Today I read a paper titled “Topological robotics: motion planning in projective spaces”

The abstract is:

* We study an elementary problem of topological robotics: rotation of a line, which is fixed by a revolving joint at a base point: one wants to bring the line from its initial position to a final position by a continuous motion in the space.*

*The final goal is to construct an algorithm which will perform this task once the initial and final positions are given.*

*Any such motion planning algorithm will have instabilities, which are caused by topological reasons.*

*A general approach to study instabilities of robot motion was suggested recently by the first named author.*

*With any path-connected topological space X one associates a number TC(X), called the topological complexity of X.*

*This number is of fundamental importance for the motion planning problem: TC(X) determines character of instabilities which have all motion planning algorithms in X.*

*In the present paper we study the topological complexity of real projective spaces.*

*In particular we compute TC(RP^n) for all n<24.*

*Our main result is that (for n distinct from 1, 3, 7) the problem of calculating of TC(RP^n) is equivalent to finding the smallest k such that RP^n can be immersed into the Euclidean space R^{k-1}.*