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!

Dendritic structures

Mr. Horibe made a number of dendritic fullerenes which are similar to the Kepler's stellated polyhedrons. By using the Euler theorem, it is quite straightforward to show that, in a cage-like fullerene without hole, N₅-N₇=12, where N₅ and N₇ are the number of pentagons and heptagons, respectively. In the typical fullerenes where the local Gaussian curvature is positive everywhere, we should have N₇=0. The total number of pentagons is 12. Another way to put it is to introduce the so-called topological charge, 1 for a pentagon and -1 for a heptagon. So the topological charge of a cage-like fullerene is 12.

Heptagons always generate a negative Gaussian curvature. For a cage-like fullerene, whenever we introduce an extra heptagon, we have to include a pentagon in order to satisfy the identity N₅-N₇=12.

One can replace a pentagon in a Goldberg icosahedron (icosahedral fullerene) by 6 pentagons (a hemisphere of C20) and 5 heptagons to get a spike. So the topological for this area is still 1 after replacement. Similarly, we can do the same replacement for all other 11 pentagons to get a dendritic structure.



Of course, we can use the same kind of trick to "grow" a spike (a carbon nanotube endcapped with the hemisphere of a C₂₀) along the normal direction of any pentagon on any kind of graphitic structure.

Incidentally, this structure without the C20 caps is just the inner part of the high-genus fullerenes we have done before.

Bead model of small stellated dodecahedron

Another class of bead models made by Mr. Horibe is the small stellated dodecahedron, which is one of the Kepler–Poinsot polyhedra.



The basic idea of making this kind of dendritic dodecahedra is to choose a suitable size of Goldberg polyhedron and then grow a short segment of endcapped carbon nanotube along each pentagon. In this particular case, the endcapped CNTs are just hemisphere of C20s. Similar trick to make dendrite-like structures is used in many of Mr. Horibe's work.

High-Genus carbon surface

Beaded model for the high-genus carbon surface as suggested by Terrones a few years ago.