Beaded Compound of five tetrahedra 五個正四面體所構成的十二面體串珠複合體

本作品由國立臺灣大學醫工系兩位大學生 李尹希與 Jeffrey Yu-Lin Chu製作，加上助教何厚勳不可或缺的設計， 入選 Bridges 2024 數學藝術展覽。

作品簡介

「五個正四面體所構成的十二面體串珠複合體」是一件基於數學幾何結構的藝術創作，該作品的靈感來自於五個四面體複合體（compound of five tetrahedra），並透過串珠技術來呈現其對稱性與結構美學。

創作背景

這件作品源於國立臺灣大學開設的「分子美學」課程，在助教 何厚勳 的指導下，學生們對幾何結構的分子模擬產生興趣，並決定透過串珠技術來重現數學上的五四面體（tetrahedra）複合體。

什麼是五四面體複合體？

五四面體複合體是一種由五個正四面體彼此交錯排列而成的幾何結構，具有高度的對稱性。該結構是由數學家 Arthur Cayley 在 19 世紀研究正多面體的組合方式時提出的，它屬於柏拉圖多面體的複合體之一。

作品特色

• 材質： 珠子、魚線
• 尺寸： 12.0 × 12.0 × 12.0 公分
• 創作年份： 2023 年
• 數學概念： 正多面體的對稱性、分子結構的幾何模擬

數學藝術與分子結構

在數學與化學的交會下，某些分子結構的排列方式與數學中的幾何結構高度相似。例如，碳納米管 和 富勒烯（Fullerene） 等碳分子結構，與許多正多面體或其變體有緊密的關聯性。這件作品正是希望透過可視化的方式，讓數學與科學的交互關係更為直觀。

相關連結

• Bridges 2024 數學藝術展
• 五四面體複合體（英文維基百科）
• 數學藝術（Mathematical Art）

這件作品不僅展示了數學與藝術的結合，也讓我們看到幾何結構如何影響科學領域的發展，尤其是在分子建模與材料科學方面的應用。

© 2025 珠璣科學 |五個正四面體所構成的十二面體串珠複合體

Enantimorphs for compounds of five tetrahedra

Left- and right-handed compounds of five tetrahedra built by a student, 資工系 徐衍新, in my class, Molecular Aesthetics 2020.

Bead model of Kaleidocycle (萬花環)

Kaleidocycle or a ring of rotating tetrahedra was invented by originally by R. M. Stalker 1933. The simplest kaleidocycle is a ring of an even number of tetrahedra. The interesting thing about the Kaleidocycle is that you can twist it inwards or outwards continually. The geometry of kaleidocycle has been studied by many people from different fields in the last 80 years:

1. Stalker, R. M. 1933 Advertising medium or toy. US Patent 1,997,022, filed 27 April 1933 and issued 9 April 1935.
2. Ball, W. W. Rouse 1939 Mathematical recreations and essays, 11th edn. London: Macmillan. Revised and extended by Coxeter, H. S. M.
3. Cundy, H. M.; Rollett, A. R. 1981 Mathematical models, 3rd edn. Diss: Tarquin Publications.
4. Fowler, P. W.; Guest, S. Proc. R. Soc. A 461(2058), 1829-1846, 2005.
5. 全仁重, Motivation Behind the Construction of Maximal Twistable Tetrahedral Torus.
6. HORFIBE Kazunori, Kaleidocycle animation.

Typically, people use paper or other solid materials to make this kind of toy. A few months ago, I discovered that you can easily make this toy by tubular beads through the standard figure eight stitch (right angle weave).

This particular model consists of 8 regular tetrahedra. You can easily extend rings that contain 10, 12, ... tetrahedra.

The procedure I used to make this 8-tetrahedra Kaleidocycle is by the standard figure-eight stitch (right angle weave) in which one just keep making triangles. Of course, some care should be paid on the sequence of these triangle.

Tetrahedral C28 and related structures

There are only three tetrahedral fullerenes with number of carbon atoms less than that of buckyball. They are C₂₈, C₄₀, and C₄₄, respectively. The spiral code for the smallest tetrahedral fullerene, C₂₈, is [1 2 3 5 7 9 10 11 12 13 14 15]. Following this code, we can easily make its bead model using the standard figure-eight stitch. We can see that, in this molecule, there are 12 pentagons, 3 in a group located at a vertex, and 4 hexagons located on the four faces of the tetrahedron. If we replace these pentagons by heptagons, we get a tetrapod-like structure, in which tri-pentagon vertices become tri-heptagon necks as shown in the following figure.

Using these tetrapods as building blocks, we can get the following diamond-like structure. In fact, this is exactly the structure Mr. Horibe put in the postcard. OK, if we start from other tetrahedral fullerenes such as C₄₀ and C₄₄, we can find out a lot more diamond-like structures.

A postcard from Mr. Horibe

A nice postcard I got from Mr. Horibe:

More carbon tetrapods

In the appendix of "Periodic Nanostructures", M. V. Diudea and C. L. Nagy gave an extensive list of tetrapod-like structures of carbon nanotubes. Here are some bead models based on the structures listed in the book.

C76

There are two isomers of C₇₆ that satisfy the isolated pentagon rule (IPR). Here is the bead model for C76 with T_d symmetry just constructed by Yuan-Chia Fan.

The spiral codes for these two isomers of C₇₆ are

C76:1 [1 7 9 11 13 18 26 31 33 35 37 39] D₂
C76:2 [1 7 9 12 14 21 26 28 30 33 35 38] T_d

Building blocks for pseudo D-type Schwarzite

I just noticed that I already have a picture of punctured C₈₄ in the logo of this blog the other day. I checked my photo library and found another picture of this model which was taken almost five years ago. I possibly made this bead model after reading the paper on Nature with title "Energetics of negatively curved graphitic carbon" by Lenosky et al. (1992 vol. 355, 333-335).

However, we now know this is not a true triply periodic minimal surface. The correct D surface should partition the space into two congruent parts. It is not possible to get such a structure if one use heptagons.

Of course, we now know how to make a correct D-type Schwarzite which exactly partition the space into two identical regions. Wei-Chi made this beautiful bead model of D-type Schwarzite a few years ago.

C84 with three and four heptagonal holes

As I said in the previous post, one can puncture holes on a C₈₄ and use the resulting structures as basic units for building larger graphitic structures. Here I show two C₈₄-derived bead models with three and four holes surrounded by heptagonal.

The Schlegel diagram and a simple beading path for three-hole structure can be worked out easily.

Similarly, here is the Schlegel diagram and a beading path for a four-hole (tetravalent) unit.

Of course, one can use these building blocks to make many interesting structures. I am planning to make a dodecahedron consisting of 20 tetravalent units. It is not hard to see the angle between two holes in a unit is about 109 degree which is very close to 108 degree for the inner angle of a pentagon. So one can expect that these units should be happily fitted in the resulting dodecahedron without too much distortion.

C84 - a tetrahedral fullerene

I posted a few bead models of C₈₄ before, but I had never given the detailed beading procedure for this molecule. C₈₄ is the smallest achiral fullerene with tetrahedral shape that satisfies the independent pentagon rule (i.e. no two pentagons are connected).

Here is another bead model of C84 consisting of 8mm beads I made yesterday.

The easiest way to make it is to follow the beading path as shown in the following Schlegel diagram. Note that this path does not correspond to the path give by a spiral code.

In principle, one can puncture holes on this molecule and use the resulting structures as building blocks to create more complicated structures or super fullerenes (fused C84 in this case). I will show how this can be done later.

One more super carbon tetrahedron

I made another super carbon tetrahedron (超級碳正四面體) consisting of four fused C60s with 8mm beads. Unlike the previous one, the two neighbored C60s are connected by a ring of hexagons.

Super carbon tetrahedron

We can make a super carbon tetrahedron (超級碳正四面體) with four C60 building blocks that have three holes drilled on the three pentagons surrounding the same hexagon. Of course, four equal CNTs with suitable length are required to connect these four punctured C60s. Here, in addition to heptagons (blue), one also creates three octagons (purple) and one nonagon (red) on the C60 at each vertex. Of course, this structure was created by Mr. Horibe first.

Building blocks:

Corresponding Schlegel diagram and weaving path for creating a single vertex (punctured C60):

VB diagram with T symmetry

There are many possible VB diagrams for C60. Many of them are still symmetric. The bead model I showed in the previous post corresponds to the kekule structure of single hexagon rotated by one beads, which leads to a fusion of three pentagons into a trefoil. One can also perform this operation on the remaining 9 pentagons at suitable 3-fold axes to produce an interesting pattern which looks like a tiling of four trefoils on a sphere.

I made the following two bead models of C60 to show this pattern. (left column: north hemisphere; right column: south hemisphere)



It is easier to see this pattern with four trefoil tiles using a schematic plot.


(I would be happy to recommend this chiral pattern with T symmetry on a football for the FIFA next time!)

Sierpinski Tetrahedron (Level 3)

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.

[Strange, this post disappear automatically.]

The Bead Model of Sierpinski Tetrahedron

Previously, I have shown the Sierpinski buckyball made by Chuang (莊宸) and his classmates (Class 2007, Chemistry, NTU) about three years ago. It is also straightforward to make other Sierpinski Platonic solids. Here is the simplest one: the beaded Sierpinski tetrahedron.

Au20: a molecular tetrahedron

It is known that the geometry of an Au₂₀ cluster is a tetrahedron with all of 20 gold atoms on the surface. It is quite straightforward to make this structure with beads. Since most of gold atoms in this structure have coordination higher than 3, so I have chosen rod-shaped beads. The resulting beaded structure seems to mimic the tetrahedron shape pretty well.

Tetrahedron

Large Truncated Tetrahedral and Octahedral Fullerenes

Chuang Chern made this amazing truncated tetrahedron which consists of more than a thousand beads. I guess he probably spent more than 10 hours on this gigantic fullerene. This structure is essentially a natural extension of the system with T or Td symmetry as I shown before. All fullerenes belonging this catagory can be completely specified by two vectors, a and b. The first vector, a, denotes the relative arrangement among the three pentagons at each vertex as shown in the right figure below. When these three pentagons are not nearest neighbours with each other, the resulting tetrahedron becomes a truncated one, just like the one I have posted below.

The second vector, b, specifies the separation between two pentagons, belonging to two neighbouring vertices separately.

Approaching the Octahedral Limit

Now let's imagine we have a continuous Achimedean solid just like our truncated beaded tetrahedron which can be described by the same two vectors, a and b. Now let the vector b gradually changes to zero, then this tetrahedron will be continuously transformed into an octahedron. The following figure taken from Wikipedia demonstrates this can be done graphically.



However, this is not possible in fullerene cages due to the intrinsic physical properties of the graphitic sheet. The smallest possible value of b is when these two pentagons are separated by one hexagon. We can't fuse two pentagons into one, the Euler theorem will be violated then. Additionally, since there are six vertices in an octahedron, every vertex of an octahedron must contain two pentagons. So the four faces around any one of the six vertices cannot be all the same. The two triangular faces lies along the direction defined by these two pentagons, the direction of vector b, will be different from the other two faces lying perpendicular to this direction. But when another vector, a, is very large, the truncated tetrahedron will approach the octahedron if the detailed structure around the vertices is ignored!

In general, it is not possible to have a fullerene with the octahedron symmetry, if only 5- and 6-member rings are allowed.

最近新拍的照片 (A few new pictures)