From pentaexcavated Icosidodecahedron to Mackay polyhedra

Quasicrystal (準晶)

形狀美麗的分子（Chemistry & Chemical Industry, Vol. 62, 2012）

Prof. Sonoda (園田高明) of Kyushu University sent me a special issue on "Beautifully Shaped Molecules" of "Chemistry & Chemical Industry" published by Japanese Chemical Society. There are five articles by famous Japanese chemists in this issue. The photos I showed here are from an article on "Self-Assembled Polyhedra" (my translation based on the Kanji in the title, I don't know Japanese, though) by Makato Fujita (藤田 誠) of Tokyo University.

Chiral cube - the generalized Goldberg polyhedra

Chern Chuang discovered that one can generalize the Goldberg polyhedra to those polyhedra consisting of square grids, instead of triangular or hexagonal grids. Accordingly, the following polyhedron can be classified as the generalized Goldberg vector (3,1); while the cuboctahedron and rombicuboctahedron as shown in the previous posts can be classified as (1,1) and (2,0), respectively.

Cuboctahedron & Rhombicuboctahedron

C960 - Goldberg vector (4,4)

The number of atoms in a Goldberg polyhedron is given by the formula N=20(h² + hk + k²). If h=k, one has N=60*h²=60*4² = 960 atoms in (4,4)-Goldberg polyhedron.

Collection of a few Goldberg polyhedra

C720 - Goldberg vector (6,0)

Goldberg Polyhedron (5,0)

Here is the bead model for the Goldberg polyhedron (5,0) The number of vertices in this model is 20(5²) =500. The number of beads (6mm, faceted) used is 500*3/2=750.

C540

I made another giant fullerene with Goldberg vector (3,3) last week.


Goldberg vector (3,3) gives the position where to put the next pentagon. With this in mind, one can create easily any fullerene specified by the Goldberg vector (i,j).

Fullerenes belonging to icosahedral group

It is straightforward to make bead models for higher fullerenes with icosahedral symmetry. The simplest way is to use the Goldberg vector (see the following figure) to specify the relative position between two pentagons. Goldberg vector is very similar to the chiral vector used for defining carbon nanotubes.

Suppose we have the first pentagon located at orgin, (0,0), then we can ask where the next pentagon can we put? The answer is that any coordinate specified by (i,j) as shown in the following figure gives a unique fullerene with icosahedral symmetry. For instance, if the next pentagon is located at (i,j)=(1,1), we have a C60. It is not hard to show that the number of carbon atom for the fullerene specified by the Goldberg vector, (i,j), is N=20(i² + ij + j²).


(Figure 8 in "Jin, B.-Y.*; Chuang, C.; Tsoo, C.-C. “The Wonderful World of Beaded Molecules. 串珠分子模型的美妙世界” CHEMISTRY (The Chinese Chemical Society, Taipei) 2008, 66, 73-92, in chinese.")

A bead model of C60, Goldberg vector (1,1).



Icosahedral fullerene specified by the Goldberg vector (2,1) has 140 carbon atoms. This is the smallest chiral fullerene with icosahedral symmetry.




The following beaded fullerene is specified by (4,0) has 320 carbon atoms.

Wooden bead C60

This is a C60 constructed by eaglewood buhdda beads (烏木佛珠). I bought these beads in ChengDu, Sichuang last year.

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.

A nice glass cuboctahedron

Alberto has this beautiful gass box with cuboctahedral shape in his living room.

Another photo of beaded icosahedron

Yet another bead model of rhombicuboctahedron

Another rhombicuboctahedron using beads with long-aspect ratio:

Bead model of an Icosidodecahedron

Bead model of an Icosidodecahedron:

Beaded Zongzi

Tomorrow is the Duanwu Festival (端午節), also known as Dragon Boat Festival, a traditional and statutory holiday in China. We eat Zongzi (粽子) during this Festival. Zonzi is a traditional Chinese food, made of glutinous rice stuffed with different fillings and wrapped in bamboo or reed leaves. The shape of zongzi ranges from being relatively tetrahedral in southern China (including Taiwan) to cylindrical in northern China. It is not a bad idea to make bead model of zonzi. Since there are 12 pentagons in a cage-like fullerene, one has to put three pentagons around each vertex of a tetrahedron.

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Octahedron with six nodes on each face

Another set of beaded models of platonic solids

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