Dr. Barbour,
thank you for thought provoking essay. I read it today the second time, because of Daniel Waagner Fonteles Alves, whose essay I found more accessible than yours. The concept of shape dynamics is new to me but my immediate instinct is that this is the right way to go.
You write, " A closed geometry is needed to model a universe as a whole. A closed three-geometry is much harder to imagine, but is mathematically possible. "
I happen to visualize 3-geometry very well, even though call it 4D geometry, because I insist on considering not just the 3D surface, but the 4D object as a whole. My approach to physics is visual and, in line with your holistic approach, I see the universe as a hypersphere, on the 3D surface of which we live. I found that this 4D perspective dispels the paradoxes that plague contemporary physics and makes the workings of the universe, from its smallest components to largest structures, appear to make perfect sense.
I especially appreciate your holistic view, because I conceive of space, energy and time as 3 aspects of one and the same, a process, with either one being the expression of the other two. To me it appears that shape dynamics is the best way to model such processes evolving and interacting locally.
I wonder if Alpha shape is a hypersphere -? because, topologically, 4D allows for most symmetries in comparison to all other spaces, and as a space with even number of dimensions, also permits more rotations than the next runner-up, 3D.
I very much hope that you would find time to comment on my essay, even though the ratings are already closed ( http://fqxi.org/community/forum/topic/1547 ) I would very much value your feedback.
Congratulations on making the finalists list!