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New 3D Leapers. Leapers restricted to the diagonal and triagonal patterns of the 3D field.[All Comments] [Add Comment or Rating]
Charles Gilman wrote on Thu, Sep 2, 2004 07:20 AM UTC:Having studied these ideas offline, I can see your 'simple' diagonal-pattern pieces for the compounds that they are. In cubic-cell geometry, all the orthogionals are the same angle to each other; the diagonals are not. Turning between the move of two steps diagonally and that of 1 differs immensely depending on whether the turn is through 90°, 120°, or 60°. The first is indeed the dual of the usual Knight's move, and so a piece with such moves is indeed a Camel. The second is the dual of the move defining a hex Knight in Glinsky terminology, and the third that of the move defining a hex Knight in Wellisch terminology (which Glinsky would presumably consisder a Ferz). Would you consider the 'true' hex Knight (if such a thing exists!) the piece combining the moves of both hex pieces?
Charles, Your application of angles is in relation to the orthogonal pattern. If you wish to be bound by such restrictions, feel free. In this simple treatise, the diagonal and triagonal patterns are being considered as seperate entities. And all their planar and cubic potentials are thus expressed in such regard. Allow me to state that these suggestions are not to be taken as absolute, either in their regulation or nomenclatures. They are primarily attempts to aid developers in thinking outside the box. The logic of the 90 degree restriction collapses when applied to the cubic move within the diagonal pattern and with the simple planar move within the triagonal pattern.
Here's an example of a d-Ferz, or a 1x2x2 planar piece in a diagonal pattern: [ ][ ][ ][ ][ ] [ ][X][ ][X][ ] [ ][ ][X][ ][ ] level 3 [ ][X][ ][X][ ] [ ][ ][ ][ ][ ] [ ][X][ ][X][ ] [X][ ][X][ ][X] [ ][X][ ][X][ ] level 2 [X][ ][X][ ][X] [ ][X][ ][X][ ] [ ][ ][X][ ][ ] [ ][X][ ][X][ ] [X][ ][O][ ][X] level 1 [ ][X][ ][X][ ] [ ][ ][X][ ][ ] In relation to the orthogonal pattern, there are several apparent 'leaps'. A developer might disallow those planar moves which result in adjacent diagonal cells and create an interesting new leaper for the 3D playing field. A composite of an o-Dabbabah and an o-Hippogriff. Or accept all its potential planar moves and thus have the o-Ferz moves also present. Most interesting.
Charles Gilman wrote on Mon, Sep 6, 2004 07:38 AM UTC:Good ★★★★Having looked into thesse pieces further, I am starting to warm to them. I notice that where the orthogonal-oblique piece is a two-coordinate leaper, m:n say, with a SOLL of m²+n², the diagonal-oblique piece has the three SOLLs {2(m²+n²[±mn])}, that is to say, the SOLL of the of the element of the di-ob piece that is itself another or-ob, and the numbers 2mn either side of it. For example, the standard Knight is a 2:1 leaper, SOLL 5, so the SOLLs of the d-Knight are {2(5[±2])} = {10, 6, 14}. 10 is the Camel SOLL, twice that of the standard Knight, and 6 and 14 are twice the 3 of the Wellisch hex Knight and the 7 of the Glinsky hex Knight.
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