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# Man and Beast 10: The Hybrid Diagonal

This series of articles considers four kinds of radial: the orthogonal (Rookwise), found on all uniform geometries; the standard diagonal (SD, Bishopwise), with twice the orthogonal's Squares Of Leap Length (SOLLs), found in square and cubic geometries; the nonstandard diagonal (ND, Unicornwise), with thrice the orthogonals SOLLs, found in hex and cubic geometries and commonly called triagonal in the latter; and a third kind of diagonal coexisting with the rest on a hex-prism geometry, with the even SOLLs of the orthogonal. As mentioning this direction caused confusion I separated it out from other radials to simplify earlier articles. This article deals with coprime Straight pieces using the direction, which as it has both rectilinear and hex elements I term the hybrid diagonal (HD).

A Bishop's move can be seen as two simultaneous Rook moves at right angles at the same rate, in the hex-prism case one along a hex board and one between hex boards. A Unicorn move on a cubic board can be seen as three Rook moves with the same restrictions, or as a Rook and Bishop move at right angles by the same number of steps. The latter definition also describes the diagonal of the hex planes of the xyrixa geometry, as featured in Mark Thompson's Tetrahedral Chess, and therefore also of a 2d hex board or a hex-prism board's hex planes. The third kind of diagonal move on a hex-prism board may be considered an interhex Rook move and a hex Unicorn move at right angles by the same number of steps. It has some affinity with the cubic-board Unicorn move, but cannot inherit the same piece names as the hex-prism board allows pure-hex pieces of those names. The number of forward/sideways/backward directions depends on orientation. If hex boards are ranks the numbers are 6/0/6 (same numbers as SD), if they are Wellisch-orientated levels a very balanced 4/4/4, if they are Glinski-orientated levels 2/8/2. Pieces marked # are specifically hex-level, as hex-ranked boards lack forward Unicorn diagonals.

The acknowledgments, overview, and glossary to the series can be found here. Established pieces have a link to their Piececlopedia page. The images used in this series, and indeed in most of my pages, can be found here.

## Pieces

Continuing from MAB 01: Constitutional Characters, I start by naming the symmetric short-range piece after the dessert RUMBABA - something sharing the Dabbaba's leap length but in a "rum" direction. It is bound to one in three cells and switches, like the Ferz, between odd and even hex boards. Rejecting a pattern name like Point/Cross/Saltire as "forward" directions vary with orientation, I term the forward-only (FO) piece a RUMCHICK by analogy to Dabchick. As a Rumbaba is a coprime radial piece (it cannot pass through the centre of cells where it cannot stop) its rider logically requires a distinct name, just as a Rook is not called a Wazirrider. The name DICORN reflects the affinity with the cubic-board Unicorn, and the fact that for every move hex Unicorns have on the same level or rank, Dicorns have two - up and down or forward and backward. Likewise its FO form DILEMMA, after the pair of notional "horns of a dilemma", regarding levels, although hex-ranked boards which cannot have Horns at all have 6-direction Dilemmas.

For compounds with corresponding orthogonal pieces the next rank of nobility down from each corresponding orthogonal+ND compound - Marquis below Duke, Viscount below Count - befits these titles' intermediate nature in a nobility too swollen to make do with just Dukes, Counts, and Barons. Thus General+Rumbaba=GRANDMARQUIS, Rook+Dicorn=MARCHIONESS, Wazir+Rumbaba=MARQUIS, Wing+Dilemma=VISCOUNTESS, Point+Rumchick=VISCOUNT. Compounds with SD pieces follow a similar "demoting" rule: Bishop+Dicorn=PONTIFF (chief priest defined in relation to head of state, reversing Governor), Ferz+Rumbaba=LANDLORD (property owner not necessarily a constitutional lord), Mitre+Dilemma=MEDIATOR (metaphorical sense of Pontiff as "bridge-builder"), Cross+Rumchick=HOST (term often applied to landlords of inns but also to others). All these compounds are unbound as the orthogonal or SD component is itself unbound on a hex-prism board. The Landlord and Host are also switching, between odd and even hex planes. Compounds with corresponding hex-diagonal pieces retain the binding to 1 in 3 cells as the Unicorn binding is the intersection of this and a single hex board. I use the sum 1+2=3 to give Unicorn+Dicorn=TRICORN and Horn+Dilemma=TRILEMMA# to reflect having the same-level, up, and down - or same-rank, forward, and backward - groups of directions. In the short range, Viceroy+Rumbaba=BELLMAN means a town crier often in a tricorn hat, and a character from Lewis Carroll's Hunting of the Snark into the bargain. Saltire+Rumchick=BELLBOY# means someone relatively menial answering a bell rather than ringing his own.

For compounds with both square-board radials I look to modern elected leaders and their entourages: King+Rumbaba=PRESIDENT, Queen+Dicorn=FIRSTLADY, Prince+Rumbaba=PREMIER, Princess+Dilemma=SPOKESWOMAN, Princeling+Rumchick=SPOKESMAN; for compounds with both hex radials, leaders of transition from old to new: Grandduke+Rumbaba=FOUNDER (of new rÃ©gime, short for a founding president such as George Washington), Duchess+Dicorn=SUFFRAGETTE (woman campaigner for womens' rights), Duke+Rumbaba=REVOLUTIONARY, Countess+Dilemma=DISSENTRESS#, Count+Rumchick=DISSENTER#; and for compounds of all three kinds of diagonal without the orthogonal, the less democratic of modern rÃ©gimes: Governor+Dicorn=DICTATOR, Baron+Rumbaba=LOYALIST, Missionary+Dilemma=APPOINTEE#, Heir+Rumchick=SYCOPHANT#. Note the female titles for pieces with a Rook or Wing move. Desperate characters in this paragraph reflect a touch of desperation on my own part!

Images for pieces on this page were driven by my 3d 5 level variants. The Rumbaba shares an image with the MAB 06 Commuter for the same reason as the Viceroy does with the Tripper, while the Dicorn and Dilemma share the Giraffe and Blotch ones as they are a large-small pair with two horns. The compound pieces I attempt to coordinate as similarly-modified Rook/Bishop/Queen images in the long-range cases, and Wazir-Ferz-Prince ones in the short-range ones. Images for capturable and royally-restricted hex-prism pieces with the same move are not distinguished as I do not expect both to occur in one variant. Here is a table with the Dicorn/Marchioness/Pontiff/Tricorn/Firstlady/Suffragette/Dictator on the first row, their short-range counterparts on the middle one, and where I have them the FO long-range ones at the bottom:

Landlord, Host, Bellman, and Bellboy can, like fellow short-range pieces Baron and Heir, be feminised for mixed-range Rook/Wing compounds: Rook+Landlord=LANDLADY, Wing+Host=HOSTESS, Rook+Bellman=BELLWOMAN, Wing+Bellboy=BELLGIRL#. For Rook+Rumbaba itself I settled on HOUSEWIFE, a woman with a smaller home than the Chatelaine's but her notional equal under the modern classlessness of the previous paragraph's politicians. For Wing+Rumchick HOUSEMAID is the obvious subordinate. Enhanced Bishops and Mitres continue a religious, though not necessarily Christian, theme: Bishop+Rumbaba/Marquis/Bellman=DRUID/THAUMATURGE/EXCOMMUNICATOR, Mitre+Rumchick/Viscount/Bellboy=BARD/MARTYR/BOUNCER. Thaumaturges are workers of miracles, excommunicators expel people from the church as an institution, and bouncers expel them literally from buildings. Enhanced Unicorns are disciplinarians and enhanced horns their subordinates: Unicorn+Rumbaba/Marquis/Landlord=DOMINATOR/MARTINET/BARRISTER, Horn+Rumchick/Viscount/Host=INSTIGATOR#/VITUPERATOR#/SOLICITOR#. A barrister also appears in the Hunting of the Snark. For pieces whose long-range component is itself compound Queen/Duchess/Governor+Rumbaba=DIVA/DOMINATRIX/SUPREME, Princess/Countess/Missionary+Rumchick=ORACLE/INSTIGATRIX#/SUPERIOR#. Supreme is a superlative qualifier for the sovereign's governorship, and Superior the corresponding comparative which is also a religious title. Where the long-range element includes the Dicorn/Dilemma move I mix traditional, modern, subversive, and punning names: Dicorn+Wazir/Ferz/Prince/Viceroy/Duke/Baron=CHAMBERLAIN/CONGRESSMAN/RENOUNCER/MAYOR/SENATOR/BROADCASTER, Dilemma+Point/Cross/Princeling/Saltire/Count/Heir=COURTIER/JURYMAN/DENOUNCER/CHAIRMAN#/CANDIDATE#/JOURNALIST#. Renouncer is intended to mean a Prince renouncing his title. Adding a Rook/Wing move again feminises to give Marchioness+Ferz/Viceroy=CONGRESSWOMAN/MAYORESS, Viscountess+Cross/Saltire=JURYWOMAN/CHAIRWOMAN#. The rest are Pontiff+Wazir/Viceroy=JUSTICE/MAXIMUS, Mediator+Point/Saltire=MAGISTRATE#/MAJOR#, Tricorn+Wazir/Ferz=MINISTER/EXECUTIVE, Trilemma+Point/Cross=SHADOWER#/MIDDLEMAN#. In Lewis Carroll's Pig Tale the birds feed "justices with jam". Maximus is a superlative qualifier for Pontiffs corresponding to Supreme for Governors, Major the corresponding comparative, Magistrate a part-time judge, and Shadower intended as shorthand for Shadow Minister, a spokesman in an Official Opposition on the same subject as a minister. I have piece images only for symmetric pieces in this group. The Housewife and Druid use modified Rook and Bishop images and the Chamberlain, Congressman, and Renouncer ones are shared with corresponding Giraffe compounds:

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Continuing from MAB 02: Shield Bearers, FO pieces capturing on the HD suit names close to the most exotic cubic-cell divergent piece with the same non-capturing move: BREAKER (orthogonal), WIELDER (SD), OUTLANDER# (ND, abbreviated to OUT in compounds). All may be specified as Ancient, European, Warhead, et cetera reagrding what initial double-step moves they have or do not have. They extend to long-range as BREAKRIDER, WIELDRIDER, and - to both follow the Lowrider/Highrider pattern and be a real word - OUTRIDER#, and expand to all directions of each kind as FOOTMAN, COOK, and GHILLIE - all continuing the serving theme of Steward, the first feminisable, and the last specifically Scottish. Lewis Carroll's Pig Tale mentions "interesting books to be read by cooks", and the Duchess in his first Alice book employs a cook. Extended and expanded they become FOOTWOMAN (obviously), CHEF (a restaurant cook), and BAILIFF (who in Scotland has a certain notoriety). The Ghillie and Bailiff can exist on hex-prism boards however orientated, but can change hex plane only by capturing. Compound names are formed in the order of Breaker between Broker and Yeoman, Wielder between Warder and Low, and Out after Saltire. Thus a Point capturing only as a Host is a PAWNBREAKER, one capturing as a Bellboy a BROKEBREAKER, and their riders a PAWBRERIDER and BROBRERIDER. A Prince capturing as a Bellman and a Queen capturing as a Dicorn are are worthy of names as they are used here, as Waiter+Cook and Waitress+Chef compounds, For this reason I term them RESTAURATEUR and RESTAURATEUSE. Note that it is a common error to put an N in these two words.

Where the non-capturing move is on the HD I consider the 3d figure of which all moves together suggest a 2d picture and suffix with -CHICK: HEXCHICK (orthogonal, suggesting vertices of cube), OCTCHICK (SD, suggesting edges of octahedron lined up as Star of David), FLATCHICK# (ND, necessarily capturing in a flat plane, hex boards not being ranks). These too can be prefixed according to initial move. Compound names are formed in the order, following Out above, of Hex/Oct/Flat/Rum. Naturally they extend to HEXRIDER/OCTRIDER/FLATRIDER#, expand to CONTRAFOOTMAN/CONTRACOOK/CONTRAGHILLIE, and both to CONTRAFOOTWOMAN/CONTRACHEF/CONTRABAILIFF. The Contraghillie and Contrabailiff can exist on hex-prism boards however orientated, but can change hex plane only without capturing. A Bellman capturing as a Prince and a Tricorn capturing as a Queen are CONTRARESTAURATEUR and CONTRARESTAURATEUSE.

Pieces divergent between long- and short-range moves take the MAB 02 suffixes. Thus a Rumbaba capturing as a Dicorn is a RUMBASNATCHER and vice versa a RUMBAPACIFIER, a Rumchick capturing as a Dilemma a RUMCHSNATCHER and vice versa a RUMCHPACIFIER, a Wazir capturing as a Dicorn a FOOTMSNATCHER and vice versa a CONTREFOOTMPACIFIER, a Wing capturing as a Rumchick a BREAKPACIFIER and vice versa a HEXSNATCHER, a Cross capturing as a Dilemma a WIELDSNATCHER and vice versa an OCTPACIFIER, a Bishop capturing as a Rumbaba a COOKPACIFIER and vice versa a CONTRACOOKSNATCHER, and many others. Glinski-ranked boards have some cells from which AVERSE BREAKERS and AVERSE WIRLDERS have no noncapturing, and AVERSE HEXCHICKS and AVERSE OCTCHICKS no capturing, moves.

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Continuing from MAB 04: Generalised Generals, generals adding the Rumchick move to a Wazir, Ferz, Viceroy, Prince, Duke, or Baron I prefix AMBER-, PEARL-, INDIGO-, WIND-, STORM-, and ISLAND-. Each has a full rider, one partial rider ending in -DILEMMA and one along the lines of AMBERROOK, INDIGOUNICORN, WINDQUEEN (itself combining the extensions giving WINDROOK and WINDBISHOP), and the three royally-restricted versions AMBERRULER/WINDRULER/STORMRULER (effectively General/King/Grandduke+Rumchick). Those adding a Point, Cross, Saltire, Princeling, Count, or Heir move to the Rumbaba I prefix BRONZE-, RUBY-, MAGENTA-#, DESERT-, LAVA-#, and FIRE-#. Each has one partial rider ending in -DICORN and one along the lines of RUBYMITRE, DESERTPRINCESS (itself combining the extensions giving DESERTWING and DESERTMITRE). Both groups have the usual full rider, with the prefix curtailed if longer than 5 letters e.g. INDIGRIDER, BRONZRIDER.

Gold, Brass, Silver, Steel, Azure, and Jade themselves can have HD moves added: the Rumchick to give generals that I prefix DAWN-, SAND-#, DUSK-, SNOW-#, CLOUD-, and MIST-; or the Rumbaba to give generals that I prefix OASIS-, DUNE-#, LAKE-, SWAMP-#, RAIN-, and ICE-. All have riders. Mixed-range pieces end less obviously: -ROOK/-MEDIATOR, -ROOK/-TRILEMMA, -BISHOP/-VISCOUNTESS, -BISHOP/-TRILEMMA, -UNICORN/-VISCOUNTESS, -UNICORN/-TRILEMMA, -MARCHIONESS/-MITRE, -MARCHIONESS/-HORN, -PONTIFF/-WING, -PONTIFF/-HORN, -TRICORN/-WING, and -TRICORN/-MITRE. Royally-restricted versions are the DAWNRULER, SANDRULER, OASISRULER, and DUNERULER. Well-argued cases for bettering my prefixes, either with new ones or by swapping them around, are welcome.

Generals using all four kinds of radial I give a radical and perhaps controversial prefix: the first five letters of the general's symmetric component to give e.g. WAZIRGENERAL#, RUMBAGENERAL#, PRINCGENERAL#, DUKEGENERAL. What's the controversy? It gives WAZIRRIDER# and FERZRIDER#, theoretic but never-used synonyms for Rook and Bishop, quite different meanings: a Rook or Bishop also moving any distance on any hex-prism forward radial! Pieces extending only the symmetric component sound tautological: WAZIRROOK#, FERZBISHOP#, and even more so PRINCQUEEN#, DUKEDUCHESS, MARQUMARCHIONESS#, ARCHDEMPRESS - though note lesser partial riders such as the PRINCROOK# and PRINCBISHOP#. Pieces extending only the FO component are the more self-explanatory likes of WAZIRAPPOINTEE#, FERZDISSENTRESS#, PRINCTRILEMMA#, DUKEMEDIATOR, MARQUMISSIONARY#, ARCHDDILEMMA. Note the royally-restricted ARCHDRULER, PREMIRULER#, REVOLRULER, PRINCRULER#, DUKERULER, MARQURULER#, and WAZIRRULER#. On a hex-ranked board the VICERGENERAL and its full and partial riders in between can move forward and along ranks but not backward.

Divergent pieces can also be so prefixed. The AMBERBREAKER, PEARLWIELDER, and INDIGOOUT# are a Wazir, Ferz, and Viceroy capturing as (and only as) a Rumchick, and the AMBERHEX/PEARLOCT/INDIGOFLAT# vice versa. The BRONZEBREAKER, RUBYWIELDER, and MAGENTAOUT# are a Point, Cross, and Saltire capturing as a Rumbaba, and the BRONZEHEX/RUBYOCT/MAGENTAFLAT# vice versa. Likewise long-range pieces such as the AMBERBREAKRIDER, a Wing capturing as a Dilemma. Divergent compounds include the likes of Footman+Hexchick=AMBERFOOTMAN, Contrafootman+Breaker=BRONZEFOOTMAN, Cook+Octchick=PEARLCOOK, Footwoman+Hexrider=AMBERFOOTWOMAN, Contrachef+Wieldrider=RUBYCHEF. The INDIGOGHILLIE and INDIGOBAILIFF can change hex plane only by, and the MAGENTAGHILLIE and MAGENTABAILIFF only without, capturing. Among part-symmetric snatchers and pacifiers a Point capturing as a Dicorn is an AMBERBREAKSNATCHER and vice versa an AMBERHEXPACIFIER, a Rook caturing as a Rumchick a BRONZEBREAKPACIFIER and vice versa a BRONZEHEXSNATCHER, a Ferz capturing as a Dilemma a RUBYWIELDSNATCHER and vice versa a RUBYOCTPACIFIER, a Mitre capturing as a Rumbaba a PEARLWIELDPACIFIER and vice versa a PEARLOCTSNATCHER, et cetera.

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Finally I "top and tail" the part-symmetric pieces with a WORLDGENERAL moving one step in all 38 radial directions - or WORLDRULER if royally-restricted - and a DAYGENERAL# in all forward ones. WORLD- can prefix Rook, Bishop, Unicorn, Dicorn, and their compounds, indicating long range along their directions; likewise DAY-# can prefix Wing, Mitre, Horn, Dilemma, and theirs. The WORLDRIDER moves any distance in all 38 directions and the DAYRIDER# any distance in all 9 forward ones - a different 9 depending on whether Glinski- or Wellisch-level. Here are the piece images for Worldruler, Worldrider, and Worldqueen:

## Notes

Oblique pieces specific to hex geometries have their own article, MAB 14: Oddly Oblique, and hex-specific pieces that are wholly or partly coprime theirs, MAB 17: Hex Heavies. Pieces with a radial and an oblique component, one or both forward-only or Curved or Crooked or hex-specific, are covered in MAB 19: The Vice Squad, and non-Straight and part-Straight pieces in hex-specific directions in MAB 20: Far From Square.

The intermediate nature of the Marquis and Viscount in the nobility is comparable to that of Brigadier/Major/Lieutenant/Corporal in an army too large for just Generals, Colonels, Captains, Sergeants, and Privates. Many individuals have coined Trilemma independently, mostly in the non-Chess context of being torn between three decisions.

Historically Pontiffs need not be Christian, and actually precede that religion. Gaius Julius Caesar held the rank, producing the Julian calendar in that capacity, before rising to higher things. The general definition is the highest priest of whatever is the established religion in the sovereign state whose capital is Rome. Technically speaking the office was abolished in the year 1984 when the Italian Republic became secular, and any reference to Popes since then (or before the emperor Constantine) as Pontiff is anachronistic.

Vicergeneral's similarity to the religious title Vicar-general is accidental, but Princbishop's similarity to the old Prince-Bishops of Durham could be thought less so as the Princ- is genuinely from the Prince piece.

The combination of Cook with Waiter and Chef with Waitress to form Restaurateur and Restaurateuse reminds me of my joke about the man who tried to set up a restaurant as a one-man business. He was a chef first and a... (the pause is key to the pun) waiter second!

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By Charles Gilman.
Web page created: 2008-02-28. Web page last updated: 2015-05-16﻿