Thanks to both Francesco Montesano and Benjamin Root.  I have done some 
reading.  And I have made some progress, though I am not quite where I 
want to be yet.

So the problem appears to be that the only backend for which I had 
suitable Python 3 libraries was agg.  It only requires libpng, which I 
have.  I can render a Matplotlib canvas, but it appears that the only 
output that agg offers is in the form of PNG files to disk.  I cannot 
create a live window on the screen.

Reading more, I realize that the way I was getting GUI output previously 
(with Python 2.7 and Matplotlib 1.1) was through wxPython.  
Unfortunately, it appears that wxPython's star is fading, and a Python 
3-compatible version will not be written.  In fact, wxPython hasn't 
released a new version in nine months.

The other choices for Matplotlib GUI output on Linux appear to be 
through GTK, PySide, and PyQt.  I am not familiar with GTK, but I know 
that it is widely-used.  Also, GTK appears to be Python 3-compatible, 
and so that is where I need to go.

I'm going through a trial and error process.  Unfortunately, the names 
of the repositories in Ubuntu are not very helpful.  I installed a few 
GTK and python-gtk related packages that I thought were relevant.  On my 
first build attempt I got no errors, but also, I didn't get a GTKAgg 
backend.  Upon re-reading, I saw that I should modify matplotlib's 
setup.cfg file to force a GTK build attempt, and to report errors if it 
fails. That's what it does.  In the "optional backend dependencies" 
section I am not seeing any GTK libraries listed, even though I have 
installed python-gtk2-dev (2.24.0), python-gobject-2-dev (2.28.6), 
libgtk2.0-dev (2.24.10), libglib2.0-dev (2.32.3), python-gi-dev (3.2.2), 
python-gobject-dev (3.2.2), python3-gi (3.2.2), and a few DOZEN packages 
on which these depend.

If anyone knows the way forward from here, I would appreciate your 
advice.  Thanks again.

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