If Γ is a 2-Moufang generalized n-gon for n ≤ 6, then Γ is Moufang. © 2004 Elsevier Ltd. All rights reserved. MSC: 20E42; 51E12; 05C25

We say that a polygon inscribed in the circle is asymmetric if it contains no two antipodal points being the endpoints of a diameter. Given n diameters of a circle and a positive integer k < n, this paper addresses the problem of computing a maximum area asymmetric k-gon having as vertices k < n endpoints of the given diameters. The study of this type of polygons is motivated by ethnomusiciolog...

The rotations are the symmetries preserving the (cyclic) ordering of vertices. Thus, a rotation g is determined by the image gv, so the subgroup N of rotations has n elements. A reflection is an order-2 symmetry reversing the ordering of vertices. Imbedding the n-gon in R, there are n axes through which the n-gon can be reflected, so there are n reflections. Since |G| = 2n, every symmetry is ei...

BACKGROUND Local anaesthetic blocks of the greater occipital nerve (GON) are frequently performed in different types of headache, but no selective approaches exist. Our cadaver study compares the sonographic visibility of the nerve and the accuracy and specificity of ultrasound-guided injections at two different sites. METHODS After sonographic measurements in 10 embalmed cadavers, 20 ultraso...

To form the proximal-distal axis of the C. elegans gonad, two somatic gonadal precursor cells, Z1 and Z4, divide asymmetrically to generate one daughter with a proximal fate and one with a distal fate. Genes governing this process include the lin-17 frizzled receptor, wrm-1/beta-catenin, the pop-1/TCF transcription factor, lit-1/nemo-like kinase, and the sys-1 gene. Normally, all of these regul...

Curves which can be rotated freely in an n-gon (that is, an regular polygon with n sides) so that they always remain in contact with every side of the n-gon are called rotors. Using optimal control theory, we prove that the rotor with minimal area consists of a finite union of arcs of circles. Moreover, the radii of these arcs are exactly the distances of the diagonals of the n-gon from the par...

We give the first polynomial-time algorithm for the problem of finding a minimumperimeter k-gon that encloses a given n-gon. Our algorithm is based on a simple structural result, that an optimal k-gon has at least one “flush” edge with the ngon. This allows one to reduce the problem to computing the shortest k-link path in a simple polygon. As a by-product we observe that the minimum-perimeter ...

A generalized n-gon, n 2, is a rank 2 geometry whose incidence graph has diameter n and girth 2n, and each vertex has valency 3. If the latter condition is not satisfied , then we have a weak generalized n-gon. In this paper, we will always consider generalized n-gons with n 3 (generalized 2-gons are trivial geometries). They are the irreducible spherical buildings of rank 2. A generalized poly...