Surface structure of i-Al(68)Pd(23)Mn(9): An analysis based on the T*(2F) tiling decorated by Bergman polytopes

A Fibonacci-like terrace structure along a 5fold axis of i-Al(68)Pd(23)Mn(9) monograins has been observed by T.M. Schaub et al. with scanning tunnelling microscopy (STM). In the planes of the terraces they see patterns of dark pentagonal holes. These holes are well oriented both within and among terraces. In one of 11 planes Schaub et al. obtain the autocorrelation function of the hole pattern. We interpret these experimental findings in terms of the Katz-Gratias-de Boisseu-Elser model. Following the suggestion of Elser that the Bergman clusters are the dominant motive of this model, we decorate the tiling T*(2F) by the Bergman polytopes only. The tiling T*(2F) allows us to use the powerful tools of the projection techniques. The Bergman polytopes can be easily replaced by the Mackay polytopes as the decoration objects. We derive a picture of ``geared'' layers of Bergman polytopes from the projection techniques as well as from a huge patch. Under the assumption that no surface reconstruction takes place, this picture explains the Fibonacci-sequence of the step heights as well as the related structure in the terraces qualitatively and to certain extent even quantitatively. Furthermore, this layer-picture requires that the polytopes are cut in order to allow for the observed step heights. We conclude that Bergman or Mackay clusters have to be considered as geometric building blocks of the i-AlPdMn structure rather than as energetically stable entities

Embedding Stacked Polytopes on a Polynomial-Size Grid

A stacking operation adds a $d$-simplex on top of a facet of a simplicial $d$-polytope while maintaining the convexity of the polytope. A stacked $d$-polytope is a polytope that is obtained from a $d$-simplex and a series of stacking operations. We show that for a fixed $d$ every stacked $d$-polytope with $n$ vertices can be realized with nonnegative integer coordinates. The coordinates are bounded by $O(n^{2\log(2d)})$, except for one axis, where the coordinates are bounded by $O(n^{3\log(2d)})$. The described realization can be computed with an easy algorithm. The realization of the polytopes is obtained with a lifting technique which produces an embedding on a large grid. We establish a rounding scheme that places the vertices on a sparser grid, while maintaining the convexity of the embedding.Comment: 22 pages, 10 Figure

Facets of secondary polytopes and Chow stability of toric varieties

Chow stability is one notion of Mumford's Geometric Invariant Theory for studying the moduli space of polarized varieties. Kapranov, Sturmfels and Zelevinsky detected that Chow stability of polarized toric varieties is determined by its inherent {\it secondary polytope}, which is a polytope whose vertices correspond to regular triangulations of the associated polytope \cite{KSZ}. In this paper, we give a purely convex-geometrical proof that the Chow form of a projective toric variety is $H$-semistable if and only if it is $H$-polystable with respect to the standard complex torus action $H$. This \emph{essentially} means that Chow semistability is equivalent to Chow polystability for any (not-necessaliry-smooth) projective toric varieties.Comment: 13 pages, to appear in Osaka Journal of Mathematics Vol. 53, No. 3, (2016

Few smooth d-polytopes with n lattice points

We prove that, for fixed n there exist only finitely many embeddings of Q-factorial toric varieties X into P^n that are induced by a complete linear system. The proof is based on a combinatorial result that for fixed nonnegative integers d and n, there are only finitely many smooth d-polytopes with n lattice points. We also enumerate all smooth 3-polytopes with at most 12 lattice points. In fact, it is sufficient to bound the singularities and the number of lattice points on edges to prove finiteness.Comment: 20+2 pages; major revision: new author, new structure, new result