Ausfüllung der Ebene durch Kreise

On the λ-system of circles

“…In 1960 A. FLORJAN [4] proved that s(q, 1)=s(q, q, 1; 1). These upper bounds seem to be precise for many values o f q ;~ 1 (Fig.…”

“…4 We associate r ~ with any circle Ci the set Si of points P lying ,,nea er to 2C i than to any othre circle 2Cj, i.e. d(P, 2C,)< d(P, 2Cj), i~j (Fig, 3).…”

Solid Coverings of the Euclidean Plane with Incongruent Circles

“…4), which we write in the form δ(x, r, w) = π + 2(wr 2 − 1) arctan(y/(x + r )) 2y(x + r ) , ( 11) where x is the distance of P from O 1 O 2 , and y = √ 1 − x 2 . We assume that wr 2 − 1 = 0 (this will be satisfied in the examples discussed below).…”

“…Packings and coverings of the plane with incongruent circles are the subject of several papers (see, e.g., [5] and [9]- [11]) and of the comprehensive monograph [7]. One of the results is that the density of any packing (covering) with circles of not too different radii is not greater (not less) than the maximum (minimum) density of packings (coverings) with congruent circles.…”

Some Densest Two-Size Disc Packings in the Plane

“…The upper bounds given by L. Fejes Tóth and Molnár [FM] for the least upper bound δ(1, r ) of the density of a packing of unit discs and discs of radius r < 1 have been sharpened by Florian [Fl1], who proved that the density cannot exceed the packing density within a triangle determined by the centers of mutually touching circles of radius 1, r and r. Unfortunately, such packings do not tile the plane for any value of r, thus this general bound is never sharp.…”