The verb behind Lachesis, lanchano, means to obtain something as one's lot, and its noun, lachos, is the share that falls to a person when a set of shares is distributed. Greeks used the word for concrete things first: land divided among settlers, spoils divided after a raid, priesthoods and offices assigned by drawing tokens from a jar. Allotment of that kind was not thought of as random. It was a recognised and public way of settling who received what, and it removed the decision from human favouritism without handing it to nothing at all.
Lachesis is that procedure as a goddess. Where Clotho makes the thread and Atropos ends it, Lachesis fixes how much of it there is. That places her at the centre of the sequence in more than order: the length of a life is the part of the arrangement that a Greek audience found genuinely interesting, because it is the part that visibly differs from person to person and that no one can account for.
Her iconography follows the function rather than a story. Ancient art gives her a rod or a measure, sometimes a globe or a scroll, and she is shown reading or marking rather than working with her hands. The Roman name Decima, the tenth, ties the group to the months of a pregnancy counted inclusively, so that the middle sister again holds the span rather than its beginning or its end.