George Duke wrote on Tue, Oct 19, 2010 03:23 PM UTC:
(I) Infinite the tesselations there are whatever the chosen polygon, Frolov said the last comment. Yet equilateral triangle is the right root expression. Just as OrthoChess is/has been square of squares, so it could be better equilateral of equilaterals,
http://www.chessvariants.org/index/displaycomment.php?commentid=25264. [ChessboardMath12 has 18 comments, so this one
. is put here of variety.] Drawn to left is 3x3x3 or 3^2
. . in adaptable notation. The sizes correspond squares to
. . . triangles: 9,16,25,36,49,64,81,100.... Square is 10x10
. . . . and triangle is 10x10x10 in 2-d. As Frolov intimates,
__*Classical Tetraktys at left above*__
which is more fundamental, square or triangle? Which is the portal out of dark forbidding one-dim to two-dim? Must not even superior Mayans have triangulated to achieve the 26000 earthly precession cycle accurately? How many fundamental pieces are there in more perfect triangles, regularly equiangular pi/3? Three (which suggests fundamental pieces equal number of sides in any uniform tesselation, for follow-up, 3 -> 3; 4->4...)
(II) Restrictive term 'triangulate' means something different in Chess play itself of course. All OrthoChess so far knows is that King or Queen triangulate, return in three, but variantists discover others. Now as sounding irony, natural basic triangular omni-Pawn, moving/capturing through side, on 8^3 or 9^3 or any of them, is unable to ''triangulate.'' The Pawn (who travels like Frolov's Rook here one-stepping) can neither quadrangulate nor quintangulate. Once that (Centennial Chess Steward-like) Pawn steps from his triangle, he can only circumnavigate in six. Far more likely if he returns, it is a backstep not an hexangulation.
http://www.chessvariants.org/index/displaycomment.php?commentid=25264
(I) Infinite the tesselations there are whatever the chosen polygon, Frolov said the last comment. Yet equilateral triangle is the right root expression. Just as OrthoChess is/has been square of squares, so it could be better equilateral of equilaterals, http://www.chessvariants.org/index/displaycomment.php?commentid=25264. [ChessboardMath12 has 18 comments, so this one . is put here of variety.] Drawn to left is 3x3x3 or 3^2 . . in adaptable notation. The sizes correspond squares to . . . triangles: 9,16,25,36,49,64,81,100.... Square is 10x10 . . . . and triangle is 10x10x10 in 2-d. As Frolov intimates, __*Classical Tetraktys at left above*__ which is more fundamental, square or triangle? Which is the portal out of dark forbidding one-dim to two-dim? Must not even superior Mayans have triangulated to achieve the 26000 earthly precession cycle accurately? How many fundamental pieces are there in more perfect triangles, regularly equiangular pi/3? Three (which suggests fundamental pieces equal number of sides in any uniform tesselation, for follow-up, 3 -> 3; 4->4...) (II) Restrictive term 'triangulate' means something different in Chess play itself of course. All OrthoChess so far knows is that King or Queen triangulate, return in three, but variantists discover others. Now as sounding irony, natural basic triangular omni-Pawn, moving/capturing through side, on 8^3 or 9^3 or any of them, is unable to ''triangulate.'' The Pawn (who travels like Frolov's Rook here one-stepping) can neither quadrangulate nor quintangulate. Once that (Centennial Chess Steward-like) Pawn steps from his triangle, he can only circumnavigate in six. Far more likely if he returns, it is a backstep not an hexangulation. http://www.chessvariants.org/index/displaycomment.php?commentid=25264