The other day, I came up with a problem, but I do not know how to approach it with a solution.
The problem:
Suppose, I have a string dipped in paint. Now I place that string onto a piece of paper arbitrarily and pull one end of the string in a fixed direction via a fixed point on the paper. (Note that the point and the direction is fixed for all such trials.) What are some properties of the geometry so traced out by the paint-smeared string, that remains invariant no matter which way I place the string?
Here is a YouTube video for a rough idea of string paintings.
Here are some concerns regarding the problem:
- Is this a valid problem?
- How can one approach such a problem?
- Are there any generalisations that can be made without changing the properties of the plane resting underneath the string?
I am an undergraduate and have a background in linear algebra (Hoffman), analysis (Baby Rudin), abstract algebra (Herstein) and general topology (Munkres). Thanks in advance.