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Search: MSC category 51M20 ( Polyhedra and polytopes; regular figures, division of spaces [See also 51F15] )

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1. CJM 2011 (vol 64 pp. 123)

Lee, Jae-Hyouk
Gosset Polytopes in Picard Groups of del Pezzo Surfaces
In this article, we study the correspondence between the geometry of del Pezzo surfaces $S_{r}$ and the geometry of the $r$-dimensional Gosset polytopes $(r-4)_{21}$. We construct Gosset polytopes $(r-4)_{21}$ in $\operatorname{Pic} S_{r}\otimes\mathbb{Q}$ whose vertices are lines, and we identify divisor classes in $\operatorname{Pic} S_{r}$ corresponding to $(a-1)$-simplexes ($a\leq r$), $(r-1)$-simplexes and $(r-1)$-crosspolytopes of the polytope $(r-4)_{21}$. Then we explain how these classes correspond to skew $a$-lines($a\leq r$), exceptional systems, and rulings, respectively. As an application, we work on the monoidal transform for lines to study the local geometry of the polytope $(r-4)_{21}$. And we show that the Gieser transformation and the Bertini transformation induce a symmetry of polytopes $3_{21}$ and $4_{21}$, respectively.

Categories:51M20, 14J26, 22E99

2. 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 maps
Categories:51M20, 52B15, 05C25

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 polyhedra
Categories:51M20, 05C25, 05C10, 05C30, 52B70

4. 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 space

5. 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 polytopes
Categories:51M20, 20F55

6. CJM 1998 (vol 50 pp. 426)

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