Flat Zipper-Unfolding Pairs for Platonic Solids

We show that four of the five Platonic solids' surfaces may be cut open with a Hamiltonian path along edges and unfolded to a polygonal net each of which can "zipper-refold" to a flat doubly covered parallelogram, forming a rather compact representation of the surface. Thus these regular polyhedra have particular flat "zipper pairs." No such zipper pair exists for a dodecahedron, whose Hamiltonian unfoldings are "zip-rigid." This report is primarily an inventory of the possibilities, and raises more questions than it answers.Comment: 15 pages, 14 figures, 8 references. v2: Added one new figure. v3: Replaced Fig. 13 to remove a duplicate unfolding, reducing from 21 to 20 the distinct unfoldings. v4: Replaced Fig. 13 again, 18 distinct unfolding

Some Polycubes Have No Edge Zipper Unfolding

It is unknown whether every polycube (polyhedron constructed by gluing cubes face-to-face) has an edge unfolding, that is, cuts along edges of the cubes that unfolds the polycube to a single nonoverlapping polygon in the plane. Here we construct polycubes that have no *edge zipper unfolding* where the cut edges are further restricted to form a path.Comment: 11 pages, 10 figures, 9 references. Updated to match the version that will appear in the Canad. Conf. Comput. Geom., Aug. 202

The combined AFM manipulation and fluorescence imaging of single DNA molecules

A combined fluorescence microscope/AFM set-up was constructed, which enabled the real-time manipulation of single DNA molecules. Fluorescence images of these TO-PRO-3 intercalated strands could be taken, while they were stretched with an AFM tip on silanised or polylysine covered glass surfaces. Characteristic AFM force spectra of single DNA molecules were achieved and a statistical analysis of the rupture forces, plateau heights and rupture lengths was compiled. The wide-field fluorescence images indicated that the DNA underwent condensation on polylysine to form aggregated rods and globular structures. Due to the strong unspecific adhesion of the DNA to the polylysine surface, AFM tip manipulation frequently led to strand scission. In addition, it was possible to “write” nm-sized letters of fluorescent DNA by unraveling agglomerated strands from the tip onto the surface. In contrast, DNA strands on silane showed far less condensation. Extended single chains were bound to the surface by the termini or at specific sites along the double helix. These fixed and straightened strands could be overstretched laterally to ca. 1.6 times the original contour length. Chain rupture occurred at the tip position, but occasionally mid-strand rupture was also observed. An analysis of the chain fluorescence intensity for different stretching lengths revealed that the dyes remain intercalated up to the end of the DNA B-S overstretching transition