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I am going to have to get back to you after I have examined this in more detail offline.
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.
Yes, you've correctly identified the truncated octahedron with the desired property. Its hexagons are regular. (If you truncate all the way down to triangles, you have the cuboctahedron, dual to the rhombic dodecahedron.)
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.)
Like each Betza, each Gilman could take a monograph. (I) In the beginning (we are going to review all 20 Man and Beasts CCs) ''The Rook's directions are called orthogonal,'' he writes. Already CVs have their own vocabulary, because nowhere else is orthogonal used quite that way, being reasonable enough adaptation, though Rook's move has no right angle as such, until there are two. Bishop is just as ''orthogonal'' really, whose moves are perpendicular. Are mega-city street patterns orthogonal? Yes, because there are more than one direction at once, and rectilinear or straight are more current, or just square for the blocks. Radical philosopher Georges Sorel (1847-1922) pointed out that deliberate re-laying of streets by authorities to square-patterns in 19th century Paris was designed to thwart, or contravene, anarchists and revolutionaries, who could no longer easily hide out in corners, byways and alleys to terrorize, behind that curved dark lurking corner. (II) Ending the article under ''Notes'' here Gilman links *Fusion Chess* that is now being analysed by Jeremy. Gilman's Primate is Duniho's Pope. Maybe. Primate seems to be Bishop + Wazir, whereas Pope is Bishop + King. Could Charles please clarify. ''Primate'' of course is the clerical term, not the biological one.
Thanks for keeping me on my toes, I can see how ambiguous 'the same' can be now that you put it that way. I have modified that sentence and hopefully removed any ambiguity.
Further to my last adjustment I have also clarified what I understand to be the reason behind the term 'orthogonal'. I have also moved the definitions of the basic types of radial direction to the introduction, to mark them out from the definitions of individual pieces.
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.
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.
Colourbound. Obvious for Bishop and Ferz on squares, which have two bindings for them, dark and light. Hex has three bindings, because whatever hexagon your piece is on, there are three directions not two. Cubic has 3 planes and 6 standard diagonals, because opposite edges are one and the same diagonal, and there are 12 edges per starting cube. Cubic has 4 nonstandard diagonals because opposite corners are one and the same diagonal (going the opposite directions), and there are 8 corners (vertices). Bishop in cubic passing through edges the six ways possible can still never reach any cube sharing a face; therefore there are two bindings in cubic for Bishop. In other words, you'll need another Bishop to reach the other half of cubes. In nomenclature of piece-types, Unicorn is the nonstandard cubic-diagonal logical mover (triagonal) through vertices; and there are four bindings, as experimentation shows with a dozen blocks on your work table. This is more important stuff than Betza's Chess Unequal Armies for the 22nd century. Like proliferation itself, CUA has no possible logical end, whereas basics of colourboundness and bindings, divergence and symmetry, are being universally applicable on the 3 boards: hex, cubic, square. And therefore the best forms can eventually surface and all the rest, being infinite, scrapped into the dustbin of history.
Switching. Bishop and Knight (and Falcon) switch both rank and file each move. Rook switches only one or the other. 'Triangulate' even Kasparov has heard of and uses descriptively. Triangulation is a regular chess term. King triangulates because he can return to the same square in three moves. Most CV pieces can return in two (feel free to make up a name for it) just by the reverse move, but there are exceptions, such as Shogi Fragrant Chariot or Lance (FO Rook). Other exceptions are both Falcon and Hunter of Karl Schultz' Falcon-Hunter from 1940s. Queen triangulates, but Knight, Bishop and Rook cannot. For an anti-monarchist Gilman is yet partial to royal names, preserving them or inventing them. See for yourself Diarch, Dowager, Regent, Grandduke, Duchess. More in keeping with realpolitik are Gilman's USURPER, HERETIC, and BESEIGER, UNICORN enhancements all. Here are two already intermediate questions: <1> Does enhancement of Unicorn by Ferz to HERETIC enable him to reach only 1, or 2, or 3, or all 4 bindings? <2> Does enhancement of Unicorn by Prince to USURPER enable her to reach 2, 3, or all 4 bindings? (Hint of thought process: Unicorn is the nonstandard cubic diagonal mover of one binding only.)
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.
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.
(1) I corrected the root-3 ND cubic in brackets because I thought of it right away. (2) Gilman answers the two questions that the example compounds are unbound. That means they have all 4 bindings and can reach every cube. Unicorn alone reaches only 1 in 4, and so it takes 4 ordinary Unicorns properly spaced to ''cover'' the board. Usually in a CV one would conceive of implementing plain old Unicorn. Now actually compounding Unicorn generally will get the piece to reach every cell. The answer is in the text last paragraph before Notes. Hey, test questions come from texts. However, Gilman has not addressed that compounds here are not inevitably unbound, for examples, with well-chosen leaper that does not change binding of the Unicorn, or some Forward Only enhancement you could think of.
Think of a starting cube and the cube in the plane above where Bishop one-steps through an edge. Then think of the same starting cube and the cube in the plane above where Unicorn one-steps through a vertex. Choose the ones above so they have faces adjacent. The distance travelled center to center in the first case is root-2 and second case root-3. Now think of the position of the first destination cube sliding gradually to that of the adjacent second (Unicorn) cube, one cube sliding into the other of the same plane. Keep an imaginary line center to center from the below departure cube, and the line segment increases from position to position. Then every value for distance travelled between 1.414 and 1.732 is assumed along the way. Every possible value along the infinite Cantorian continuum, until we reach root-3. And at that position the cube is locked into where Einhorn one-steps.
Later articles cover other compounds. MAB 04 covers compounds of a symmetric and a forward-only MAB 01 piece in different kinds of directions - all are unbound on a cubic board. MAB 08 covers compounds of a radial and oblique piece - Bishop+Camel=Caliph and Unicorn+Elf=Leprechaun retain their components' bindings but Bishop+Elf=Levite and Unicorn+Camel=Cafila are unbound.
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.
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?
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.
Now you've hit on something. All right, if an aversion produces a mixed-cell board that's a bonus. Making pieces averse to a single Elephant binding gives squares alternating with pentagons, and making them averse to a Trebuchet binding gives a curious mixture of squares in clusters surrounded by pentagons. Making them averse to the binding of a Veering or Backing Knight (see Man and Beast 12) gives a third pattern entirely of pentagons, an asymmetric one with the boundaries forming an unfortunate swastika motif. Returning to the Dabbaba binding, I see that the paths of Bishops unaffected by the restriction suddenly look like hex Rook paths! Of course hex boards can also have pieces averse to a Dabbaba binding, and for them the board turns into the following rhombic pattern: ____ ____ ____ ____ /_/\_\/_/\_\/_/\_\/_/\_\ \_\/_/\_\/_/\_\/_/\_\/_/ /_/\_\/_/\_\/_/\_\/_/\_\ \_\/_/\_\/_/\_\/_/\_\/_/ /_/\_\/_/\_\/_/\_\/_/\_\ \_\/_/\_\/_/\_\/_/\_\/_/ /_/\_\/_/\_\/_/\_\/_/\_\ \_\/_/\_\/_/\_\/_/\_\/_/
It looks like the piece moves aren't showing up in the Game Courier diagrams in this page for some reason. I used Web Archive to read an older version of this article (from 2016) and it does show the piece moves in the diagram.
https://web.archive.org/web/20160601094057/https://www.chessvariants.com/ideas/man-and-beast-01-constitutional-characters
Ben Reiniger wrote on Wed, Sep 20, 2023 10:21 PM UTC in reply to Anonymous Lurker from 01:10 AM:Thanks for the reminder, this has long been on a list of things to do when I have the time.
The diagram designer changed behavior around movement markers after Charles had made all of these, breaking them. I need to go through with a regex replacement. Doing that through the site's forms would take a while; doing it through the database would be faster, but then revision history wouldn't be updated, so I'd want to more carefully check that the changes worked and didn't break something else.
https://www.chessvariants.com/index/listcomments.php?id=42393
https://www.chessvariants.com/index/listcomments.php?id=37824
Ben Reiniger wrote on Sun, Sep 24, 2023 07:17 PM UTC in reply to Ben Reiniger from Wed Sep 20 10:21 PM:OK, this page is done; please let me know if anything remains to fix here.
(There were two diagrams here too in which a piece name started with a period and the surrounding brackets had been URL-encoded, which made the first pass of the regex replacement convert too much. It was easy enough to spot and fix. I can't just begin the process by converting the URL-encoded %7B and %7D back to brackets, because then the main regex will stop too soon. Maybe I can adjust the main regex to exclude matches to %7B.?)
Bn Em wrote on Mon, Sep 25, 2023 10:23 AM UTC:There's one main thing left to fix, though it's probably Charles' error: The diagram for the Primate (WB, top right in the last block of diagrams) has a spurious sideways rook move, of a kind which isn't introduced until Alternative Fronts. Idk whether we consider that within the editors' remit though, esp. with Charles gone
Incidentally, the thing with the mathematical expressions (which you note here and also affects this page's degree signs ⟨°⟩) is the same problem as affected this page, among many others across the site. In principle worth fixing though it may be more difficult to do thoroughly, especially if people have edited affected pages in the interim
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I see several principal radial directions, each generally with its own SOLL:
- Wazir-wise
- Ferz-wise
- Viceroy-wise
- Rumbaba-wise
You observe each of these, and treat on them at length (though as it appears in only one geometry, #4 gets a separate article).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.