Introduction. A pseudoline is formed from a line by stretching the plane without tearing: it is the image of a line under a homeomorphism of the plane [13]. In arrangements of pseudolines, pairs of pseudolines intersect at most once and cross at their intersections. Pseudoline arrangements can be used to model sorting networks [1], tilings of convex polygons by rhombi [4], and graphs that have ...

We generalize the Hadwiger theorem on line transversals to collections of compact convex sets in the plane to the case where the sets are connected and the transversals form an arrangement of pseudolines. The proof uses the embeddability of pseudoline arrangements in topological affine planes. In 1940 Santaló showed [8], by an example, that Vincensini’s proof [9] of an extension of Helly’s theo...

Let G be a graph embedded in the plane and let A be an arrangement of pseudolines intersecting the drawing of G. An aligned drawing of G and A is a planar polyline drawing Γ of G with an arrangement A of lines so that Γ and A are homeomorphic to G and A. We show that if A is stretchable and every edge e either entirely lies on a pseudoline or intersects at most one pseudoline, then G and A have...

It is desirable to fabricate colorful coatings that have nonfading properties and are environmentally friendly. In this study, a novel approach for creating structural color coatings from monodisperse silica particles is presented. The structural color coating films, formed from an array of silica particles with a small amount of black additive, are easily prepared by a simple electrophoretic d...

The beautiful structural colors in bird feathers are some of the brightest colors in nature, and some of these colors are created by arrays of melanin granules that act as both structural colors and scattering absorbers. Inspired by the color of bird feathers, high-visibility structural colors have been created by altering four variables: size, blackness, refractive index, and arrangement of th...

Refer to Figure 1. A monotone matching is a set of n segments, each the portion of a unique pseudoline, and each spanning a unique slab, such that the left endpoint of each segment is above the right endpoint of the segment in the previous slab. In addition, the point on the first vertical line is below the left endpoint of the first segment, and the point on the last vertical line is above the...