Comments on type-II high-genus fullerenes

It is of interest to compare the bead model of a dodecahedron fused by 20 C60 in the previous post to high-genus fullerenes we discussed before. In fact, one can also view that bead model as a high-genus fullerene with genus=11, which is obtained by substracting one from the number of faces in a dodecahedron. The most important difference between these two types of high-genus fullerenes is the orientations of the TCNT necks relative the surface of the polyhedron. In the original high-genus fullerenes, which I will call the type-I high-genus fullerene from now on, the orientations of the TCNT necks are along the normal of the polyhedron. Here in the type-II high-genus fullerene, the orientations of the TCNT necks, which are created by fusing two neighbored C60, are lying on the surface of polyhedron (dodecahedron here) and along the directions of its edges.

Dodecahedron consisting of 20 fused C60s (still incomplete)

I am working on a beaded dodecahedral structure which is going to consist of 20 fused C60. The construction strategy is similar to certain bead models made by Mr. Kazunori Horibe. I called those models as "Carbon poti donut （波提甜甜圈）".
In the carbon poti donut, every C60 is connected to two neighbored C60 through two particular 5-fold axes. It is easy to see which two 5-fold axes I am talking about. Here I am trying to connect every C60 to three other C60 such that all 20 C60 locate on the vertices of a dodecahedron. But the bond angles generated by the trivalent C60 do not match with the angle of a dodecahedron. So the whole structure distorts quite significantly.

This model is very similar to a dodecahedron consisting of 20 dodecahedra (Clathrate cluster). I tried to make this model last month. But because each dodecahedron in this model is highly strained, I gave it up.

Quasicrystals with Zometool

Yuan-Chia Fan (范原嘉), Hsin-Yu Ko(柯星宇) and Mu-Chieh Chang (張慕傑) made a beautiful quasicrystal consisting of two types of rhombohedra with zometool last Friday. They put this model in the main lobby of our chemistry department for the alumni reunion last Saturday.

In addition to the tiling approach to the quasicrystal, one also constructs it using the cluster approach. In this sense, the carbon onion we made before is also a quasicrystal.

Originally, we didn't order enough red sticks to connect different shells of this structure. So only a few red sticks are used to hold neighbored Goldberg polyhedra along one five-fold axis (z direction). We now have enough red sticks for all twelve (or six) 5-fold axes emanating from the central dodecahedron (Goldberg vector (1,0)). So libration motion of each shell is quenched.

Yuan-Chia, Hsin-Yu, and Mu-Chieh also designed a wonderful support around pentagon at the bottom to enhance the stability for the this six-layer Goldberg polyhedra.

D surface constructed from four helical strips

There is another way to build a D-type triply periodic minimal surface (TPMS) with beads. Chern has told me previously that one can not only use helical strips to build G-type TPMS, one can also use exactly the same helical strips to build D-type TPMS. If one examine two helical strips carefully, one can find that there are exactly two different ways to put them together. One gives a D-type TPMS, the other one gives a G-type surface!

The following pictures are a bead model of D-surface consisting of four helical strips. Two of them are left handed, the other two are right handed. To build a D-surface, one has to put two helical strips together in an arrangement such that two neighbored strips are mirror-symmetric to each other. So the overall structure of D-surface is not chiral.

It is useful to look at other posts with the keyword helical strip, especially the one on the G-surface created by patching two helical strips with one strip shifted by half pitch.

A new display case

I have a new display case for some of Chern's and my beadworks in the chemistry building of National Taiwan university.

Four face-sharing pentagonal dodecahedra

E. A. Lord, A. Mackay, and S. Ranganathan described in their book, "New geometries for new materials", a simple cluster consisting of four face-sharing pentagonal dodecahedra arranged in a tetrahedral configuration (pp.48). Here is a bead model of this cluster.

In their book, there are more clathrate structures that one might be able to construct with beads.

Clathrate cluster

I bought some more green rice-shape beads last week and managed to finish this interesting clathrate cluster of 60 dodecahedra in the last weekend. One can still see deformation of many dodecahedra in this clathrate cluster though.

Three beadworks for the Joint Mathematical Meeting

Chern and I submitted three beadworks, P-, D- and G-TPMSs, for
the mathart exhibition of Joint mathematical Meeting which is going to be held in Boston next January.

Here I took the two photos from the JMM site:
the first one is the G-TPMS view from another angle:

Beaded Fullerene of Schoen's G Surface
18.5cm x 18.5 cm x 20cm
Faceted plastic beads and fish thread
2011

and also the D-TPMS

Beaded Fullerene of Schwarz's D Surface
23cm x 21cm x 18 cm
Faceted plastic beads and fish thread
2008 (constructed by my former student Wei-Chi Wei)

Two more posters for my exhibition

I made two more A3-size posters with the bead model of gyroidal graphitic surface for the exhibition, "The Fabulous World of Beaded Molecules 串珠幾何的異想世界", on Wednesday.


1. Top view:


2. Side view:

Exhibition : The Fabulous World of Beaded Molecules 串珠幾何的異想世界

In conjunction with the special Marie Curie’s exhibition for the international year of chemistry, I am going to have a joint exhibition, "The fabulous world of beaded molecules (串珠幾何的異想世界)", for my beadworks here in the chemistry department of National Taiwan University from 10/9-10/20.




I made a few posters for this event.

P-surface:


D-surface:




G-surface: