Yet another weird SF fan
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 Yet another weird SF fan

### Four-Dimensional Undecidable “Elementary” Geometry

A few years ago, I realized (with another update here) that the elementary geometry of points, lines, and circles becomes undecidable when it includes screws or spirals. You can think of lines and circles as the one-dimensional connected uniform curves in a two-dimensional Euclidean space and you can think of spirals, lines, and circles as the one-dimensional connected uniform curves in a three-dimensional Euclidean space. I'm still not sure of what a complete set of such curves in a four-dimensional space would be like, but it would include some very strange objects.

For example, consider the curve parameterized by $$(w,x,y,z)=(\sin t,\cos t,\sin \sqrt{2}t,\cos \sqrt{2}t)$$ where $$t\in[-\infty,\infty]$$. It is easy to see that this is a dense subset of the Clifford torus that's the product of two unit circles centered at the origin (in the $$(w,x)$$ and $$(y,z)$$ planes). Unlike the similar curves in two- and three-dimensional space, this isn't closed.

Question: Would it make more sense to focus on closed, uniform, connected subsets of Euclidean spaces? In two dimensions that would include the empty set, points, lines, circles, and the entire plane. In three dimensions that would include the empty set, points, lines, circles, helices, planes, spheres, cylinders, and the entire space. In four dimensions …

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