The generation and classification of tile-k-transitive tilings of the Euclidean plane,the sphere and the hyperbolic plane

Algorithms based on the theory of Delaney-Dress symbols are discussed that can be used to recursively produce all possible equivariant types of tile-k-transitive tilings of the Euclidean plane, the sphere and the hyperbolic plane, for any (reasonable)k. A number of results can be obtained using computer implementations of the algorithms.     

Two finiteness theorems for periodic tilings of d-dimensional euclidean space

Consider the d -dimensional euclidean space E ^d . Two main results are presented: First, for any N∈ N, the number of types of periodic equivariant tilings that have precisely N orbits of (2,4,6, . . . ) -flags with respect to the symmetry group Γ , is finite. Second, for any N∈ N, the number of types of convex, periodic equivariant tilings that have precisely N orbits of tiles with respect to the symmetry group Γ , is finite. The former result (and some generalizations) is proved combinatorially, using Delaney symbols, whereas the proof of the latter result is based on both geometric arguments and Delaney symbols. <lsiheader> <onlinepub>7 August, 1998 <editor>Editors-in-Chief: &lsilt;a href=../edboard.html#chiefs&lsigt;Jacob E. Goodman, Richard Pollack&lsilt;/a&lsigt; <pdfname>20n2p143.pdf <pdfexist>yes <htmlexist>no <htmlfexist>no <texexist>no <sectionname> </lsiheader> Received September 5, 1996, and in revised form January 6, 1997.     

Periodic Delone Tilings

Given a periodic point set in 3-dimensional Euclidean space, an algorithm is described for computing the corresponding Delone tiling (and its Delaney symbol). Examples of applications in tiling theory and crystallography are discussed.     

Pion and nucleon dissociation in π−p→π−π+π−p at 205 GeV/c

In a 48 000-picture exposure of the Fermilab 30-inch hydrogen bubble chamber to a 205 GeV/cπ^?beam, we have measured 169 events of the reaction, π^?p→π^?π⁺π^?p, with a cross section of 635 ± 61 μb. This reaction proceeds almost entirely via low mass π^? → 3π and p → pππ dissociation. Factorization is satisfied for p → pππ dissociation in πp and pp interactions.     

Atomic force microscopy analysis of wool fibre surfaces in air and under water

Wool fibre surfaces have been treated by solvent cleaning which leaves the native covalently bound surface lipid layer intact, and by alcoholic alkali which removes the lipid layer. The resultant surfaces have been analysed by atomic force microscopy (AFM), with particular emphasis on force–distance (F–d) methods. Methodologies were developed for investigation in situ in water of both the surface topography and the characteristics of the lipid layer. Longitudinal surface texturing was resolved in images of wool fibre surfaces in air; the texturing remained prominent after exposure to water. High resolution F–d curves revealed features associated with the lipid layer. A simple formalism was used to show that the layer had a thickness of a few nm, and an effective stiffness of some 0.12±0.01 N/m. Strong adhesive interactions, equivalent to a pressure of 0.1 MPa, acted on the tip at the tip-to-substrate interface. The methodology and formalism are likely to be relevant in the broad field of thin-film analysis and for fibre technology.     

Nucleation of polypropylene by cyclic oligomers present in poly(ethylene terephthalate)

It is shown that cyclic oligomers present in poly(ethylene terephthalate) are excellent nucleating agents for the crystallization of polypropylene. However, rigorous purification of poly(ethylene terephthalate) showed that these oligomers are not responsible for the ability of the polymer to induce transcrystallinity when cooled in contact with polypropylene melts.