A proximity drawing of a graph is a straight-line drawing where adjacent vertices are represented by points that are deemed to be close according to some proximity measure. A rectangle of innuence drawing is a proximity drawing where the measure is based on the concept of rectangle of innuence. Given two points u and v, the rectangle of innuence of u and v is the axis-aligned rectangle having u...

A pattern of 12,860 short (0.15 x 0.05 deg) texture lines contained an orientation texture-defined (OTD) rectangle of aspect ratio a/b (a was the rectangle's height and b was its width). All the lines within the rectangle had the same orientation and all the lines outside the rectangle had the same orientation. These two orientations were theta deg symmetrically clockwise and anticlockwise of t...

Given a right triangle and two inscribed squares, we show that the reciprocals of the hypotenuse and the sides of the squares satisfy an interesting Pythagorean equality. This gives new ways to obtain rational (integer) right triangles from a given one. 1. Harmonic and Symphonic Squares Consider an arbitrary triangle with altitude α corresponding to base β (see Figure 1a). Assuming that the bas...

The cyclohedron Wn, known also as the Bott-Taubes polytope, arises both as the polyhedral realization of the poset of all cyclic bracketings of the word x1x2 . . . xn and as an essential part of the Fulton-MacPherson compactification of the configuration space of n distinct, labelled points on the circle S1. The “polygonal pegs problem” asks whether every simple, closed curve in the plane or in...

The combination of an amide containing a linear ligand (L1) and an organometallic molecular "clip" (clip-1) leads to the self-assembly of a Pt4 nanoscopic framework representing the first example of a Pt-based molecular rectangle (rectangle-1) incorporating amide functionality. A complementary approach was also followed to prepare a Pd(II)-based molecular rectangle (rectangle-2) by reaction of ...

The cyclohedron Wn, known also as the Bott-Taubes polytope, arises both as the polyhedral realization of the poset of all cyclic bracketings of the word x1x2 . . . xn and as an essential part of the Fulton-MacPherson compactification of the configuration space of n distinct, labelled points on the circle S1. The “polygonal pegs problem” asks whether every simple, closed curve in the plane or in...

We prove a general Russo–Seymour–Welsh (RSW) result valid for any invariant bond percolation measure on Z2 satisfying positive association. This means that the crossing probability of long rectangle is related by universal homeomorphism to short rectangle.

Polyomino Venn (or polyVenn) diagrams are Venn diagrams whose curves are the perimeters of polyominoes drawn on the integer lattice. Minimum area polyVenn diagrams are those in which each of the 2n intersection regions, in a diagram of n polyominoes, consists of exactly one unit square. We construct minimum area polyVenn diagrams in bounding rectangles of size 2r×2c whenever r, c ≥ 2. Our const...

We consider a long-range growth dynamics on the two-dimensional integer lattice, initialized by a finite set of occupied points. Subsequently, a site x becomes occupied if the pair consisting of the counts of occupied sites along the entire horizontal and vertical lines through x lies outside a fixed Young diagram Z. We study the extremal quantity μ(Z), the maximal finite time at which the latt...

Rectangle (square) packing problems involve packing all squares with sizes 1 × 1 to n × n into the minimum area enclosing rectangle (respectively, square). Rectangle packing is a variant of an important problem in a variety of real-world settings. For example, in electronic design automation, the packing of blocks into a circuit layout is essentially a rectangle packing problem. Rectangle packi...