Wednesday, April 29, 2015

Torus knot (1,2)

Kazunori showed me this beautiful torus knot (1,2) he made a few days ago. This structure can be classified as a torus knot, or more specifically a twisted torus without knot at all. The space curve that this tubular structure approximates can be described by the parametric equations for the torus knot (q=1, p=2). Therefore, it is reasonable to call it as the torus knot (1,2).

To me, this structure seems to be a perfect example to show the influence of the particular operation, Vertical Shift, described in the following papers:
1. Chuang, C.; Jin, B.-Y. Torus knots with polygonal faces, Proceedings of Bridges: Mathematical Connections in Art, Music, and Science 2014, 59-64. pdf
2. Chuang, C.; Fan, Y.-C.; Jin, B.-Y. Comments on Structural Types of Toroidal Carbon Nanotubes, J. Chin. Chem. Soc. 2013, 60, 949-954.
3. Chuang, C.; Fan, Y.-C.; Jin, B.-Y. On the structural rules of helically coiled carbon nanotubes, J. Mol. Struct. 2012 1008, 1-7.
Another related operation is the Horizontal shift, which is not used in this structure. Applying these two operations carefully (usually nontrivial), one can mimic the bending and twisting of many space curves in an approximate way.


Sunday, April 19, 2015

Circular helix winding around a central torus

Horibe-San just constructed another beautiful beadwork, a circular helical carbon nanotube (or circular carbon spring) winding around a toroidal carbon nanotube.


Friday, April 3, 2015

Bead model of the Chen-Gackstatter surface of genus 1

I made a bead model which approximates the minimal surface, Chen-Gackstatter surface of genus 1, for the spring break.
Other Chen–Gackstatter surfaces can be made with mathematical beading, in principle!