What is the meaning of the word "plane"?

Definitions:

a carpenter's hand tool with an adjustable blade for smoothing or shaping wood noun
- the cabinetmaker used a plane for the finish work
a level of existence or development noun
- They lived on a worldly plane but dreamt of ascending to a more spiritual one.
an aircraft that has a fixed wing and is powered by propellers or jets noun
- the flight was delayed due to trouble with the airplane
in mathematics, an unbounded two-dimensional shape noun
- We will refer to the plane of the graph as the XY plane.
make even or smooth, with or as with a carpenter's plane verb
- plane the top of the door