Beaded Hilbert Curve (Step Two)

A couple of years ago I made a (half of a) Hilbert curve, as I recall it was inspired by a conversation with Bih-Yaw. And soon I forgot about this and went on to other beaded molecules. It was at the Bridges this year that I came across the 3D-printed sculptures of Dr. Henry Segermen. Then I decided to make another beaded model for this amazing mathematical figure.

I finished submitting this, together with a short introduction on both the novelty of the beading technique and the curve itself, to the Joint Mathematics Meeting 2013, which will be held in San Diego next January. I don't think I could physically be there at the meeting, though.

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.

Hilbert's space filling curve

Chern showed me this bead model of Hilbert curve he made a few years ago when he was still an undergraduate student.

two more pictures of an sp3 structure

Longuet-Higgins' Rhombo blocks

When reading the book, "King of Infinite Space", a bibliography about the Canadian mathematician Donald Coxeter, a magnetic toys called Rhombo blocks mentioned in pages 251-252 caught my eye. According to the description in the book, it is some kind of toys very similar to the one, "ball of whacks", I bought a few months ago.

M. Longuet-Higgins was the inventor of Rhombo blocks. I checked Rhombo blocks after I got accessed to the internet (see the picture below) this morning. Indeed, the magnetic design of this toy is probably similar to the "ball of whacks". I guess that the difference is the shape between the two mainly. I am curious about who invented this kind of design first.

Again the Rhombo block is quite expensive, though.




One can also build a Kepler ball (Rhombic triacontahedron) using Rhombo blocks. Based on the following picture, it seems to me that only 10 blocks are needed.

sp3 hybridization

The structure shown in the right of following figure has demonstrate that one can, in general, make 3-D structrures that contain the sp3 bonding. Of course, it is harder to weave this type of structures.

From beaded fullerene

Chuang made them.

Ball of whacks

I bought this ball of whacks in the Smithsonian store at Dullus airport, Wahsington D.C. last month.
This ball is basically a rhombic triacontahedron consisted of 30 magnetic design blocks.