Comments/Ratings for a Single Item

Then you'd probably want to try such things as the regular tiling of the
octagon and square. And why not play on the centers and intersections of
the tilings? The octagon and square would give quite a scope for pieces and
effects, even without using edges. I've found trying to use edges, too, is
overcrowded if not overkill.

When playing on borders, how do you define an adjacent location?  Is it a
border that shares an endpoint with the current one?  I suspect a lot of
these ideas will wrap around onto one another (for instance, the duality
between triangular and hexagonal boards, the self-duality of rectangular
boards, etc.)  I think also that when you pass between these viewpoints,
some of the pieces may be defined differently from their standard.  (That
is, if you define what you think pieces should do in a hex corner game,
perhaps they will move differently than what their counterparts in
triangular chess usually do.)  This may or may not be a good thing.  I
generally like it, but it's nice too if they work out to be consistent.

I've used the corners of a square-gridded board to increase the density
of locations on a gameboard - you get 2x the spots to move onto for no
physical increase in size. As far as basic geometry, all this does is
shrink the squares and rotate them 45 degrees. The connectivity is the
same. It works like this: a square has 4 'nearest' neighbors, the 4
orthogonally adjacent squares. But a square also touches another 4
diagonally, so it has 8 touching neighbors. The diagonal neighbors are
farther away than the orthogonal ones by a factor of 1.4... but there are
still 8 neighboring/touching locations, a 2 different distances, 1 & 1.4.

Analogously, we look at hex boards and see 6 nearest neighbors [distance
of 1], and no others touching. We see 12 next nearest neighbors, all at a
distance of 2. But we see that some hexes [6] are orthogonally and
linearly connected to the center hex, but others [6] are not. They are connected in a zig-zag pattern of hex center to hex center, and a straight line passes between other hexes, along hexsides. This is what we consider a 'hex diagonal', and it works analogously to 'square diagonals'. 

Look at this board: http://chessvariants.wikidot.com/universal-board
This uses the octagonal-square tiling. An octagon has 8 nearest neighbors,
and a square has 4. If you use figure centers and intersections, the
octagon has 16, with the 8 intersections becoming the 'nearest' [in
physical, measured distance between figure (geometric) centers and
intersections], and the 8 centers now representing 'diagonals'. The
square here gets 8, its 4 corners as the 4 nearest, and the 4 octagons it shares an edge with. If you look at the square-octagon 'edge line diagonal', analogous to the hex 'diagonal', you see there are 4 squares at the ends of the 4 lines radiating from the corner of any square. 

Now look at the octagons, using the octagon-square 'edge-diagonal'. Each
octagon has only 4 edge-diagonal neighbors to go along with its 8
'orthogonal' neighbors, for a total of 12 possible moves, if I'm figuring
this right. [Heh, I'm not sure I want to read what I just wrote!] Anyway,
that's only 3/4 of the 16 you get from using figure centers and vertices,
so this pattern doesn't transform. [Lol, Ben, rescue me here...]

One way to very concretely describe piece movement on a given board is to
use a (combinatorial) graph:  each vertex is a location available, and
there are several types of edges between these vertices.  Each piece is
allowed to move from vertex to vertex, provided that there is an edge of
the appropriate type between them.  This is good for simple pieces, but
becomes a little complicated even if we just want to allow sliders.

So the question is how do our traditional notions of 'topological' boards
translate into actual game mechanics, i.e. graph play.  The octagon-square
tiling that Joe has presented brings up some interesting questions.  That
type of tiling allows us to choose different sizes for the sides of the
octagon, so we can make the squares larger or smaller.  It seems most
natural to have all edges the same length, but do different side length
promote (in our mind, looking at the board) different movements?

Ah, that reminds me of another thing I was thinking about.  Start with a
triangular board, then consider playing on the edges.  Each edge has six
adjacent edges, but two of them lie along the same line as the given edge
(and are, under the Euclidean metric, further away).  So we have two
reasonable ways to play.  We can literally treat all adjacent edges as
'orthogonal' moves, which I think should turn the game into fairly
standard hex movement.  Or we could exclude these two funny edges, perhaps
making them into a new type of move.  Then we have a hex game which singles
out certain kinds of orthogonal moves as special (but these special
directions don't seem to be universal; without a drawing I'm having
trouble seeing how they work together...)

Oops, that triangular-edge comment was mistaken.  An edge has 10 adjacent
edges!  There are the four 'closest' ones and the two weird ones I
mentioned, but there are also four more, between the first four and the
weird two.  So I guess playing on edges purely by adjacency can create
weird games...