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64-   .            Artificially, Squares have that extra
 cell  ._.           unnecessary side, making four.  Sans
board ._._.  2-4     rank and sans file, Triangles have Cell
     ._._._. III     and Level, redundantly for convenience.
    ._._._._. IV     Corner Bishop triangles are 1,2,4,5,7,9,10...
   ._._._._._. V     With one compulsory horizontal side, there are
  ._._._._._._.      only two orientations and so two types of cells,
 ._._._._._._._.     up and down.  Corner Bishop's are /_\ and
._._._._._._._._.    Interior Bishop's the opposite, down, at a 
(Level VIII 50-64)   glance. For annotative intricacy, if (Cell + Level) is even, it is corner Bishop cell. Such as 'Level VII Cell 48' is odd and
belongs to interior Bishop. Interior Bishop reaches 28 and corner Bishop
36. That fits in that the board is 8^8^8 and 36-8 = 28. How about a
colourbound Knight on these primitive superior equiangular triangles? 
Rather than two Knight-types, additionally allow Knight diagonal through
vertex one-step at option -- making one pure Knight piece-type and eliminating
the prospect of detracting colourboundedness.  The only other uses of that
Knight-vertex-diagonal one-step are King himself and Pawn capture-mode, since Bishop (and Queen) has different actuation. Then Knight, like Rook, reaches all the cells and becomes of somewhat more value than either Bishop in triangles.  R > N > corner B > interior B. That the omni-Pawn has higher value than interior Bishop is yet unconfirmed, but they are pretty close. The better CV here appears to dispense with one Rook, as KQRBBNNPPPPPP. Six Pawns and
one Rook.  Promotion to R,N,B at opposite side excluding the Cell 1 Level
I. Kings start at 50 and 64. Array-placement mobilization is by order:
Black put 1/2 pieces, White 1/2, Black 1/2, White 1/2, White Pawns, Black
Pawns. White moves first. All else strict standard F.I.D.E. -- sans squares.