Zhen-Ting Liu; I-Heng Dai; Po-Ju Chen; Hou-Hsun Ho
The first three authors are undergraduate students who took "Molecular Aesthetics", a general education course at National Taiwan University by Professor Bih-Yaw Jin. The goal of this course to show ways in which traditional craft can draw inspiration from the molecular sciences and ways in which chemistry can use of art to make it more intelligible and interesting, by utilizing hands-on activities based on mathematical beading. The two bead sculptures of 3D immersion of Klein Bottles are our term project for this course, with important contributions provided by Hou-Hsun Ho, the TA of the course.
The Klein Bottle is a closed non-orientable surface of Euler characteristic 0 that has no inside or outside. The Euler theorem puts the constraint that the numbers of pentagons and heptagons are the same for the corresponding fullerene graph. To facilitate the construction procedure, we separate the Klein Bottles into three parts, the long neck of the bottle is stretched through its side by intersection and its end is joined to a hole in the base. The intersection of the neck and the side of the bottle is handled by tetravalent connections, the rest of the surface is treated as the standard trivalent fullerene surface.
The bead model for the Triple Klein Bottle is roughly an equilateral triangle. The necks that connect three bases are short linear carbon nanotubes, which make its construction simpler than the single Klein Bottle.