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

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.

ε Keggin structure

δ Keggin structure

γ Keggin structure

β Keggin structure

α Keggin structure

From pentaexcavated Icosidodecahedron to Mackay polyhedra

Perovskite, 4 stella octangula

Oval shaped beads

I bought some plastic oval shaped beads a few months ago. I only made a buckyball with this kind of beads and never used them again. I saw these beads this afternoon accidentally and decided to make some more simple models with these oval shaped beads. So, here are five Platonic solids and a rhombic triacontahedron I made.

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.

Buckyball made of 60 dodecadedra

I made this structure with beads last weekend. Still unfinished. The finished structure should have 60 dodecahedra arranged like a buckball. One has two ways to interpret this structure:
1. If every dodecahedron represents a carbon atom, we have a standard C₆₀.
2. If we still use beads to represent CC bonds, then we have a giant molecule, C₇₅₀. In this molecule, 450 carbon atoms are sp³ hybridized or tetra-valent and 300 atoms are sp² hybridized or trivalent. But I suspect these sp² hybridized carbon atoms are not energetically favorable, so it is better to have hydrogen atoms connected to these sp2-carbons. Then we get C₇₅₀H₃₀₀!



The bead model of this structure (see here) might be first constructed by Emilie. She asked me to comment about this structure in my blog long time ago (I couldn't find the exact location though).

I decided to make one from beads after I saw the same structure made by a toy designer, Dick Esterle, who actually invented this kind of toys, in the Bridges conference last week.

Another photo of the smallest HG fullerene

Photo taken by J.-H. Hwang.

Yet another bead model of rhombicuboctahedron

Another rhombicuboctahedron using beads with long-aspect ratio:

Sierpinski Tetrahedron (Level 3)

Bead model of an Icosidodecahedron

Bead model of an Icosidodecahedron:

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.

Space-Filling Polyhedra Based on a Truncated Octahedron

I made this structure during the spring break. Chuang has made a similar structure with several truncated octahedron before. But his structure has three unit cells separated connected. So the first unit cell is not connected to the third unit cell directly. My original goal is to make a structure with six unit cells connected to each other, something like a 2x2x2 cluster. However, the structure seems to be sensitive the small deviation at each local connection. Although truncated octahedron can fill the space. the structure I made seems to have pretty large strain and distorion.

Octahedron with six nodes on each face