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1. CJM 2011 (vol 63 pp. 1254)

D'Azevedo, Antonio Breda; Jones, Gareth A.; Schulte, Egon
 Constructions of Chiral Polytopes of Small Rank An abstract polytope of rank $n$ is said to be chiral if its automorphism group has precisely two orbits on the flags, such that adjacent flags belong to distinct orbits. This paper describes a general method for deriving new finite chiral polytopes from old finite chiral polytopes of the same rank. In particular, the technique is used to construct many new examples in ranks $3$, $4$, and $5$. Keywords:abstract regular polytope, chiral polytope, chiral mapsCategories:51M20, 52B15, 05C25

2. CJM 2010 (vol 62 pp. 1293)

Kasprzyk, Alexander M.
 Canonical Toric Fano Threefolds An inductive approach to classifying all toric Fano varieties is given. As an application of this technique, we present a classification of the toric Fano threefolds with at worst canonical singularities. Up to isomorphism, there are $674,\!688$ such varieties. Keywords:toric, Fano, threefold, canonical singularities, convex polytopesCategories:14J30, 14J30, 14M25, 52B20

3. CJM 2009 (vol 61 pp. 1300)

Hubard, Isabel; Orbani\'c, Alen; Weiss, Asia Ivi\'c
 Monodromy Groups and Self-Invariance For every polytope $\mathcal{P}$ there is the universal regular polytope of the same rank as $\mathcal{P}$ corresponding to the Coxeter group $\mathcal{C} =[\infty, \dots, \infty]$. For a given automorphism $d$ of $\mathcal{C}$, using monodromy groups, we construct a combinatorial structure $\mathcal{P}^d$. When $\mathcal{P}^d$ is a polytope isomorphic to $\mathcal{P}$ we say that $\mathcal{P}$ is self-invariant with respect to $d$, or $d$-invariant. We develop algebraic tools for investigating these operations on polytopes, and in particular give a criterion on the existence of a $d$\nobreakdash-auto\-morphism of a given order. As an application, we analyze properties of self-dual edge-transitive polyhedra and polyhedra with two flag-orbits. We investigate properties of medials of such polyhedra. Furthermore, we give an example of a self-dual equivelar polyhedron which contains no polarity (duality of order 2). We also extend the concept of Petrie dual to higher dimensions, and we show how it can be dealt with using self-invariance. Keywords:maps, abstract polytopes, self-duality, monodromy groups, medials of polyhedraCategories:51M20, 05C25, 05C10, 05C30, 52B70

4. CJM 2005 (vol 57 pp. 844)

Williams, Gordon
 Petrie Schemes Petrie polygons, especially as they arise in the study of regular polytopes and Coxeter groups, have been studied by geometers and group theorists since the early part of the twentieth century. An open question is the determination of which polyhedra possess Petrie polygons that are simple closed curves. The current work explores combinatorial structures in abstract polytopes, called Petrie schemes, that generalize the notion of a Petrie polygon. It is established that all of the regular convex polytopes and honeycombs in Euclidean spaces, as well as all of the Gr\"unbaum--Dress polyhedra, possess Petrie schemes that are not self-intersecting and thus have Petrie polygons that are simple closed curves. Partial results are obtained for several other classes of less symmetric polytopes. Keywords:Petrie polygon, polyhedron, polytope, abstract polytope, incidence complex, regular polytope, Coxeter groupCategories:52B15, 52B05

5. CJM 1999 (vol 51 pp. 1230)

Hartley, Michael I.; McMullen, Peter; Schulte, Egon
 Symmetric Tessellations on Euclidean Space-Forms It is shown here that, for $n \geq 2$, the $n$-torus is the only $n$-dimensional compact euclidean space-form which can admit a regular or chiral tessellation. Further, such a tessellation can only be chiral if $n = 2$. Keywords:polyhedra and polytopes, regular figures, division of spaceCategory:51M20

6. CJM 1999 (vol 51 pp. 1240)

Monson, B.; Weiss, A. Ivić
 Realizations of Regular Toroidal Maps We determine and completely describe all pure realizations of the finite regular toroidal polyhedra of types $\{3,6\}$ and $\{6,3\}$. Keywords:regular maps, realizations of polytopesCategories:51M20, 20F55