Synthetic maps of human gene frequencies, which are maps of principal component scores based on correlations of interpolated surfaces, have been popularized widely by L. Cavalli-Sforza, P. Menozzi, and A. Piazza. Such maps are used to make ethnohistorical inferences or to support various demographic or historical hypotheses. We show from first principles and by analyses of real and simulated data that synthetic maps are subject to large errors and that apparent geographic trends may be detected in spatially random data. We conclude that results featured as synthetic maps should be approached with considerable caution.