This investigation concerns the comparison of twoand three-dimensional data obtained on a cellular material. By quantitative micrography techniques and spatial measurements, the cellular structure corresponds most closely with the shape of pentagonal dodecahedra, twelve-faced cells having five edges per face. The cell volumes have a normal distribution. The areal distributions of planar section...

We show that semisimple Hopf algebras having a self-dual faithful irreducible comodule of dimension 2 are always obtained as abelian extensions with quotient Z2. We prove that nontrivial Hopf algebras arising in this way can be regarded as deformations of binary polyhedral groups and describe its category of representations. We also prove a strengthening of a result of Nichols and Richmond on c...

The paper describes a method for generating combinatorial complexes of polyhedral type. Building blocks B are implanted into the maximal simplices of a simplicial complex C, on which a group operates as a combinatorial reflection group. Of particular interest is the case where B is a polyhedral block and C the barycentric subdivision of a regular incidence-polytope K together with the action of...

Electron microscope studies have been made of a number of phages of lactic streptococci, seven of which were phages of Streptococcus lactis C10. Two of the phages are thought to be identical; five have been classified by the method of Tikhonenko as belonging to group IV (phages with noncontractile tails) with type III tail plates; one belongs to group V (phages with tails possessing a contracti...

We show that the presentation of affine T-varieties of complexity one in terms of polyhedral divisors holds over an arbitrary field. We also describe a class of multigraded algebras over Dedekind domains. We study how the algebra associated to a polyhedral divisor changes when we extend the scalars. As another application, we provide a combinatorial description of affine G-varieties of complexi...

Let G be a polyhedral group, namely a finite subgroup of SO(3). Nakamura’s G-Hilbert scheme provides a preferredCalabi-Yau resolutionY of the polyhedral singularity C3/G. The classical McKay correspondence describes the classical geometry of Y in terms of the representation theory of G. In this paper we describe the quantum geometry ofY in terms of R, an ADE root system associated to G. Namely,...