33/ and it's an elementary geometric proof that the angle that the edge makes vs circumference A is the same angle it makes vs circumf B
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Replying to @MorlockP
34/ bc the circumference at point A is a parallel line to the circumference at point B (non Euclidean space, obviously!)
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Replying to @MorlockP
35/ to whit, the old "a line that intersects paralllel lines creates corresponding angles" proof (not axiom?) http://www.mathnstuff.com/math/spoken/here/2class/260/trans.htm …
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Replying to @MorlockP
36/ which still holds in the simple non Euclidean space we're talking about.
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Replying to @MorlockP
37/ So we can create the thread pitch we want with simple compass work (bisection is legal, trisection is not), but we can also get 60°
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Replying to @MorlockP
38/ and we can then align a straightedge along this angle and wrap it, and then we get a constant thread. Now it's a "simple" matter of
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Replying to @MorlockP
39/ building some sort of pantograph machine which can take these angles and straight edges off references and apply to spinning workpiece
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Replying to @MorlockP
40/ which is actually a tricky bit of design and machining, and will add inherent errors to output...but it is, in principle, doiable
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Replying to @MorlockP
41/ Now I want an NEA grant to fund me building such a thing. I'd put it in a gallery. Patrons can turn a crank and slowly advance cutter
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Replying to @MorlockP
42/ Which, over time, turns a rough piece of rod into a toolroom-quality screwthread.
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43/ "The Recreation of the Machine Age Among the Eloi"
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