# Comments/Ratings for a Single Item

You probably should mention that restriction explicitly. The article already considers in passing a board with two shapes of cells! A Dabbaba-averse square board has two different cell shapes -- square for the Dabbaba-binding a Ferz step from the removed one, pentagonal for the other two. The article covers at least five of the eleven Dual-Uniform tilings, which are my above list restricted to use only one kind of cell. Of the other six, at least the Rhombille and the Floret pentagonal tiling can be handled as averse boards.

It looks like the tri-hexagonal geometry is a kite that can fly (pause for groans). As well as the Rook curving round on itself, I notice that the Knight and Camel can triangulate. Assuming the three-player-variant rules for Bishop movement where six quadrilaterals meet, the Bishop and Ferz can reach every cell and the Ferz too can triangulate. The Elephant, Tripper, et cetera appear bound to single diagonals as there is no continuation of three-player-variant diagonal steps. Another group of pieces that look interesting in this geometry are those that are non-Straight in the square geometry (see Man and Beast 09). The Girlscout has a straighter path than the circuitous Rook, and alternates not between two Bishop paths but between one Bishop path and one path alternating orthogonal and diagonal steps. The Boyscout has a very strange path, bifurcating at each meeting of six cells. The Rhino moves seemes to get cut a bit short in its Crooked form, but takes a curious oblong circuit, still of eight steps, in its Curved (Leaping Bat Chess) form. Note that I am not considering geometries with cells of different shapes - should I say this explicitly, do you think?

With the addition of the 'pentagonal' boards, this article now covers five of the most familiar of the nineteen uniform and dual-uniform tilings. Several of the others -- Floret pentagonal comes to mind -- can be addressed using the AVERSE system, so coverage is likely much better than that.

I'd like to suggest the quadrilateral tilings on this list for further study. Quadrilateral tilings tend to suggest obvious interpretations of FIDE pieces, but when the quadrilaterals are not rectangles, those pieces may behave oddly. On the deltoidal trihexagonal tiling, the obvious Rook travels in a twelve-step circle.

While pieces may be bound to one of any number of mutually exclusive sets of cells, switching is always between two such sets - in the case of ranks, odd and even. The Bishop can move from odd to odd or even to even rank, as well as between the two, and so is not switching. Pieces that can move within a rank certainly do not switch ranks - although some like the Wazir switch other things. The pieces that always switch rank and file in 2d are the Ferz here, the Camel (and everything else -mel) in MAB 03, and the Bear in MAB 06 - pieces which always move an odd number of both. On a cubic board this is no longer the case as they can move within a rank. The matter of this page's compound pieces being unbound on a cubic board I hsve covered. The Primate, Pope, Besieger, and Usurper all have a Wazir move and so are clearly unbound. The Moderator and Heretic are unbound because they can move to an adjacent cell in two moves - by making a 1:1:1 move but retracting it in only two dimensions. All geometries' nonstandard diagonals have steps of length root-3 - the description asserting their common identity amounts to a root-2 and root-1 step at right angles.

Terms for the lead article will recur. Squares have SD, cubes have both SD and ND, and hexagons have ND only. Square diagonal one-step and cube diagonal one-step (whether SD or ND) are root-2. None of them have to be one-step only, they can keep going, and these are all root-2. [Correction added in these brackets later is that ND cubes of course is root-3. Gilman's comment above cites his text couple of sentences explaining the rationale for identity of all NDs.] However hex ND is root-3. Switching can be used describing ranks or files or differently bindings. EINHORN (UNICORN) is 100 years old with Raumschach. That summarizes MB01:CC. Eduardo Punset: Life requires imperfection. At what speed do the atoms move? Nicolas Garcia, nanotechnologist: In the atmosphere, the atoms move at 1500 metres a second. But in the case of this figure, they jump every nanosecond. E.P.: Is this really the essence of life? N.G.: The essence of life might be the possibility of finding places, defects, where atoms group together to form new structures.

Gilman's introductory vocabulary words are essential and many of them appear already here: standard diagonal (sd), nonstandard diagonal(nd), colourbound, (hex-prism), switching, triangulate, symmetric pieces, bindings and their independence, divergent, co-prime, non-coprime. They are all here, so get a grasp of them here, and all the follow-up Gilmans fall into place. The ones least likely to be understood on rereading, I will take up over the next comments, since we cannot cover all of them, though Gilman tends to shorten definitions, thinking readers fully understand from the context. Hey, that's good professorial practice to anyone remembering the real intent of introductory lectures, to weed out the less interested and able. What would you say by way of explanation or example about any of the 10 or 11 terms above? Here's a helpful hint. Think of Gilman nomenclature of piece-types as nonessential but for entertainment mostly instead. Then everything else besides naming piece-types, i.e. all the other terms, becomes important CV analysis in Man & Beasts.

I agree that the truncated octahedral system seems to complicate matters unnecessarily. I suspect there's an advantage hiding in there somewhere but I haven't found it. I would tend to consider the hexagonal direction the single step, and the square direction a funny sort of diagonal - but then the board has the likely undesirable property that no two of its eight main axes are perpendicular.

Hmmm. That is an interesting geometric property. Note also that in the coloring of a board for this partial knight, a cell adjoins each of the four other colors exactly once on the orthogonals, and once on the diagonals.

PS for 3-d visualization I suggest vZome. http://www.vorthmann.org/zome/ It's very good at the cubic and rhombic dodecahedral systems, I find. (It's not much good at hex-prism just yet, but I'm still talking to Scott Vorthmann about that.)

You're right about the piece that you correctly define as the MAB 14 Sexton. I have added something to that effect to the article. I can visualise two kinds of truncated octahedron. Both have 6 square faces but one has 8 triangles and the other 8 hexagons, depending on how far you truncate. It seems to be the type with hexagons that tesselates (please confirm or correct this deduction), and I have already considered hex prisms as a cell mixing hexagonal and rectilinear faces. Hex prisms, however, can be sized so that the distance between cells adjoining on both shapes of face are the same. With truncated octahedrons, the distance between cells adjoining on hexagonal faces seems to be root (3/4) times the distance between those adjoining at square faces. That calculation is based three lots of half the latter distance at right angles. If so it complicates things as not every SOLL is measurable as an integer multiple of the shortest. Another interesting geoimetric property that I have recently noticed is that oblique leapers restricted to alternate directions (so that all moves are at right angles) are (like radial leapers ordinarily) bound to the proportion of the square-cell 2d board reciprocal to their SOLL.

I see several principal radial directions, each generally with its own SOLL:

- Wazir-wise
- Ferz-wise
- Viceroy-wise
- Rumbaba-wise

Their unique characteristic, setting them apart from other directions, appears to be that as moved in space they do not move through cells other than the one they're leaving and the one they're entering (though they may pass along edges, as Viceroy, or faces, as Rumbaba).

In the rhombic dodecahedral board, I observed an interesting direction. (I tried reading Oddly Oblique to see if you'd named it there, but I couldn't tell; your construction of the pieces for a rhombic dodecahedral board was difficult to follow without more knowledge than I have of your cubic pieces and of your reducing process.) Its SOLL is 6, analogous to your cubic Sexton, and it can be found by making three mutually perpendicular Ferz steps (not possible in cubic) or equivalently a Ferz step and a perpendicular Dabbabah step. (I am guessing it is the one that Oddly Oblique calls a Sexton?)

The interesting property is that it, too, does not pass through any other cell. It passes along the edges of six other cells (three a Wazir step away, three a Viceroy step away) but the first cell it enters is the destination.

I've illustrated it here. The starting square is red; green is a Wazir step, while blue is a Ferz step. Yellow is a Viceroy step. The direction in question is included in purple.

Only distantly related, but also worthy of note, the family of shapes that brought us cubic and rhombic dodecahedral has a third member. The truncated octahedral tiling is as yet unexplored in chess variants, to my knowledge. It is found by removing three-quarters of the cells of a cubic; if we consider the chess pieces as residing on points instead of in cells, it is a body-centered cubic crystal system (as compared to a simple cubic, a hexagonal, or rhombic dodecahedral's face-centered cubic). I don't think its properties are as interesting as those of the other varieties, but it could be worth investigating.

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