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I've finally realised what kind of topology Quadruple Besiege boards have. It's a torus with a twist. If you join the array shown end to end to form a cyclinder, you then curve the cylinder further as if to make a normal torus, but twist one end by 180° just before actually making the join. Not a Moebius strip, not a Klein bottle, but a torus with a twisted tesselation.

16.February.2011 the 47th day generates cv#47, Decimal Quadruple Beseige. Two moves a turn might be considered here for subvariant. There are so many pieces because the King can never be cornered, and in fact all pieces always have 8 adjacent squares -- same as to say each square has 8 squares adjacent, even if ostensibly at an ''edge.'' There is no edge. All the piece-types are regular because Ace is just Amazon. 'a6' joins to 't6', and so on, like a Cylindrical board. Player does not need to think of Torus or Torus with twist to understand the rest of the connectivity in order to play. For example, Pawn-e9 proceeds along one rectilinear pathway without capturing e9-e10-p1-p2-p3, where he promotes, assuming 'p2' and 'p3' had already cleared out. Gilman in the last comment describes Torus with a Twist, made after the evident Cylindrical board is formed, to complete the D.Q.B. connectivity. Think of extremely long cylinder, and instead of joining t1 to natural counterpart t10, twist 180 degrees, more or less, and then join to make the one finished Torus/donut intended here, by putting t1 orthogonally adjacent rather a10, s1 to b10, r1 to c10 and so on. Or is it in order k10, l10, and m10? What becomes is not an imaginary object at all but real board still having its entire surface covered, or accounted for. However, the playing surface is best visualized by human players exactly the way pictured in the article, not intruding functional 2-d board into 3-d perspective. (Pawn may have added capture mode not yet clear, so above subject to revision.) This cv for clarity, Gilman needs to reference some actual cells not just define leap lengths, which in the text are not all compatible with this account. There are other earlier cvs of related board concept that are better to learn before D.Q.B. of year 2009, and then D.Q.B. can be rated.

Okay this is mostly a correction before Charles or Ben. Okay, folding a newspaper confirms a1 slots into k10, j1 to t10, t1 to j10, k1 to a10, and so on. Continuing correcting the last comment, s1 meets i10 and r1 h10. The shift is a steady 180 degrees. (162 or 144 degrees could be another geometry.) Pawn of the last comment traverses e9-e10-o1-02-03 instead. The long leap lengths of the article are not useful compared to just seeing the geometry. Probably the leaps are all accurate, but only the Knight-components cannot be blocked. (Gilman once stated epiphanicly he finally ''saw'' Tetrahedral fully, so that lies ahead.) D.Q.B. wants it that way, that is all of its joins with the key 1/2 fold, because it is appropriate for Pawns to promote towards the opposition. Bishop still has same 4 directions. However, from departure to arrival square, Bishop can actually make it without two blocks, not one, strategically intervening. That is tantamount to saying Bishop is two-path to any square. In Cylindrical, Bishop is not two-path; whereas Torus with pi-twist, Bishop is two-path; and in Torus, Bishop is so sometimes only. Examples on Torus pi-twisted: (1) c6-d7-e8 (2) c6-b5-a4-t3-s2-r1-g10-f9-e8; (1) c6-d5-e4 (2) c6-b7-a8-t9-s10-h1-g2-f3-e4. So also is Rook two-path in Torus with pi twist. Can humans better than machines visualize these ''morphologies'' in order to move counters on the board sharply. Not likely. Now earlier cvs in their turn of related geometry can show how original the boards are with Gilman cvs, but for now ignore the D.Q.B. Intro links.