Here you go... traditional British knitwear designs applied to a sock.
Specifically, these are gansey patterns. Ganseys (a.k.a. guernseys) are traditional fisherman's sweaters -- English and Scottish rather than Irish. The patterning uses a wide variety of clever combinations of knits and purls, with a few small cables for accents.
Ganseys are usually worked in the round, but in a peverse twist I rather like, a typical design feature is a fake seam - a 2 or 3 stitch wide section of purls or seed stitch, to replicate the seaming of a garment worked flat. A gansey usually has strong, deep ribbing, and the patterning is often confined to the yoke. A defining ridge of purl sts is worked between the plain and patterned sections. And the plain section would sometimes have the wearer's initials worked into it. (Allegedly, this was so that a drowned fisherman might be identified. I don't know whether this is true or not, but it makes a lovely detail of personalization.)
The two classic books on these designs are Beth Brown-Reinsel's Knitting Ganseys, and Gladys Thompson's Patterns for Jerseys, Guernseys and Arans. I can't recommed them enough. The Thompson book is more of a catalogue of the regional patterns and design details. Brown-Reinsel's provides a good history of the design, and a full breakdown of the traditional construction elements.
I've long been a fan of these designs, and have been noodling on a gansey-inspired sock design for a while. And here it is... It's got a seam in seed stitch which runs down the leg, and splits down the heel and along the foot. It's got the traditional deep ribbing, a definition ridge, and a plain area with my initial. The patterning uses a small cable offset with garter stitch, and a classic "steps" and ridge combination.
And it comes in multiple sizes -- so it did require some math!