For example, a triangle with side lengths 2, 3, and 1.5 can only have one shape, whereas a parallelogram with side lengths 2,4,2,4 can be a rectangle, or a rhomboid, or you could keep pushing it down from a rectangle, to a rhomboid, to a rhomboid without much height, to almost a line segment, to a line segment.
Also: Is there a name for thinking about geometry this way - a field/subfield of math that discusses these issues? Or some terms to look up?
For example, is there a name for this ability to "fold down" or crush a polygon into a line segment, or a line segment with a triangle sticking up out of it? I figure that all polygons can be crushed into either a line segment or a line segment with a triangle somewhere along, without lengthening or shortening any side. But I don't know if that's proven. Instead of posting a whole bunch of questions about these issues, it would be great to be able to google some terms instead.
If you wanted a name for this based on real life, it might be something like "scaffolding math" rather than "origami mathematics", since it presumably parallels at least one reason scaffolding (and other structures) get diagonal bracings to convert an unstable parallelogram into two stable triangles. But it could cover questions of objects made of connected line segments that can be rotated around one another at the joint (i.e. Where the angle between two line segments formed at the connection point can vary.).
I also don't know if this quality of collapsability for all but triangles is a 2D case of some phenomenon that goes up and down through other dimensions. The closest I could think of was that one can use this foldability/collapsibility idea to collapse crates into something close to a plane segment after the top and bottom are removed, without needing to remove or separate any other sides of the crate. Or that it feels similar to the idea of constructing a net that has no filled in areas when looked at from a certain angle after the final polygon is assembled. (e.g. One could make a simple net of three squares in a straight line that could have its edges joined to make a triangular prism with the two triangle shaped sides missing, which would then stand up rather than collapsing.)
I've tried googling around a bit, and have not found much of use. Wikipedia, for example, does not appear to me to list collapsibility as a way to categorise polygons: https://en.wikipedia.org/wiki/Polygon#Number_of_sides
There's mathematics like this that doesn't seem to me to go beyond the purely practical into abstraction: http://homesteadlaboratory.blogspot.com/2014/06/gate-brace-math.html
Perhaps what I'm looking for is sort of the mathematics of 2D origami (using connected line segments in 2D space rather than origami using connected planar segments in 3D space)? Perhaps topology comes into this?
Have not found anything at a scan through https://en.wikipedia.org/wiki/Mathematics_of_paper_folding
Also curious as to whether any theorems use the idea of considering a line segment as a collapsed polygon to work out something else. Or of a triangle as a bump on a line segment with both triangle and line segment made of a collapsed polygon.