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Man and Beast 03: From Ungulates Outward. Systematic naming of the simplest Oblique Pieces.
George Duke wrote on 2008-08-08 UTC
Among many others Gilman defines radial pieces in his first sentence as ''pass(ing) through the centre of a cell or only its borders.'' Radial, or line, means Bishop and Rook and their ilk, as opposed to oblique. Oblique covers Camel, Zebra, Knight and their ilk.

George Duke wrote on 2009-06-16 UTC
Oblique.(4,5) is Rector; he leaps 4 then 5, or 5 then 4, same difference if with oppositely orthogonal start to the path, since it is a leap, like Knight (1,2) or (2,1), which could mean the very same square. 4 and 5 share no multiplicative factor, so mathematically they are coprime, prime to each other. Why leave out pairs like (2,4), which are not coprime because they have common factor of 2? Because (2,4) is a Nightrider square, and we would rather get there another way than a direct leap. So would (2,6) be Camel-rider second step not coprime. Gilman is omitting non-coprime, because they would be redundant. (8,10) would be Rector-rider and needs no other name. Surrounding 13x13 set of squares to its central departure square is illustrative. We figure the zero(0) squares of Rook either way within one quadrant (the Rook is the border) as coprime for convenience. So, within 13x13 we are only omitting, with the order mattering, (2,2) (3,3) (4,4) (5,5) (6,6) (2,4) (2,6) (3,6) (4,6) (4,2) (6,2) (6,3) out of 48 squares one quadrant, and that's 25% of them. As we go farther out, about the same ratio obtains. All together oblique directions increasingly outnumber the mere two radial ones, but the omissions to naming would stay nearly the same percentagewise, because of the rider-phenomenon, also able to be called non-coprime redundancy. So Gilman is going ahead with naming most of the squares in principle (he is not likely to proceed beyond 15- or 17-block) or approving already-existing names. [See Pritchard's 'ECV' under Pieces for ancient schema for 7x7 block of squares around a central starting square]

Charles Gilman wrote on 2009-06-17 UTC
'We figure the zero(0) squares of Rook either way within one quadrant (the Rook is the border) as coprime for convenience.' Far from it, as MAB 06 confirms. Just as 1 being the only self-coprime number renders the Elephant, Tripper, Commuter &c non-coprime pieces, 1 being the only number coprime with 0 renders the Dabbaba, Trebuchet, Cobbler &c non-coprime pieces. 'So, within 13x13 we are only omitting, with the order mattering,' and regarding 0:0 as already occupied: 0:2, 0:3, 0:4, 0:5, 0:6, 2:0, 2:2, 2:4, 2:6, 3:0, 3:3, 3:6, 4:0, 4:2, 4:4, 4:6, 5:0, 5:5, 6:0, 6:2, 6:3, 6:4, 6:6. The comment '(he is not likely to proceed beyond 15- or 17-block)' at least is true, as I only went as far as I did in response to interest in such pieces in this article's previous incarnation.

George Duke wrote on 2009-06-17 UTCExcellent ★★★★★
Actually Gilman has gone to at least surrounding set of squares 27x27 for some incomplete naming. Giraffe is (1,4) leaper pre-Gilman. Two Giraffe moves at right angles reach (3,5) Gimel, and those pieces are called duals with SOLL 17 and 34. It always works that way 1:2 in same sort of pairing as Knight-Camel and Wazir-Ferz. It's interesting. The omitted ''-rider'' squares are important too because they have basal unit starting with Knight etc. And to be addressed later I think multi-path Scorpion 3, Dragon 4 squares (not pathways), Phoenix 6, and Roc 7 are more aesthetic organization of squares. For example, using block of squares 23x23, a larger board than any formally in CVPage, PARVENU is (PARSON + (11,1)PAMEL). Those are duals, Parson and Pamel, as Parson is regular (6,5) leaper and its dual is (1,11) because SOLL of Parson is (36+25) or 61, and the one with SOLL twice that, 122, is (1,11). Expect PARVENU to be able to triangulate, as with any compound of duals. You can sort of play around with duals in that you have more information than you need. Either SOLL alone or the piece-type coordinates can get you to one's dual for each oblique co-prime. It becomes over-determined who the various counterparts are, oblique piece, dual, and compound. // (1) The first prime is 2 not 1. (2) Gilman has mentioned '06' and '08' and we will get to those.

Charles Gilman wrote on 2009-06-18 UTC
At first sight 'Scorpion 3, Dragon 4 squares (not pathways), Phoenix 6, and Roc 7' seemed to omit 5, but perhaps you meant (after Moo 2 and Falcon 3) 'Scorpion 4, Dragon 5, Phoenix 6, and Roc 7'. Is this correct. I do not recall seeing your Phoenix and Roc mentioned before, but once you clarify I will endeavour to add them to the list at the end of MAB 13.

George Duke wrote on 2009-06-20 UTC
Okay. With Gilman's indulgence we keep it still basic (and only square-based), to win a few more converts to systematizing. The duals will all end in -el, because Camel and Knight are duals, and Camel has been around a long time. Move from a starting square twice as Knight and back to same square as a Camel: duals: duals triangulating back. Oblique triangulators end in -u, because Gnu has been around for a long time, and Camel + Knight is Gnu, meaning Gnu is sure to be the second compound triangulator. What is the first one? Why non-royal King, compound of Wazir + Ferz; 1,2,3. Gilman's is a system of nomenclature. Some amateur-night designer may already be using Gnu(C+N) once in a while in fantastical back-rank for no other reason than he wants more choice in jumping. He can well go through life never knowing about duals or triangulation and still make lots of CV art. For smart people let's try Zebra + Zemel = Zebu. 2,3 has leap length of root-13 and SOLL 13. Twice that is 26 and sure enough there are corresponding squares 25 and 1, so Zemel is 5,1. (I am surpirsed 5,1 is not previously named) Anyway now a piece can move from b1 to d4 to g2 and back to b1, triangulating, and call that compound of duals Zebu. All ratings of a M&B will be 'Excellent' but some few chapters will just not have rating for minor protest, nothing in between. One weakness throughout is lack of dates of invention. Highly sensitive to cultural history is Gilman, unlike lesser designers who are anti-historical let alone ahistorical. My instinct is to refer to any pieces the first time by year or period. For example, 13th century Gryphon, 17th century Centaur(BN), 1907 Unicorn, circa 1912 Nightrider, 1940s Hunter and Falcon of Schultz, 1970s Angel of Conway, 1992 Falcon, 2005 Promoter. A major question later will be, to what extent does all this knowledge algebraic and geometric make better rules-sets mind to mind?

George Duke wrote on 2009-06-22 UTC
M&B03 is pivotal. Square of leap length. 1,2 Knight's SOLL is 5, that's odd, so Knight is not colourbound. 1,3 Camel's SOLL is 10, that's even, so Camel is colourbound. Knight switches from one Bishop binding to the other each move, right? Therefore N cannot return to the same square in three, only two, four etc. Compounds of duals triangulate, but compound of any two single leapers not duals may or may not. For ex., Gazelle (N+Zebra) colourswitches, like each of its legs, and so needs 4 (or 2 trivial case) to return. 5,2 leaper half-ungulate Satyr has SOLL 29, Saturn's year in fact. 25+4=29. x2=58, which requires 49+9 and obvious 7,3 for dual Samel. Among the dozen and more duals (depending how far out we go), 6,1 Flamingo and 7,5 Famel are called triangulating compound Flambeau, good one, that will fit onto even 8x8, bound to the double outer perimetre for my Problem Themes, only unable to reach the centre 4x4, a safe haven from burning by the Flambeau. The math is 36+1=37. x2=74, and 7^2 + 5^2=74. FLAMBEAU is a torch such as carried by Shakespeare's Lady Macbeth. --'Tis unnatural, even like the deed that's done. On Tuesday last, a Falcon, towering in her pride of place, was by a mousing owl hawk'd at and killed. Act II scene iii Macbeth.

George Duke wrote on 2009-06-22 UTC
If you do not know these, any more you are rowing blindly, thrashing in thin air (remember flat earth?), designing in the void, to be condemned forever and stockaded for now. Duals include Ferz->Wazir, N->Camel, Zebra->Zemel, Giraffe->Gimel, Antelope->Namel. They should become second nature and their relationships. For example, Antelope 4,3 = 25 SOLL. x2=50. 50= SOLL Namel 7,1. Giraffe 4,1 = SOLL 17, x2=34 -> as 25+9, 5,3 Gimel. Triangulating is not the norm for oblique leaper. That is why we compound them, like Carrera compounds in 1617, not so long ago, B and N to Centaur, who triangulates. Coprime oblique leapers in fact none of them triangulate, because in squares they switch. They either switch colour, or they switch rank as to odd or even from origination. // (1) Still being thought-processed: Non-coprime NN 2,4 cannot triangulate 2-D either but keeps both colour and ranks' binding odd/even; not switching here = not triangulating? (2) Determining triangulation in cubes depends on the board's 3 dimensions with respect to the coordinates of the leaper.

George Duke wrote on 2009-06-23 UTC
All oblique coprime leapers are switching in 2-D, either changing rank and file from odd to even or vice versa, or changing binding (the latter the odd-SOLL ones). How about SOLL in cubes? We found one-step Bishop there root-2 and one-step Unicorn triagonal root-3, making their basic SOLLs 2 and 3. Covering in detail oblique cubes' movers will be ''M&B05: Punning by Numbers'' later. For Forward Only FO, Gilman names Shogi Knight HELM. FO Flamingo 6,1 is of course MALLET 6,1 through 'Alice in Wonderland' croquet. FO 3,1 HUMP, merely a Camel FO, is the one causing so much trouble 5 years ago, as to appropriateness, for members' then, as now, carelessly reading, or not reading, Gilman's (or any quality producer's) straightforward definitions. Gilman concludes 'M&B03': ''No piece's long coordinate exceeds 13. This is partly as simple pieces with a longer leap are weak on any reasonable sized-board, partly for want of names for triangulating compounds.'' Adrian King's Jupiter is 16x16 and Gilman should name at least all the 15s. Their long dimension would leap from back-rank to back-rank of Jupiter, not too much to ask. If you can name the 13s, you can name the 15s. And ''weak''? In general, of course the longer solitary leap loses strength, but there are tradeoffs, not least the board. Namel 7,1 was critical even on regular 8x10 to solve one Problem Theme in 2007. No other shorter leaper would have worked. Think of Ramayana board of only 84 with outlier squares in Archipelago and expand it. A long leaper can sometimes get in one jump 11, 12, or 15 away it takes sliders several moves to reach, if they ever get there at all. The particular 13,5 TOMEL or unnamed 15,11 may be of greater piece-value than Zebra or Giraffe on a case basis. Nomenclature should be open-ended and ever-growing.

Charles Gilman wrote on 2009-06-24 UTC
While my names cover everything out to max coordinate 9, my coverage beyond that to 13 is fairly patchy. I can identify 10:2 Zrene, 10:4 Sharolais, 10:6 Grine, 10:8 Rherolais, 11:1 Pamel, 11:3 Bemel, 11:5 Humel, 11:7 Lamel, 12:2 Pharolais. 12:3 Ghimois, 12:5 Zoetrope, 12:8 Zeltrap, 12:9 Nhamois, 12:10 Rherolais, 13:1 Cumel, 13:3 Gamel, 13:5 Tomel. Colourswitching pieces to fill just that in would require 12 distinct starting letter pairs - for 10:1/11:9, 10:3/13:7, 10:7, 10:9, 11:2/13:9, 11:4, 11:6, 11:8, 11:10, 12:1/13:11, 12:7, and 12:11. Going out to 15 would require further ones. There are a few pairs left - bo by ce cy fy gy hi ho hy ja je ji jo ju jy ly ma mu my qu ri ru sy ti tu ty vi vu vy wa wu wy xa xo xu xy za zi zu zy - but few are promising and I welcome suggestions. Best to avoid tu on account of its rude Bishop compound. Nor can I see how to continue the sequences ending in Albatross and Deacon. Then names with a C and A would need devising for 10:5, 12:4, 12:6, 14:7, and 15:5 leapers. Is it all worth it for pieces that would be very weak even on a 16x16 board? Personally I doubt it, but again suggestions are welcome.

Charles Gilman wrote on 2009-06-29 UTC
I have now extended what I now realise was an unnecessarily small range of 2d MAB 14 leapers to larger coordinate 7, using up Ho- and Mu- among the prefixes available. I have decided to rethink 10:n leapers' names as it has dawned on me that they share their SOLLs with 8:6:n ones and that would add a whole tranche of extra new names if I allowed reversibility without duplication. Names ending in -er are of course unsuitable for this because of the Rector, likewise -ry hecause of the existing Rytas. Some of these might be reused elsewhere.

Charles Gilman wrote on 2009-08-17 UTC
[comment withdrawn]

George Duke wrote on 2009-10-06 UTC
ChessboardMath11 Quiz 3.September.2009 #(1) asks ''Identify Umbrella and Mallet.'' This is technical even for Charles Gilman. Umbrella is forward only 5,4 leaper version of Rector, and Mallet is forward only leaper version of 6,1 Flamingo. That's the answer, appearing here after the paragraph beginning ''Shogi...'' In practice they might become compound with some Pawn, such as Jeremy just lists from Gilman's other work. The ten/eleven questions of the quiz are being answered periodically every few days. Then they can be summed up back at ChessboardMath11 later. #(2) already answered is that at ''Worse Than Worthless'' Betza describes Nattering Nabobs of Negativity for CDA choice. http://www.chessvariants.org/index/listcomments.php?subjectid=chessboardmath11

George Duke wrote on 2011-01-27 UTC
How many of the 21 Man & Beasts have you learned well? 1? 10? All 20? No easy answers? Start with M&B03 here, the easiest. A piece moves, and there is a leap length. Now which of the compounds of Duals, always bi-compounds, deliver Mate with King alone? (not addressed much by Gilman's anti-problemist bent, let's say) Gilman notwithstanding, sufficient mating material is standard Chess criterion to separate major/minor piece-types, right? Answer: only the first two, Man(W+F) and Gnu(N+Camel), as experimenting on board shows. The formula algebraically is two times the sum of SOLLs, to find the other one that matches for each compound. If already lost, read the article and other comments. By the third one and outward the fourth, fifth, the piece-type does not achieve it. So only the first two compounds of Duals deliver mate. That means all the other compoundings of Duals are rather weak units made as they are of two long-range leapers, known to be low piece-value. All that seems to recommend them then after those first two Dual-compounds is that they triangulate. Triangulate of course is to return in three, the way Queen or King do. Now other paired leapers, for example (Dabbabah + Camel), triangulate without being Duals; so triangulating is not really all that unique. Bi-compound duals are okay as mnemonic and organizing thought as far as that goes, but not great use to put into CVs once at and beyond Camel and Zebra distances. They are essential mostly for nomenclature. For follow-up, does Gilman really stress duals in his 200 CVs? Or mostly just in the Man & Beasts? Rather than compounds of duals, I think Gilman and other designers implement long-range component with short-range atom N,F,W,D, or A. That creates a piece-type of more useful point value 3.5 to 6.0.

jumbods63 wrote on 2017-02-03 UTCExcellent ★★★★★

There's a typo in this article: you say that "Vine+8:3 VUMP=VANDYMAN", but shouldn't that be 7:3 VUMP?