Expand all Collapse all | Results 1 - 6 of 6 |
1. CJM 2011 (vol 64 pp. 123)
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)
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)
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)
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 Category:51M20 |
5. CJM 1999 (vol 51 pp. 1240)
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)