacioli's 1509 book "The Divine Proportion", in which they described "elevated" forms of many polyhedra. In the Seoul Bridges meeting this year, Rinus Roelofs presented a beautiful paper on the similarities and differences between Da Vinci's elevations and Kepler's stellations.

For details, check the following pdf file:

Rinus Roelofs, Elevations and Stellations, Proceedings of Bridges 2014: Mathematics, Music, Art, Architecture, Culture, 235-242.

Figure 1 and 2 in the paper are original illustrations made by Leonardo da Vinci:

It is interesting that bead models for the five elevated regular polyhedra can be built easily with great effects. Among them, elevated cube and dodecahedron are more flexible as expected.

Also, these five elevated Platonic solids can be viewed as nonconvex deltahedra with the names, triakis tetrahedron, tetrakis hexahedron, triakis octahedron (stella octangula), pentakis dodecahedron, and triakis icosahedron, respectively.

Additionally, the elevated icosidodecahedron was also illustrated beautifully by Da Vinci in the book.

The corresponding bead model can also be built!

Two icosahedral complexes derived from an icosahedron

Starting from a bead model of icosahedron, one can make a few beautiful rigid polyhedral complexes by adding more regular octahedra and tetrahedra surrounding the central icosahedron. Here are two examples:

Icosahedron + Icosidodecahedron

Icosahedron + Icosidodecahedron + Rhombic Hexecontahedron