Is a plane parallel with itself?
depends on the definition you're using ...
"Two planes that do not intersect are said to be parallel." ... this definition would say (to me, anyway) a plane is not parallel to itself.
Parallel Planes -- from Wolfram MathWorld
... there is probably another definition somewhere which could be interpreted to say it is parallel to itself.
The answer depends on the definition of parallel one uses. Here are two.
- If $\Pi_1~\&~\Pi_2$ are two planes the statement $\Pi_1\|\Pi_2$ parallel means they do not intersect.
- If each $\Pi_1~\&~\Pi_2$ is a plane the statement $\Pi_1\|\Pi_2$ parallel means both are perpendicular to the same line.
#1 is the one Skeeter has given you. And that means that a plane is not parallel to itself.
#2 is used in vector space(analytic geometry). The wording is key. Under #1 we must have two planes.
But in #2 it is possible that $\Pi_1~=~\Pi_2$ so a plane is parallel to itself. Having taught mostly set theory&logic or vectors & geometry when I first read your question I thought. 'well of course a plane is parallel to itself. I am use to using set-theory textbooks that use parallelism as an example of an equivalence relation. Therefore a plane is related to itself(reflective). Moreover, we do want an element to belong to the equivalence class it determines.