In a series of highly influential publications, Cavalli-Sforza and colleagues used Principal Component (PC) analysis to produce maps depicting how human genetic diversity varies across geographic space. Within Europe, the first axis of variation (PC1) was interpreted as evidence for the demic diffusion model of agriculture, in which farmers expanded from the Near East ~10,000 years ago, and replaced the resident hunter-gatherer populations with little or no interbreeding. These interpretations of the PC maps have been recently questioned, as the original results can be reproduced under models of spatially co-varying allele frequencies without any expansion. Here, we study PC maps for data simulated under models of range expansion and admixture. Our simulations include a spatially realistic model of Neolithic farmer expansion and assume various levels of interbreeding between farmer and resident hunter-gatherer populations. An important result is that under a broad range of conditions the gradients in PC1 maps are oriented along a direction perpendicular to the axis of the expansion, rather than along the same axis as the expansion. We propose this surprising pattern is an outcome of the "allele surfing" phenomenon, which creates sectors of high allele frequency differentiation that align perpendicular to the direction of the expansion.