Beaded sculpture of SARS-CoV-2 新冠病毒串珠模型

這件作品由國立臺灣大學的四位學生Ning-Hsiang Yin、Yu-Hsiang Chan、Zhi-Run Hsu、Wei-Shin Chen，在金必耀教授的《分子美學》課程中所製作期末計畫，並展示於 2021 年聯合數學會議（Joint Mathematics Meetings） 的 Bridges 數學藝術畫廊。 作品展示頁面： Bridges Math Art Gallery。

新冠病毒的結構與幾何特性

SARS-CoV-2（新型冠狀病毒）是一種具有正鏈 RNA 基因組的冠狀病毒，其結構具有高度對稱性與幾何特性。該病毒的主要結構包括：

• 核衣殼（Nucleocapsid, N）蛋白：包裹病毒 RNA，形成內部球狀結構。
• 刺突（Spike, S）蛋白：呈現三聚體狀，負責與宿主細胞的 ACE2 受體結合，決定病毒的感染性。
• 膜（Membrane, M）蛋白與包膜（Envelope, E）蛋白：形成病毒的脂雙層外殼。

由於病毒顆粒大致呈二十面體對稱性（Icosahedral Symmetry），數學家與病毒學家經常使用對稱群理論來研究病毒的結構與組裝模式。

二十面體對稱與幾何結構

這個串珠模型的幾何結構基於二十面體對稱（Icosahedral Symmetry），這是一種常見於病毒顆粒的對稱形式。 其數學特性包括：

• 由20 個正三角形組成，具有12 個頂點、30 條邊與 20 個面。
• 屬於柏拉圖立體（Platonic Solids），並且是擁有最多面的正多面體。
• 在群論中，二十面體對稱性對應於交錯群 A₅，包含 60 個對稱變換。
• 在病毒學中，病毒顆粒常以類似二十面體的方式組裝，以最小的能量構造最穩定的外殼。

新冠病毒串珠模型

• 尺寸： 30 x 30 x 30 公分
• 材料： 6 毫米塑膠珠、彈性線
• 製作年份： 2020 年

作品介紹

這個串珠模型採用二十面體框架作為病毒顆粒的基本幾何結構，並透過額外的突起模擬病毒表面的刺突蛋白（S 蛋白）。每個刺突蛋白由 3 顆串珠組成，以反映其三聚體結構。

為了使整體結構保持穩定，學生們使用彈性線將塑膠珠串聯，使其形成具有張力的網格結構，類似於病毒顆粒的蛋白質外殼。

這個模型不僅展示病毒顆粒的幾何對稱性，也強調了其在分子層級上的組裝原理，能夠幫助大眾理解病毒結構的數學與生物學基礎。

數學與科學應用

這個作品結合數學與生物學，展示二十面體對稱性在病毒學上的應用。其相關應用包括：

• 病毒結構建模： 幫助研究人員理解病毒顆粒的組裝機制，並可應用於疫苗與藥物設計。
• 數學對稱群應用： 透過對稱群理論來分析病毒殼蛋白的排列方式。
• 分子自組裝研究： 病毒顆粒的自組裝機制可應用於奈米技術，如奈米藥物傳遞系統。

結論

透過這個新冠病毒串珠模型，學生們將數學、幾何與病毒學結合，並透過具體的物理模型來探索二十面體對稱性在生物系統中的應用。這不僅是一個藝術作品，也是一個科學教育工具，幫助觀眾從不同的角度理解病毒的結構與行為。

© 2025 珠璣科學 |新冠病毒串珠模型

各種病毒的串珠模型

去年(2019 Spring semester)一組學生製作了四種不同型態的病毒模型：一般常見的二十面體病毒結構（Icosahedral virus）、菸草花葉病毒（Tobacco Mosaic virus）、HIV病毒、噬菌體（bateriaphage）。



還有另一組製作了另一種設計的噬菌體。



當時新冠病毒尚未流行，沒人想到要做一個模型。（Molecular Aesthetics 2019）

Bead model of coronavirus (SARS-CoV-2)

A group of students in my class, "Molecular Aesthetics", is working on a bead model of coronavirus(SARS-CoV-2). They adopt (8,0)-Goldberg polyhedron for the envelop with 60 spikes distributed on the surface. I hope they can complete it next week.

Frequency 2 Icosahedron

Two icosahedral complexes derived from an icosahedron

Starting from a bead model of icosahedron, one can make a few beautiful rigid polyhedral complexes by adding more regular octahedra and tetrahedra surrounding the central icosahedron. Here are two examples:

Icosahedron + Icosidodecahedron

Icosahedron + Icosidodecahedron + Rhombic Hexecontahedron

Workshop at BCCE, Grand Valley State Univ.

I will give one more workshop in the 2014 Biennial Conference on Chemical Education which is hosted by the Grand Valley State Univ, Michigan next week. My workshop is arranged in the afternoon (2:00-5:00PM) of Aug. 4.

It is free. Hopefully, I brought enough beads with me for every participant.

First slides for my talk in Center for Synchrotron Radiation and workshop for the Chemcamp

I have two opportunities to talk about the mathematical beading this month. The first one was given to people working at the Center of Synchrotron Radiation on Jul. 1. The first slide of the talk is given here:

Tomorrow I will give a 40-min talk before the hands-on workshop for about 30 high-school students who participate the Chemcamp these few days. The first slide of the talk is given as follows.

Sierpinski tetrahedron & Mackay icosahedron for the Bridges Seoul 2014

Dr. Tsoo and I have also submitted two beadworks for the mathart exhibition in the Bridges Seoul 2014, which can be found here: tubular bead structures.

Beadworks for the mathart exhibition (Mar. 15, 2014)

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 nice mnemonic for making beaded C60s

Prof. JT. Chen forwarded me a message from Sharon, an audience of my talk early this month. Sharon has a simple mnemonic by her son for making the beaded C60. In C60, every pentagon is surrounded by 5 hexagons, and every hexagon is surrounded alternatively by 3 pentagons and 3 hexagons. (五邊形的周圍是六邊形,六邊形的周圍是一個五邊形接一個六邊形.) One can easily create a beaded C60 by following this simple rule.

Two beaded models made by Sharon:




Indeed, one does not need spiral code to make C60. But to make an arbitrary cage-like fullerene (genus=0), spiral code is the only information we need. The shape of resulting beaded structure is always similar to the shape of the corresponding microscopic fullerene. It is quite amazing that one can create the faithful structure for an arbitrary fullerene with beads so easily. A simple explanation is that hard sphere repulsion among beads effectively mimic the valence-shell electron-pair repulsion of trivalent carbon atoms in fullerene molecules.

Additionally, if one want to make a beaded C60 with two different colors, a single color for pentagons and two different colors alternatively for hexagons. Then one doesn't need to use the mnemonic as given above. One can just pay attention to the colors only. Starting with a pentagon with a single color, then hexagons with two colors alternatively, eventually, one should get a beaded C60 correctly.

A few beaded C60s (10mm faceted beads) I made in last week:


See also a discussion in the previous post.

Another photo of beaded icosahedron

C60, C70, C80 and T120

C1500 in the preparatory problem of IChO 2010

Problem 3 in the preparatory problem of IChO 2010 asks the buoyancy property of icosahedral buckyballs. It would be interesting but time consuming to construct all beaded models for the buckyballs with I symmetry less than 1500 carbon atoms. Right now, the largest buckyball belonging to this family I made is C320.

This problem claims that the giant buckyball are molecular balloons.
To me, the Archimedean principle of buoyancy is a macroscopic law due to the pressure course-graining over large amount of tiny collisions. Nano-particles such as C1500 have sizes less than the mean free path of the gas at 1 bar. So these nano-particles should simply follow the Boltzmann distribution, P ~ exp(-mgh/kT), based on their potential energy, mgh, at the height.