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!

Oval shaped beads

I bought some plastic oval shaped beads a few months ago. I only made a buckyball with this kind of beads and never used them again. I saw these beads this afternoon accidentally and decided to make some more simple models with these oval shaped beads. So, here are five Platonic solids and a rhombic triacontahedron I made.

Sangaku problem and C20

In the book "算法助述" I got from Mr. Horibe, there is an interesting problem in pp. 51-52 as shown in the following picture. As I mentioned before, Mr. Horibe made two beautiful bead models with a large metal sphere inside. And more importantly, large and small balls have the correct ratio of radii, R/r=sqrt{5}. He told me that he gave one of these two models to Hidetoshi Fukagawa, the author of the book "Sacred Mathematica - Japanese Temple Geometry", because, through Dr. Fukagawa, he knew so much about these interesting Japanese sangaku problems.

This Sangaku problem was originally proposed by Ishikawa Nagamasa and written on a tablet hung in 1798 in Tokyo’s Gyuto Tennosha shrine. I was quite surprised by that I could understand this ancient problem without any translation. It was written in Chinese (or Kanji):

"今有以小球三十個如圖圍大球，小球者各切臨球四個與大球，小球徑三百零五寸，問大球徑幾何？

答曰：大球徑六百八十二寸。"

In English:

Thirty small balls cover one big ball where each small ball touches four other small balls and big balls.
The radius of small ball is 305 inches. Find the radius of large ball.

Answer: The radius of large ball is 682 inches.

In the page 52 of the book, there are some explanations on how to reach the answer by noting that thirty balls are located on the thirty vertices of an icosidodecahedron. It was written with some Japanese characters which I don't understand. But it is not hard to guess the rationale. One can see that if one starts from an arbitrarily chosen small ball, then moves to next ball along a fixed direction, and then the next one along the same direction. Eventually, one would trace over exactly 10 balls which are located on the large circle of a sphere with the radius equal to the sum of radii of small and large balls. Using some trigonometry, one can get the answer.

Bead valence sphere model of tetra-t-butyl tetrahedrane

The tetrahedrane derivative with four tert-butyl substituents, tetra-t-butyl tetrahedrane, was synthesized by the Austrian chemist,Günther Maier, in 1978. Here is the bead valence sphere model of this interesting molecule I made this afternoon. I think the hardest part to make molecules with many sp3 centers how to control the force evenly in the whole weaving process.

Bead VSM of tetrahedrane

I should forget another platonic alkane, the tetrahedrane. The shape of this molecule based on the valence sphere model is just like 10 spheres close packed in a tetrahedron. Which models, valence sphere model or ball-and-stick model, is closer to the true shape of a tetrahedrane molecule?

Bead VSM of dodecahedrane

In principle, we can make the valence sphere model for any molecule with beads. But in practice, it is a little bit hard to thread the Nylon cord through a bead structure with tetravalent bonds, which are common for most molecules though.
But anyway, I made a bead VSM of dodecahedrane, C₂₀H₂₀.

Bead VSM of cubane

I just made a bead VSM (valence sphere model) of cubane (C₈H₈) by myself. The structure looks neat to me. Every valence electron pair is faithfully represented by a big bead, purple for CC bond and pink for CH bond. Small beads which have no chemical meaning are used to bind the pink beads to the central carbon cube. I didn't distinguish CC bonds from CH bonds. In principle, electron pairs responsible for these two types of chemical bonds should have different momenta. So they should have different sizes of charge clouds.

A dodecahedral ball made with bamboo

Again, I saw this interesting dodecahedral ball in Alberto's living room. He told me that he bought this at Vietnam a few years ago.

Another photo of beaded icosahedron

Another set of beaded models of platonic solids

Here is another set of beaded Platonic solids made with the same type of beads.

Beaded icosahedrons

Beaded models for icosahedron are harder to make. This is because each vertex has five edges connected to it, so one has to face a pentavalent coordination. It is very difficult, if not impossible, to make a pentavalent coordinate with spherical beads. But if want use beads with long aspect ratio, stable and beautiful icosahedral structures can still be made.

Ping-Pong ball in a beaded dodecahedron

From beaded fullerene

Chuang made it.

Cube

From Dec 25, 2008

Dodecahedron

Tetrahedron

I try to experiment with the idea that using three beads to stand for an edge of polyhedron. In this case, I made a tetrahedron. Previously, I have posted some of this kind of models. The original idea came from Chuang again.

Platonic solids with cylindrical bonds

Cylindrical beads that have capsule shape are perfect pedagogical materials for creating physical models of fullerenes since they can look like the chemists' intuition of chemical bonds and, at the same time, effectively mimic the steric repulsion among different beads. To demonstrate these points, here, we use these beads to create the five Platonic solids. 

Chemical bonds as cylindrical beads

I found this kind of cylindrical beads in one of local stores (廣昕) last weekend. They look perfect for mimicking chemical bonds. I will try other structures out later this month. 

Icosahedron

Platonic Solids

I bought these elongated beads last week, on sale, of course. The shape of these beads are closer to that of chemical bonds. I have used these beads to create five Platonic solids. They look great.

Tetrahedron


Cube



Octahedron




Dodecahedron



Icosahedron

octahedron