a, Skull of Georgiacetus vogtlensis in dorsal view: to the left are conditions of characters mapped in d–h and to the right are the different possible character states for posterior migration of the maxilla (yellow) across all taxa. b, c, Skulls of Cotylocara macei and Simocetus rayi in dorsal view, the latter redrawn10, with their respective states for characters mapped in d–h. d–h, Evolution of characters that are likely to be associated with echolocation on our phylogenetic tree (Fig. 3). i, Summary of all changes detailed in d–h with convergent characters in green and non-convergent changes in black; numbers refer to characters described in the Supplementary Information. In a–f and h red is the primitive state, blue is the most derived state, and purple is an intermediate derived state. Shades of grey in g correspond to positions of the maxilla, as illustrated on the right side of a, with changes between character states (states 0 through 5) labelled below for clarity. Dashed terminal branches indicate that data are missing for that fossil taxon. Character evolution was inferred using parsimony on the entire phylogeny, although d–i only include those taxa needed to convey the pattern of character evolution at the base of Odontoceti. All evolutionary reconstructions are unequivocal except for f, which is an ACCTRAN optimization. An alternative, but equally parsimonious reconstruction for f, is discussed in the Supplementary Information. An arrow in d–i indicates where echolocation is inferred to have evolved based on its occurrence in bolded taxa: osteological evidence in Cotylocara macei and observational evidence in the crown group of Odontoceti. More details on the characters and their states can be found in the Supplementary Information.