Inspired by Gothic-influenced architectural styles, we analyze some of the circle patterns found in rose windows and semi-circular arches. We introduce a recursive circular ring structure that can be represented using a set-like notation, and determine which structures satisfy a set of tangency requirements. To fill in the gaps between tangent circles, we add Appollonian circles to each triplet of pairwise tangent circles. These ring structures provide the underlying structure for many designs, including rose windows, Celtic knots and spirals, and Islamic star patterns.