"narrow track worn or cut in the ground," as by a passing wheeled vehicle, 1570s, probably from Middle English route "way, a road, space for passage" (see route (n.)); though OED finds this "improbable." If so, it is a doublet of route.
Of the lines on the face by 1620s. The figurative meaning "narrow, monotonous routine; habitual mode of behavior or procedure" is attested by 1839 (Carlyle); earlier figurative use was as an obstacle to rapid transit (1705).
Enter an OLD LADY.
[Bosola] You come from painting now.
Old Lady. From what?
Bos. Why, from your scurvy face-physic.
To behold thee not painted, inclines somewhat near
A miracle: these in thy face here, were deep ruts,
And foul sloughs, the last progress.
There was a lady in France, that having the small-pox,
Flay'd the skin off her face, to make it more level;
And whereas before she looked like a nutmeg-grater,
After she resembled an abortive hedgehog.
[Webster, "The Duchess of Malfi"]
The verb meaning "mark with or as with ruts" is by c. 1600. Related: Rutted; rutting.
"periodically recurring sexual excitement in animals; animal mating season" (originally of deer), early 15c., from Old French rut, ruit, from Late Latin rugitum (nominative rugitus) "a bellowing, a roaring," from past participle of Latin rugire "to bellow" (from PIE imitative root *reu-). If so, the notion is of the noise made by deer at the time of sexual excitement. The noun rut "roar of the sea" (1630s) in Scottish and persisting in New England dialect is of uncertain connection.
especially of animals, "desire copulation, be under the influence of sexual passion," late 14c., ruteien, from rutei, probably an Anglo-French form of the noun (see rut (n.2)). Related: Rutted; rutting.