An interstice relationship for flowers with four petals

摘要

Given three mutually tangent circles with bends (related to the reciprocal of the radius) a, b and c respectively, an important quantity associated with the triple is the value 〈a, b, c〉:= ab+ac+bc. In this note we show in the case when a central circle with bend b0 is "surrounded" by four circles, i.e., a flower with four petals, with bends b_1, b_2, b_3, b_4 that either (b_0, b_1, b_2)~1/2 +(b_0, b_3, b_4)~1/2 = (b_0, b_2, b_3)~1/2 +(b_0, b_4, b_1)~1/2 or (b_0, b_1, b_2)~1/2= (b_0, b_2, b_3)~1/2+ (b_0, b_3, b_4)~1/2+ (b_0, b_4, b_1)~1/2 (where (b_0, b_1, b_2)~1/2 is chosen to be maximal). As an application we give a sufficient condition for the alternating sum of the 〈a, b, c〉~1/2 of a packing in standard position to be 0. (A packing is in standard position when we have two circles with bend 0, i.e., parallel lines, and the remaining circles are packed in between.)