When most of us think of geometry we either get a headache or our eyes glaze over. It's right up there with reading Beowulf as the most exciting intellectual pursuit most artists could possibly look forward to. Well, I'm not sure if he ever read Beowulf, but Leonardo Da Vinci, among his many other artistic, scientific, engineering, and mathematical pursuits, certainly found it interesting. Being a painter, he explored ways in which he could utilize his interest in lines, angles, arcs and all the other esoteric details of the subject in planning and drawing his faces and figures. I guess not too many of us think of figures and geometry in the same context, but I guess that's what made Leonardo a legendary intellect in his own time.
Geometry is more readily associated with the art and science of architecture, and Leonardo certainly had a strong interest in that. In the sixteenth century, there wasn't much in the way of books to study on the subject, except for the writings of the Roman Architect, Vitruvius. Vitruvius lived and worked in the first century BC. But more important than his buildings were his writings. It was through these that one of Leonardo's most famous drawings came to be. Perhaps you can recall it without searching it out. It's a nude male figure, arms outstretched, with what appears to be four legs and four arms around which has been inscribed a perfect square and a slightly larger circle. Leonardo called it the Vitruvian Man.
As Vitruvius described the human figure in geometric terms, if one places a compass point at the navel (ouch) of the full-grown adult male (not sure if this holds true for females or not), then places the pencil at the feet of the figure, the circle created by this configuration will also touch the tips of the fingers of the outstretched arms when raised to the level of the top of the head. Similarly, he discovered that a square could be drawn using height of the figure as one axis and the horizontally outstretched arms as the other axis. Thus, ones height (you don't have to be nude but no fair wearing shoes) equals the distance between the two opposite middle fingers when the arms are fully extended. Try it sometime, it works (give or take an inch or so).