Does "center" always mean "half"?

I'm spending the next couple months trying out some new non-dance, moving- and body-scale math lessons for my new book.  There's an activity in Mathematics Their Way (from the '70s) that I want to try out.

The lesson is ostensibly about using your whole, moving body to find the "middle" or the "center" of a space that you can't fold. I was thinking I'd start my version of the lesson by asking first and second graders "What is half?" I planned to give them a piece of 8 1/2" x 11" paper and see what they come up with.

Except, as I was getting the materials ready today I noticed some paper cut into big triangles. I took one up to my 10yo who was doing her homework. I asked her to fold it in half. This is what she did.

 Then I asked her to fold it into half again. The image below illustrates what she was doing for a split second before she said..."ah...no..."

At which point she said, "I can't fold it into half that way. But I can fold it into fourths."

I tried to get her to tell me how she knew when she had folded it in half, but she wasn't interested in talking. But it did make me wonder how I might fold this triangle into eighths.

Then I thought: Wait a minute! We've folded this triangle into eight one-eighths and still haven't found its center/middle. Or, well, maybe we did on the first fold ... but maybe there's another center we haven't yet found?

I love it when what I think I know is challenged. We need to do this for our students on a regular basis.

And, I'm rethinking my lesson for tomorrow. I'll focus on the language of "middle" or "center" and use rectangular and triangular paper. But "half"?  Half I'll save for another time.