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According to Augustus De Morgan (quoted in "A Long Way from Euclid" by Constance Reid), Euclidean ruler-and-compass contructions (based on lines and circles) should have included helices.
The elementary geometry of points, lines, and circles is very different when helices are added. The former is decidable, also covered in "A Long Way from Euclid." OTOH, the latter is undecidable. It includes a model for the Peano Axioms: the intersection of a line and a helix.
BTW, does anybody know if this is original? It seems obvious but I haven't seen it anywhere.
@jhertzli this sounds cool, but I can't even think what a helix looks like on a plane, or how a compass and straightedge could have made one.
What kind of connection does this have with Peano? Might be out of my depth at this point.
@birdman According to Augustus de Morgan: "Every inch of a straight line coincides with every other inch, and of a circle with every other of the same circle. Where, then, did Euclid fail? In not introducing the third curve which has the same property---the screw. The right line, the circle, the screw---the representation of translation, rotation, and the two combined...ought to have been the instruments of geometry."
@jhertzli @birdman https://archive.org/details/longwayfromeucli005621mbp Did de Morgan (or Reid) actually give rules for Euclid+helices, or were helices just an ad hoc proposal to trisect angles and such? In particular, wow did he propose specifying *which* intersection point of a helix and a line to use in a straightedge + compass + screw construction? What exactly is the undecidable problem?
@jeffgerickson @jhertzli this is essentially what I wanted to know too. I'm curious to see how far in depth this went