👉 In mathematics, a knoll or knoll-shaped region is an area in Euclidean space that is enclosed by a finite number of circles. Each circle touches another circle at its center and has radius equal to half the distance between their centers. For example, consider a square with sides 1 unit long. The squares inside the square are all knolls because they are bounded by circles centered on the two opposite vertices of the square. Similarly, the squares outside the square are also knolls, as