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http://www.chessvariants.org/index/listcomments.php?subjectid=ChessboardMath10
has Tetraktys and more triangles.
         .        Flowerman inquired about small Chess, and now we
 I      ._.       have a wholly new geometry. Triangles, out of
II     ._._.      ancient Tetraktys, nine-celled. F.i.d.e.-64 works
5-9   ._._._.     fine in equilateral triangles. All F.I.D.E. rules
IV   ._._._._.    apply respecting stalemate, touch move, 50-move,
 V  ._._._._._.   etc. Opposing Kings start at two corners 50 and 64.
VI ._._._._._._.  Castling is two squares towards the safer center.
  ._._._._._._._. Black King goes to cell 26, and whereever Rook 
 ._._._._._._._._.is arrayed, Rook then goes to 37 if not already 
(Level VIII, 50-64) there. Either 'King 50-37' or just '0-0' is Castle,
and can be said ''King Castles'' and then ''Rook BUNKO LEAPs.''
King towards corner is called Bunker Leap in CVs, but here it's well Bunko
Leap for Rook instead (likely) going towards corner. They must not have
already moved, nor King be or move through check. Otherwise, F.i.d.e. Royal King one-steps through any side or vertex along a Knight half-path. Realize again Knight is straight as an arrow in triangles, either Dabbabah-like or Alfil-like: 
http://www.chessvariants.org/index/displaycomment.php?commentid=25236.
Finally, six Pawns, complementing the eight lined RRNNBBQK, one-step omnidirectionally, like CENTENNIAL Chess (#13 Next Chess), with same divergence of side without and vertex with capture. No promotion at common cell 1, Black promotes at 4,9,16,25,36,49,64, and White's promotion cells are all opposite. A Chess VARIANT of Triangular F.i.d.e. follows. If principle of mobilization is agreed on, it should go in order as: Black places 1/2 pieces, then White 1/2, Black 1/2, White 1/2, White Pawns, Black Pawns. No Pawns above Level IV, and no pieces within the interior triangle 54-21-60. Besides ChessboardMath10 above, -11 has triangles: http://www.chessvariants.org/index/listcomments.php?subjectid=ChessboardMath11.