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Actually, there is infinitive number of possible triangle tesselation: any polygon can be divided into triangles (number of triangles is same as number of i's sides). In all triangle variants i know this polygon is hexagon, in tesselation i suggested, it's rectangle. And large triangles, composed of triangular cells can be arranged in same ways as cells. But there are two most interesting tessellations, i see: the own that i described before and 3 triangles arranged in one large triangle, 3 such large triangles are arranged into larger triangle, and so on (it should have 27 or 81 cells).
Another close eqivalent of standart chess: bishop is same, rook moves as spire ( http://www.chessvariants.org/index/msdisplay.php?itemid=MSdelta88chess ), knight moves as compound of my knight and ninja. Other pieces are based on them.
Another interesting triangle variant would be board, consisting of several suares, and each square consist of 4 triangles, and each triangle is playing space. I have not thought about pieces for such board yet, but idea is interesting.
This is good and the best way to use triangles to advantage in a board along with corresponding 9x9x9. Here is what I did with the same 8x8x8: http://www.chessvariants.org/index/displaycomment.php?commentid=25264. The pieces and type movements need to be compared in the two efforts as time allows. The progress also suggests in roundabout way that we should attack 64 squares as ridiculously obsolete and narrow of past centuries. Frankly standard 64 squares is considered a waste of time to vast majority of its former core audience of elite -- very much including the recently educated, one and all seeking ultimate mental challenge. That percentagewise decline holds from Brazil to Russia to Australia notwithstanding manufactured Internet numbers that can be generated. Such an argument is already spread around the CVPage and would need organization, unable to justify in few sentences of a comment or two. As the present instance, Triangles, once settling on the correct triangle boards, 2 or 3 at most, then the lesser products some things like 6x6x6 or 10x10x10, or mix of squares and triangles, or right angle triangles -- these abortions become as uninteresting as they are inferior. Do them in the privacy of your home if you have to. Or enter them into popular culture at risk of ridicule. So now, following through on Triangles as themed example, every design of different creative win condition or piece mix will naturally follow suit, staying with or reluctantly varying from triangular 8^3 and 9^3. There are still millions of 8x8x8 (of triangles) to deal with. NextChess itself, soon to include a hexagonal, and already having a 3-player, for two irregular examples still legitimate next chesses, as defined, is sustained by this breaking away from pure artwork as such towards logical irrefutable scientific directionality.
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