Feathers are arranged in a precise pattern in avian skin. They first arise during development in a row along the dorsal midline, with rows of new feather buds added sequentially in a spreading wave. We show that the patterning of feathers relies on coupled fibroblast growth factor (FGF) and bone morphogenetic protein (BMP) signalling together with mesenchymal cell movement, acting in a coordinated reaction-diffusion-taxis system. This periodic patterning system is partly mechanochemical, with mechanical-chemical integration occurring through a positive feedback loop centred on FGF20, which induces cell aggregation, mechanically compressing the epidermis to rapidly intensify FGF20 expression. The travelling wave of feather formation is imposed by expanding expression of Ectodysplasin A (EDA), which initiates the expression of FGF20. The EDA wave spreads across a mesenchymal cell density gradient, triggering pattern formation by lowering the threshold of mesenchymal cells required to begin to form a feather bud. These waves, and the precise arrangement of feather primordia, are lost in the flightless emu and ostrich, though via different developmental routes. The ostrich retains the tract arrangement characteristic of birds in general but lays down feather primordia without a wave, akin to the process of hair follicle formation in mammalian embryos. The embryonic emu skin lacks sufficient cells to enact feather formation, causing failure of tract formation, and instead the entire skin gains feather primordia through a later process. This work shows that a reaction-diffusion-taxis system, integrated with mechanical processes, generates the feather array. In flighted birds, the key role of the EDA/Ectodysplasin A receptor (EDAR) pathway in vertebrate skin patterning has been recast to activate this process in a quasi-1-dimensional manner, imposing highly ordered pattern formation.