It is relatively well-known that any arrangement of points that can be constructed with a straightedge and compass can also be constructed with an unstretchable string (of arbitrary length, negligible thickness/stiffness, etc.) However, a string is capable of generating a much larger array of curves (or surfaces) than the straightedge and compass are, and even more when multiple strings can be used in tandem. To my knowledge, there isn't a canonical set of rules for string constructions (as there are with straightedge & compass and origami constructions, although these rules aren't necessarily unique), but a provisional list of valid 'moves' is the following:
- Strings can connect a set of $n$ existing points in sequence, possibly visiting any given point multiple times, or forming a loop. The sequence of points may also include 'new' points (with unspecified/arbitrary coordinates) along the string itself. You can imagine each vertex as a 'frictionless knot' that allows segments of the string to slide between pairs of adjacent edges (or less trivially between edges that are all associated with the same string, when multiple strings are used.)
- Vertices/knots may be 'fixed in place' or allowed to move freely.
- The string can be contracted/pulled at one end so that the vertex path attains its minimum length (this minimum is unique because the objective function is convex in the zero stiffness case, where tension must be positive.)
- The string can also be pulled at a vertex at constant total length (or preventing 'influx/outflux' of slack from the terminal vertices) in a given direction, and by tracing the resulting equilibrium point (of any mobile vertex, or the pulled vertex in particular) over all possible pulling directions, one obtains a curve (or hypersurface in higher dimensions) that generalizes the 'circle' from the compass and straightedge rules.
- The intersection between two curves/hypersurfaces can be constructed by connecting two strings (with different knot structures and total lengths) at a common vertex, and either finding the equilibrium points of the vertex (in 2 dimensions) or tracing the equilibrium path/manifold in higher dimensions.
- A line can be drawn between any two constructible points, or more generally the trace of any minimum-length string configuration can be traced.
- Minimum length string configurations can be 'superposed' so that any pair of vertices coincide (and any pair of 'edges' align at that vertex.) For example, a constructible triangle can be rotated and translated so that one of its legs aligns with a constructible segment (at a vertex.)
Another rule that might be added in higher dimensions is the following:
- An $m+1$-dimensional submanifold (of $\mathbb R^n$, say) can be constructed from a one parameter family of constructible $m$-dimensional submanifolds (of $\mathbb R^n$.)
Now because the equations for these constructions all involve Euclidean distances and derivatives thereof, they satisfy a system of multivariable polynomial equations and can therefore be viewed in an algebraic geometric sense as varieties (over $\mathbb{R}$, say.) My question is, are there any special properties of these varieties, and where might I learn more about them?
