The Moser's worm problem [springer link] asks for the region of smallest area that can accommodate every plane curve of length 1. Curves can be rotated and translated and may be considered identical upto rotation and translation transforms.

What I ask is

How many such curves of there, ie What is the cardinality of the set of 'worms' as referred to in Moser's worm problem?

  • 4
    $\begingroup$ If I ever publish a paper in which I introduce a construction I refer to as a `crab' and discuss a problem about this, please do not ask about Jagy's crabs. $\endgroup$
    – Will Jagy
    Nov 14 '13 at 5:01

The cardinality of the set of plane curves of length 1 is the same as that of the continuum. It is $\le$ the continuum because every such curve comes from a continuous function $[0,1] \to \mathbb{R}^2$, and such a function is determined by its values on the rational numbers. To see that it is $\ge$ the continuum, consider curves of constant curvature (sufficiently small such that they do not form a complete circle.)

The same argument works for equivalence classes of plane curves under rotations and translations, because curves of different constant curvatures are not equivalent.

  • $\begingroup$ That explains it, will the cardinality increase if we remove the equivalence constraints of rotation and translation. $\endgroup$
    – ARi
    Nov 14 '13 at 6:33
  • 1
    $\begingroup$ @ARi No, it would still be the continuum. This is what the first paragraph of my answer shows. $\endgroup$ Nov 14 '13 at 15:04

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