T240 & C140

The aspect ratio of T240 (12mm) in previous post seems to be too large. My scanner probably has some problem to generate correct aspect ratio for large objects. So I just manually adjust it with the Illustrator.

一種碳環，兩種大小

T240 with 12mm and 4mm beads respectively.

The Three Canonical TPMS and Their Graphenoid Representations

tensegrity structures and straw

Straws can also be used to construct tensegrity structures:
Soda Straw Tensegrity Structures:



My 18" 270-Strut Tensegrity Sphere

Creative Science uses straws and suitable trivalent connector to build C60.


Nice instruction on straw icosahedron( A japanese site ) More can be found at this site.

Straw polyhedron

I found an interesting site:
Straw polyhedron and other nets


This page discusses the application of straws to the construction of polyhedron. Here are some of the poyhedron they built:

Tetrahedron


Octahedron


Icosahedron


Straw polyhedrons or straw fullerenes (restricted to trivalent structures) are formally the same as the beaded fullerenes we advocate in this blog. But there is a major difference, the straw models do not simulate the correct sp2 force field of fullerenes, while our beaded models do. Thus it is hard to create large and complicated fullerene structures with straws and string. On the other hand, based on our previous experience, the beaded models can effectively simulate arbitrary fullerenes which may be large and complicated as shown in many pictures posted in this blog.

almost 2x2x4 P surface

One more unit cell to complete a 2x2x2 P-Type TPMS model

1x2x2 P-Type surface

Finally, I have a P-Type sufrace (made with 6mm faceted beads) with four unit cell. I made a serious mistake in the weaving of the fourth unit cell. It took me another day to have this beautiful beaded model.

New Light Tent

I uploaded some photos I took. This time I removed the translucent papers off the tent.

Please take a look of my picasa album.

The photos beyond this one are took in the new light tent while the ones before it are not. I doubt that my photos are yellowish is because my cam is actually reaching its lifetime.

Real Schwarzite D-Surface Graphenoid

Later last night I finished the coding of the "real" Schwarzite D-Surface Graphenoid generation, which is done by finding the dual lattice of the following infinitely extending deltahedron:



Which is copied from Ken Brakke's site, who is the author of Surface Evolver. And his TPMS gallery always gives me a great deal of help when doing the coding of negative-curved graphenoid.

The dual of this deltahedron is then the Schwarzite D-surface graphenoid with minimal number of atoms per unit cell. (There are 8 unit cells in the figure!)



A view from high symmetric point:



Here I also present some figs of the second smaller one:






One with four unit cells:

P-type surface

T120 with shell beads

Chuang's work again.

rhombicuboctahedron

great rhombicuboctahedron

cuboctahedron

octahedron

truncated octahedron

Cube

C20, Dodecahedron: another beaded representation

truncated tetrahedron

Tetrahedron

capsule shaped beads

The new type of beaded fullerenes designed by Chuang strongly suggests that very good 3D structures of fullerenes can be constructed from capsule shaped beads.



On one hand, this kind of beads is similar to shape of a chemical bond that most chemists are familiar with; on the other hand, the steric hindrance among different beads can still effectively mimic the sp2 repulsion.



Unfortunately, capsule shaped beads are not popular in beading society. We couldn't find this kind of beads commercially available locally.

C60 with each bond built from two spherical beads and one tube

In the following figure we show the beaded model for C60 based on the composite bead technique. The final shape is not quite satisfying as we can see it is a little bit distorted. This is probably due to that the sizes and shapes of the constituent beads and tubes are not exactly the same. I believe a better structure can be constructed if we can control the sizes and shapes in a better way.

Two more beaded C20s

I just made two more dodecahedral C20 in order to make my point clearer. The model in the left of the following figure is made from the standard spherical beads. Due to the hard-sphere repulsions among nearest beads, the resulting structure is quite stable even under moderate external pressure.

To some people, particularly some organic chemists, the most confusing part of our beaded representation of fullerenes is that the beads in the model stand for bonds instead of atoms as commonly used in stick-and-ball models. Thus we need thirty beads to make a C20. Since most people are used to the concept that any spherical object in a molecular model must correspond to an atom, thus our beaded models based on spherical beads may cause some confusion. The simplest way to solve this problem is that we should use beads with large aspect ratio to build our physical models. The model in the right of the following figure is constructed with this kind of beads. The beads as chemical bonds can be seen clearly. Unfortunately, this kind of beads cannot mimic repulsions among sp2 orbitals as good as spherical beads.

An effective strategy to improve this problem is based on the type of beading introduced in previous message.

Dodecahedral C20 with correct bond shape and force field

Previously, I have shown that beads with spherical shape can effectively mimic sp2 force field of fullerenes. However, these beads represent bonds instead of atoms. This may lead to confusion for students. We can avoid this problem by using beads with large aspect ratio, but the resulting structure usually has poor mechanical stability. Here Chuang has created a nice beaded fullerene of C20 (Ih) which can clearly exhibit the bond network of fullerene and at the same time possess great stability.

Spiral Codes of Fullerenes

I re-checked my copy of the "Atlas" and found that there are actually no missing pages but with those pages arranged in reverse order. I don't know how'd that ever happen. For convenience, I made a digital copy of the spiral code table on our wiki site.

Spiral Codes of Fullerene Data Base

I only typed cases under C50 without IPR and will continue to finish the copy in a couple of days.

By the way, I found that Fowler actually left a fortran code for generating the spiral codes. I think that I'll do a Matlab version some other day.

double-wall T250 with armchair inside and zigzag outside

We have discovered a new types of D_5h torus.

New T240

串珠分子模型的美妙世界

這一年來，我們花了不少時間研究串珠分子模型的理論與建構規律，我最近正在撰寫一份有關這些工作的總結報告，其中英文摘要如下：


串珠分子模型的美妙世界

金必耀1，莊宸1，左家靜2

摘要

化學分子的實體模型是化學家與眾多從事微觀世界研究的人，在闡明複雜分子的三度空間結構上，不可或缺的工具。在本文中，我們將利用一般生活上，常用於裝飾、藝術用途上的串珠，來製作任意結構的芙類（fullerenes）分子。透過直角編織法，編織出的多圓環，可以代表芙類分子中的多碳環。由於圓形串珠的硬殼球排斥與微觀 sp2 碳碳鍵的價殼層電子對排斥非常類似，所以串珠間的排斥力場可以模擬微觀分子內之力場，因此芙類分子的串珠模型的幾何結構與真實分子的結構非常相似，這與我們所知的其他種類之分子模型，極為不同。本文並將介紹各種芙類分子結構的串珠實體模型與其建構方法，包括各類籠形構造、甜甜圈結構、螺旋管結構、沸石結構、週期最小曲面。簡言之，串珠可以說是建構芙類分子之最佳材料，而且所做出來的串珠模型，本身就是一個結構優美，極具藝術價值的展示品，其所隱含的幾何意義、化學觀念，更可引人深思，進而深入探索其中的奧妙。







The Wonderful World of Beaded Fullerenes

Bih-Yaw Jin†, Chern Chuang† and Chia-Chin Tsoo‡

Abstract

Physical models of molecular systems are indispensable tools for both chemists and practicing researchers working on various complicated molecular systems in order to understand their delicate three-dimensional structures. In this study, we discovered that beads commonly used in ornaments and decorative arts could be used to build the physical models of fullerene molecules through the so-called “Right Angle Weave” technique. We have demonstrated that these aesthetically pleasing beaded models provide faithful 3-D representation of the corresponding fullerenes due to the remarkable analogy between the classical hard-sphere repulsions among beads and the microscopic valence shall electron-pair repulsions of the sp2 carbon atoms. We have also investigated the construction rules for various kinds of fullerenes, and constructed corresponding physical models with beads.

Keywords: physical models, beading, fullerenes, Platonic solids, periodic minimal surfaces

C70 and beaded pig

C70 is a cute molecule that can be used to build a model of pig by putting some peripheral parts on it.

T120 with 10mm faceted beads

beadwork by Chuang

New beaded models of six T240

New T240 with 3mm beads

It is nice to have T240 using 3mm beads. The structure of this model is stronger than those made with larger beads. This seems to be in agreement with the scaling law.

New assorted beaded fullerenes

Some more new structures

Fluorescent T420

Starting from the inner core of type 4 torus with large inner-rim radius, one can build another beautiful carbon torus which contains 420 carbon atoms. This model is made from 3 mm beads.


beadwork by Chuang.