Three German typefaces appeared within about a year of each other and defined a category that is still with us. Jakob Erbar's Erbar came first, in 1926. Paul Renner's Futura and Rudolf Koch's Kabel followed in 1927. All three drew letters from the compass and the ruler — circles for O, C and G, triangles for A and V, straight strokes everywhere else — in the spirit of the Bauhaus conviction that form should come from construction rather than from the pen.
Futura won. Its perfectly round O and pointed apexes became shorthand for modernity, and it travelled further than any of its contemporaries — eventually to the Moon, on the plaque Apollo 11 left behind in 1969. But the pure geometric sans has a well-known flaw: circles and straight lines don't set comfortably as text. The letters fight each other for space, and long passages feel mechanical.
The second generation fixed that by cheating, gracefully. Herb Lubalin and Tom Carnase's ITC Avant Garde Gothic in 1970 kept the geometry but pushed it toward display, with famously tight spacing and ligatures. Adrian Frutiger's Avenir in 1988 went the other way: he described it as a geometric sans drawn by a human hand, with an O that isn't quite a circle and strokes that thin where they join. Avenir is the ancestor of nearly every "humanist geometric" that followed.
The current wave — Gotham, Circular, Figtree and their many relatives — inherits Frutiger's compromise and adds large x-heights tuned for screens. Vigil sits inside this tradition and makes one more adjustment: it treats the corners as a variable, not a constant.
If you want the practical version of this — how to recognise a geometric sans on sight, and which ones are worth using — that's the field guide.