Bead-Chain Building Blocks

Contained within the box is a bead chain cube, showcasing an innovative method for building tensile structures. This wooden model utilizes four pre-tensioned linear bead chains interconnected with suitable cross-links. The elastic properties of the bead strings generate tension, causing the beads to repel each other when tightened. This interaction between string tension and bead repulsion ensures the overall self-balancing of the structure.

Bead-Chain Cube (BC-Cube):

Take apart BC-Cube to obtain four chains of five beads each.

Challenge 1: BC-Cube

The first task is to reconstruct the four five-bead chains into a cube structure as depicted in the diagram below. The eight terminal beads of these chains align with the vertices of the cube, while the remaining twelve beads are distributed along the cube's edges.

Challenge 2: Giant Tetrahedron

The next task is to link the same four five-bead chains into a tetrahedral arrangement as shown in the diagram below. Each edge of the tetrahedron consists of four beads, and each triangular face contains a total of ten beads.

Basic Operation

The fundamental process in constructing a bead chain model involves "cross-linking". As illustrated in the figure below with two chains of four beads each, the method entails stretching and crossing these chains over the gap between beads. The pre-stressed elastic strings create tension, causing the beads to snugly tighten and wrap around the crossing point.

Bead-Chain Molecular Models

Bead-chain building blocks enable the construction of valence sphere models for various molecules. In these models, beads symbolize the valence electron pairs of the molecule, while the taut elastic strings provide the necessary attractive force to bind these electron pairs together within the molecule. By carefully balancing tension and compression, bead-chain building blocks effectively simulate the equilibrium structure of numerous molecules.

Cubane, with the chemical formula C₈H₈, features carbon atoms located at the eight vertices of a cube. These carbon atoms are bonded together by twelve carbon-carbon bonds, with the carbon-hydrogen bonds oriented outward from the structure.

See Also

• Chinese(中文)：珠串組合益智積木與珠立方
• Chinese(中文)：2021 數感嘉年華 之 挑戰珠立方 poster

External links

• 2019 Bridges Conference Art Exhibition: BC-Cube and BC-Dodecahedron by Bih-Yaw Jin
• Bih-Yaw Jin, Bead-Chain Construction Set and Interlocking Puzzle Inspired by Polyhedranes, Proceedings of Bridges 2019: Mathematics, Art, Music, Architecture, Education, Culture, 553–556.
• 2024 Bridges Conference Art Exhibition: Polyhedrane with Great Rhombicuboctahedron Skeleton and Bead-Chain Buckyball by Bih-Yaw Jin
• 2020 Bridges Conference Art Exhibition: Octahedral nanodiamond by Bih-Yaw Jin and Chia-Chin Tsoo
• 2021 Bridges Conference Art Exhibition: Bead-Chain Woven Sodalite by Bih-Yaw Jin
• 2022 Bridges Conference Art Exhibition: Interlocked Bead-Chain Model of Zeolite A Structure by Bih-Yaw Jin
• Facebook: 珠璣幾何 GeoBeading

A few bead models of diamondoid molecules

Mathematical beading can be used to construct any diamondoid molecule, also known as nanodiamonds or condensed adamantanes. Here, I show three such molecular systems:

Adamantane (C₁₀H₁₆);
Diamantane (C₁₄H₂₀) also diadamantane, two face-fused cages;
One of 9 isomers of Pentamantane with chemical formula C₂₆H₃₂.

Talk on the connection between bead models and chemical bonding

The three figures at the bottom are taken from the works of Su and Prof. W. Goddard III at Caltech. These figures show electron density profiles based on the so-called electron Force Field, which is essentially a simplified floating spherical Gaussian orbital method with an empirically fitted Pauli potential that takes care of the antisymmetric property of electrons.

More bead/rubber band models

I found that it is much easier to use commercial beads to make the so called styrofoam/rubber band models of L. C. King. According to Prof. H. A. Bent, every student of chemistry should make a set of this kind of models for learning the concept of chemical bond. Here, I will show how to make valence sphere models using 25mm wooden beads and rubber band with 8mm small plastic beads as endcaps.

1. To make the valence sphere model of a methane which has the tetrahedral shape, we can connect two wooden beads (two-bead unit) first as shown in the following picture:

Then cross two two-bead units with each other to get a tetrahedral methane:

2. We can easily generalize the procedure to molecules with six valencies. First make another two-bead units with rubber bands, then cross it around the tetrahedral four-bead unit we just made. Then one should get an octahedral arrangement of six beads as shown in the following picture:

3. One can also try to make a five-bead unit which represents the valence sphere model of a molecule with dsp³ hybridization.

4. Valence sphere models for molecules with more than one center such as the ethane can easily be made too. Here is my procedure for making the valence sphere model of an ethane molecule: first I connect a three-bead unit and two two-bead units as shown in the following picture.

Then, connect these three units at suitable positions, one should get the valence sphere model for the ethane.

Bead VSM of methane, ammonium, water, and hydrogen fluoride

There is no doubt that tetravalent molecules such as methane occupy an important place in the chemical bond theory.
In 1865, German chemist August Wilhelm von Hofmann made the first stick-and-ball molecular models of methane in lecture at the Royal Institution of Great Britain. It is planar!

Then, 1872, van't Hoff, then a graduate student, learned of a possible tetrahedral arrangement of the valence bonds of carbon, proposed by the Russian chemist Alexander Butlerov in 1862. He later made a set of 3-D paper models of tetrahedral molecules.

Following Prof. H. Bent's recipes, Qing Pang (龐晴) of TFGH (北一女) made several bead valence sphere models (bead VSM) for tetravalent molecules with the formula AX_nE_m, where n+m=4, n is the number of bond electron pairs and m is the number of lone electron pairs. Here she didn't use the Windsor's knot to end the Nylon thread, instead she used simply tiny beads to cap the terminal beads of this kind of tree-like structures. Also, she use blue beads to represent bond electron pairs and yellow beads long electron pairs. You can see bead model exactly realizes the valence sphere model of Bent. I will show other bead VSM (made by Qing Pang) of molecules with several centers later.

Tangent sphere model of ethane

Using beads, one can make a faithful representation of the so-called valence sphere model (VSM) or tangent sphere model for a molecule proposed by Prof. Henry Bent in the 60s. In this model, each valence electron pair in a molecule is represented by a sphere. Its diameter is determined by the de Broglie wavelength of the corresponding electron, λ = h/p, where p is its momentum and h is the Planck constant.

Here is the first bead VSM (BVSM) of ethane (C₂H₆) made by Qing Pang (龐晴) of the Taipei First-Girls High School (北一女). She used the so-called Windsor-knot technique (雙活結) to bind beads that are not parts of loops in a molecular graph. For simplicity, she used beads of the same size to build the BVSM of ethane. This is equivalent to making the assumption that all valence electrons have the same momentum. The paper below the bead model is from the manuscript entitled "Approximate Molecular Electron Density Profiles. I. Construction" that I got from Prof. Bent last month.

BTW, Prof. Bent has just published a new book entitled "Molecules and the chemical bond" which is the first book-length sequel of his early articles on tangent sphere model last year. If you want to know more about the tangent sphere model, you should read the book or the original articles published in J. Chem. Edu.
You can read parts of this book at the google book.

1. Bent, H. A. J. Chem. Edu. 1965, 40, 446.
2. Bent, H. A. J. Chem. Edu. 1965, 40, 523.
3. Bent, H. A. J. Chem. Edu. 1967, 42, 308.
4. Bent, H. A. J. Chem. Edu. 1967, 42, 348.
5. Bent, H. A. J. Chem. Edu. 1968, 44, 512.
6. Bent, H. A. J. Chem. Edu. 1967, 45, 768.