Polygonal Space Research Articles

A Case of Developmental Cystic Lung in an Infant

(1) The greater part of the right lung was involved in the cystic condition with a disorderly arrangement of the bronchi. The number of lobes was increased. (2) The lung was a mass of thin walled cysts of irregular shape and size. Many of the cysts communicated with each other and some of them led into small bronchi. (3) Histologically the cystic condition was due to an abnormal arrangement both of air spaces and of bronchi. (4) This variety of cystic lung resembled the honeycomb variety of the multiple small cyst group of Sellor's classification—that is a cyst-like formation which is mesh-like in character, exhibiting a series of polygonal spaces bounded by fine septa and showing practically nothing of the natural configuration of lung tissue.

Hexagonal (Polygonal) Spaces Between the Outer Portions of the Rods and Cones

Hexagonal (Polygonal) Spaces Between the Outer Portions of the Rods and Cones

Poncelet polygons in higher space

Poncelet polygons in higher space

On the Quantization of Polygon Spaces

Moduli spaces of polygons have been studied since the nineties for their topological and symplectic properties. Under generic assumptions, these are symplectic manifolds with natural global action-angle coordinates. This paper is concerned with the quantization of these manifolds and of their action coordinates. Applying the geometric quantization procedure, one is lead to consider invariant subspaces of a tensor product of irreducible representations of $SU(2)$. These quantum spaces admit natural sets of commuting observables. We prove that these operators form a semi- classical integrable system, in the sense that they are Toeplitz operators with principal symbol the square of the action coordinates. As a consequence, the quantum spaces admit bases whose vectors concentrate on the Lagrangian submanifolds of constant action. The coefficients of the change of basis matrices can be estimated in terms of geometric quantities. We recover this way the already known asymptotics of the classical $6j$-symbols.

The Euler characteristic of the regular spherical polygon spaces

Let $a$ be a real number satisfying $0<a<\pi$. We denote by $M_n(a)$ the configuration space of regular spherical $n$-gons with side-lengths $a$. The purpose of this paper is to determine $\chi (M_n(a))$ for all $a$ and odd $n$. To do so, we construct a manifold $X_n$ and a function $\mu: X_n \to \mathbf{R}$ such that $\mu^{-1}(a)=M_n(a)$. In fact, the function $\mu$ is different from the well-known wall-crossing function. We determine the index of each critical point of $\mu$. Since a level set is obtained by successive Morse surgeries, we can determine $\chi (M_n(a))$.