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!

One more Truss model of perovskite (or fluorite/antifluorite) structure

I made another truss model of perovskite structure with octahedra constructed by yellow and blue bugle beads.

Truss model of perovskite (or fluorite/antifluorite) structure

A pleasure to greet two friends from afar

Takaaki Sonoda and Kazunori Horibe visited me for the last few days. I arranged a few workshops for them in two local high schools, Taipei First Girls High School and Chian-Kuo High School, and a meeting on the mathematical art and games. A few photos from these activities:

Taipei First Girls High School

Chien-Kuo High School

Little Mama Bear (bead store)

Math department, Academia Sinica

Inside the cubic Kaleidoscopes

I made a Kaleidocycle and a stella octangula for them as gifts:

Unfortunately, Takaaki didn't know that they are fragile and broke the stella octangula the first day. So I made one more Kaleidocycle for him.

The opening of the Analects by Confucius and thus the first phrase of Chapter I after which the Chinese title of this book is named 學而.

學而時習之、不亦說乎。有朋自遠方來、不亦樂乎。人不知而不慍、不亦君子乎。

Isn't it a pleasure to study and practice what you have learned? Isn't it also great when friends visit from distant places? If one remains not annoyed when he is not understood by people around him, isn't he a sage?