Based on a n-regular polygon $P_n$, we show that $r_n = 1/(2 sum^{[(n-4)/4]+1}_{j=0}{cos 2jpi/n)}$ is the ratio of contractions $f_i(1 leq i leq n)$ at each vertex of $P_n$ yielding a symmetric gasket $G_n$ associated with the just-touching I.F.S. $g_n = {f_i $mid$ 1 leq i leq n}$</TEX>. Moreover we see that for any odd n, the ratio $r_n$ is still valid for just-touching I.F.S $H_n = {f_i circ R $mid$ 1 leq i leq n}$</TEX> yielding another symmetric gasket $H_n$ where R is the $pi/n$-rotation with respect to the center of $P_n$.