We show that a convex body can pass through a triangular hole iff it can do so by a translation along a line perpendicular to the hole. As an application, we determine the minimum size of an equilateral triangular hole through which a regular tetrahedron with unit edge can pass. The minimum edge length of the hole is (1+ √ 2)/ √ 6 ≈ 0.9856. One of the key facts for the proof is that no triangul...

lanthanide(iii) chloro complexes of 4[n-(4’-hydroxy-3’-methoxybenzalidene) amino] antipyrine semicarbazone (hmbaaps) (i) and 4[n-(3’,4’,5’-trimethoxybenzalidene) amino] antipyrine semicarbazone (tmbaaps) (ii) with the general composition lncl3.2l (ln = la, pr, nd, sm, gd, tb, dy or ho; l = hmbaaps or tmbaaps) are reported. all the complexes were synthesized in ethanolic medium and refluxed the ...

Number of recent technologies take Geographic Information Systems to new levels of power and usability. One of the most promising technologies that empower Geographic Information Systems is the 3D GIS modeling. In this paper, a novel robust multilevel data structure model called EBOT model (block octree tetrahedron mode Geographic Information Systems l) is presented on the basis of BOT visualiz...

The useful terms chiral and chirality were coined by W. H. Thompson (Lord Kelvin) in 1884 and are derived from cheir, the Greek word for a hand, indeed one of the most familiar chiral objects. The simplest chiral object of the three-dimensional perceptual space is, however, the chiral three-dimensional simplex, the irregular tetrahedron. As early as 1827 the famous German mathematician August F...

DNA tetrahedron as the simplest 3D DNA nanostructure has been applied widely in biomedicine and biosensing. Herein, we design and fabricate a series of circular assemblies of DNA tetrahedron with high purity and decent yields. These circular nanostructures are confirmed by endonuclease digestion, gel electrophoresis and atomic force microscopy. Inspired by rotary protein motor, we demonstrate t...

We describe a strategy of scaffolded DNA origami to design and construct 3D molecular cages of tetrahedron geometry with inside volume closed by triangular faces. Each edge of the triangular face is approximately 54 nm in dimension. The estimated total external volume and the internal cavity of the triangular pyramid are about 1.8 x 10(-23) and 1.5 x 10(-23) m(3), respectively. Correct formatio...