Conduction in a semi-infinite wall with a grooved line of contact between the wall material and convective environment is studied using series expansions. A periodic composition of semicircles is shown to result in a uniform gradient distribution at specific values of the groove radius and the convection heat transfer coefficient. Two fractal parquets exposed to natural thermal gradients are studied by the methods of complex analysis. In double periodic patterns each elementary cell is fractal (Sierpinsky＇s carpet and Sierpinsky＇s gasket) in which ＇dark＇ and ＇light＇ phases have arbitrary conductivities. The Maxwell approximation is used to calculate effective characteristics of both fractal structures by ＇homogenization＇ of the environment of an ＇inclusion＇. Solution of an exact two-dimensional refraction problem within an elementary cell including two components is used for upscaling, i.e. recalculation of effective conductivities and dissipations of subfractals of consequently increasing order.