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George Duke wrote on Wed, Jun 10, 2009 06:13 PM UTC:
Constitutional characters are the basic radial pieces. They pass through ''centres'' of squares in straight lines either orthogonal or diagonal. Knight, for example, does not belong here because in going from (0,0) to (1,2) square, there is an intermediate square whose ''centre'' the Knight does not pass throught in the completely direct route. The ''centre'' is for convenience, since the chess piece has to be standing somewhere. From the centre of any square (0,0) to the center of square (1,1), we can make a right triangle for the Ferz, and that one-step Bishop goes from (0,0) to (1,1) a distance based on Pythagorean theorem (1^2+1^2) and take the root for the hypotenuse, root-2. Thus, longer one-step for Ferz 1.4 than Wazir 1.0. There are three main geometries for boards (1) 3-D cubes like blocks one upon another (2) 2-D hexagonal spaces with no gaps (3) regular squares in rectangular board. In hexagonal, unlike squares, all the adjacent hexagons are 1.0 away from centre to centre. (In squares we always have to distinguish 1.414 and 1.0.) The second hexagonal ''orthogonally'' is 2.0 units away for convenience. But after any hexagon one-step, thee are two more hexagons sideways forward outward. Either one is the first step of a Gilman ''nonstandard diagonal'' directly from the original square without the pass-over square; Wellisch and others use the concept before. What is the distance of the first step of the nonstandard diagonal compared to the simple hex one-step across a side? Each interior angle of regular hexagon is 120 degrees. Geometry based on that makes several triangles 30-60-90, and we quickly realize that Gilman is right, the answer is root-3. So the distances across hexagons of the two logical moves precisely are 1.0 and 1.732 ctr-to-ctr. Root-3 is nonstandard diagonal one-step (it's not a jump), and 1.0 is the normal one-step hexagonally. There is no standard diagonal in hex because there is no literal point they both touch.

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