Êâàòåðíèçàöèÿ
"Âíà÷àëå áûëî Ñëîâî" – ÿçûê ïîçâîëÿåò îáîçíà÷èòü òî, ÷åãî åù¸ íèêîãäà íå áûëî, à òåõíîëîãèÿ èíîãäà ìîæåò âîïëîùàòü â æèçíü òî, ÷òî âñåãî òîëüêî áûëî îáîçíà÷åíî ïîñðåäñòâîì ÿçûêà. Êîëëåêòèâèçàöèÿ, ýëåêòðèôèêàöèÿ, êîìïüþòåðèçàöèÿ, àëãåáðàèçàöèÿ, êâàòåðíèçàöèÿ...
Ñýð Ãàìèëüòîí â 1843 ãîäó ââ¸ë êâàòåðíèîíû 1, i, j, k
1 +i +j +k +00 +01 +10 +11
+i –1 +k –j +01 –00 +11 –10
+j –k –1 +i +10 –11 –00 +01
+k +j –i –1 èëè +11 +10 –01 –00.
Ïîñëåäíåå ïðåäñòàâëåíèå, ïîçâîëÿåò îïðåäåëèòü êâàòåðíèîí, êàê èñòèííîñòíóþ XOR–òàáëèöó, êîòîðàÿ ïðè ïîìîùè äâóõ èì¸í 0 è 1 (èñòèíà è ëîæü) îïðåäåëÿåò 4 èìåíè, ôîðìàëüíî âûðàæàÿñü, îòïðàâëÿÿñü îò äâóõ ÷èñåë, ââîäèò 4 âåêòîðà: 00, 01, 10, 11, èñòèííîñòíûõ ñîñòîÿíèé, ñîîòâåòñòâóþùèõ ýòèì èìåíàì. Ñàìîå ãëàâíîå, ÷òî íåìåäëåííî áðîñàåòñÿ â ãëàçà, ÷òî ýòè èñòèííîñòíûå ñîñòîÿíèÿ èçâåñòíû ëîãèêå, êàê ëîãè÷åñêàÿ îïåðàöèÿ XOR:
XOR(00, 00) = XOR(01, 01) = XOR(10, 10) = XOR(11, 11) = 00 è ò.ä.
È ýòî íå ñëó÷àéíîñòü, â òîì æå ñàìîì ãîäó Àðòóð Êýëè îïóáëèêîâàë îáîáùåíèå êâàòåðíèîíà íà òðåõìåðíûå èñòèííîñòíûå âåêòîðà è ïîëó÷èë Òàáëèöó óìíîæåíèÿ îêòîíèîíîâ
+000 +001 +010 +011 +100 +101 +010 +011
+001 –000 +011 –010 +101 –100 –111 +110
+010 –011 –000 +001 +110 +111 –100 –101
+011 +010 –001 –000 +111 –110 +101 –100
+100 –101 –110 –111 –000 +001 +010 +011
+101 +100 –111 +110 –001 –000 –011 +010
+110 +111 +100 –101 –010 +011 –000 –001
+111 –110 +101 +100 –011 –010 +001 –000. Ó Àðòóðà Êýëè 8 èì¸í, ñîîòâåòñòâóþùèõ âîñüìè èñòèííîñòíûì ñîñòîÿíèÿì ïåðåìåííîé è â êàæäîé êëåòî÷êå òàáëèöû çàïèñàíî òðåõìåðíîå èìÿ z = x XOR y.
Îáðàòèòå âíèìàíèå, ìû òîëüêî ÷òî ïîäîøëè ê ïåðâîìó øàãó – àëãåáðàèçàöèè"
+0 +1 +2 +3 +4 +5 +6 +7
+1 –0 +3 –2 +5 –4 –7 +6
+2 –3 –0 +1 +6 +7 –4 –5
+3 +2 –1 –0 +7 –6 +5 –4
+4 –5 –6 –7 –0 +1 +2 +3
+5 +4 –7 +6 –1 –0 –3 +2
+6 +7 +4 –5 –2 +3 –0 –1
+7 –6 +5 +4 –3 –2 +1 –0
À íà ñëåäóþùåì ýòàïå íåëüçÿ íå óâèäåòü, ÷òî ÷èñëà íå ñëó÷àéíî çàïîëíÿþò "òàáëèöó óìíîæåíèÿ". Äåéñòâèòåëüíî,
+0 +1 +2 +3 +4 +5 +6 +7
+1 –0 __ __ __ __ –7 +6
+2 __ –0 __ __ +7 __ –5
+3 __ __ –0 +7 __ __ –4
+4 __ __ –7 –0 __ __ +3
+5 __ –7 __ __ –0 __ +2
+6 +7 __ __ __ __ –0 –1
+7 –6 +5 +4 –3 –2 +1 –0
 ñòîëáöå "+4" ÷èñëà òîæå ðàñïîëàãàþòñÿ íå ñëó÷àéíî: îò "+7" ââåðõ çàïèñûâàåòñÿ ïåðâûé ñòîëáåö: +7, +6 (+4 óæå ñòîèò), çàòåì ñíèçó, ñî ñìåíîé çíàêà íà ïðîòèâîïîëîæíûé, ïðîäîëæàåì ñíèçó ââåðõ: –3 (óæå çàïèñàíî), –2, –1,–0 (óæå çàïèñàíî).  ñòðîêå "+4" êàðòèíà òà æå ñàìàÿ, îò "–0" âïèñûâàåòñÿ ïåðâàÿ ñòðîêà òàáëèöû: +1, +2, +3 (óæå çàïèñàí), çàòåì ñëåâà â òó æå ñòðîêó, ñî ñìåíîé çíàêà íà ïðîòèâîïîëîæíûé, ïðîäîëæàåì: –5, –6, –7 (óæå çàïèñàíî).
+0 +1 +2 +3 +4 +5 +6 +7
+1 –0 __ __ +5 __ –7 +6
+2 __ –0 __ +6 +7 __ –5
+3 __ __ –0 +7 __ __ –4
+4 –5 –6 –7 –0 +1 +2 +3
+5 __ –7 __ –1 –0 __ +2
+6 +7 __ __ –2 __ –0 –1
+7 –6 +5 +4 –3 –2 +1 –0
Íî ýòî íå âñ¸! Îêàçûâàåòñÿ ïîñëåäíÿÿ ñòðîêà è ïîñëåäíèé ñòîëáåö òîæå çàïèñûâàþòñÿ, òîëüêî â ñòðîêó "+3" è â ñòîëáåö "+3". Äåéñòâèòåëüíî, ñòðîêà +3 çàïîëíÿåòñÿ ïîñëåäíåé ñòðîêîé îò "+7" âëåâî: –6, +5, (–4 ïðîïóñêàåì, ïîòîìó ÷òî óæå çàïîëíåíà) çàòåì ïðîäîëæàåì çàïîëíÿòü ñòðîêó ñëåâà (ñî ñìåíîé çíàêà íà ïðîòèâîïîëîæíûé): +3 (óæå çàïîëíåíî), +2, –1.
+0 +1 +2 +3 +4 +5 +6 +7
+1 –0 __ __ +5 __ –7 +6
+2 __ –0 __ +6 +7 __ –5
+3 +2 –1 –0 +7 –6 +5 –4
+4 –5 –6 –7 –0 +1 +2 +3
+5 __ –7 __ –1 –0 __ +2
+6 +7 __ __ –2 __ –0 –1
+7 –6 +5 +4 –3 –2 +1 –0.
È â çàâåðøåíèå çàïîëíÿåì ñòîëáåö "+3" ïîñëåäíèì ñòîëáöîì îò "–7" âíèç: +6, –5, (+4 óæå çàïîëíåí); äàëåå ïðîäîëæàåì ñâåðõó (ñî ñìåíîé çíàêà íà ïðîòèâîïîëîæíûé): (+3 óæå çàïîëíåí ïðîïóñêàåì), –2, +1, (–0 óæå çàïîëíåí, ïðîïóñêàåì).
+0 +1 +2 +3 +4 +5 +6 +7
+1 –0 __ –2 +5 __ –7 +6
+2 __ –0 +1 +6 +7 __ –5
+3 +2 –1 –0 +7 –6 +5 –4
+4 –5 –6 –7 –0 +1 +2 +3
+5 __ –7 +6 –1 –0 __ +2
+6 +7 __ –5 –2 __ –0 –1
+7 –6 +5 +4 –3 –2 +1 –0.
Äâà ñòîëáöà è äâå ñòðîêè ðàçáèâàþò ïîëå òàáëèöû óìíîæåíèÿ íà 4 òàáëèöû ïîëîâèííîãî ðàçìåðà (íà ÷åòûðå êâàòåðíèîíà, â êàæäîì èç êîòîðûõ íå çàïîëíåíà îäíà èç äâóõ äèàãîíàëåé). Êñòàòè, òî÷íî òàêóþ æå ðàáîòó ìîæíî ïðîäåëàòü íàä òàáëèöåé óìíîæåíèÿ îêòîíèîíîâ, îïóáëèêîâàííóþ â "Ìàòåìàòè÷åñêîé ýíöèêëîïåäèè" ñîòðóäíèêîì Ìàòåìàòè÷åñêîãî îòäåëåíèÿ ÌÈÀÍ (Ñèáèðñêîå îòäåëåíèå) Åâãåíèåì Êóçüìèíûì (òàáëèöà äðóãàÿ, (ñì. ðèñ.) îäíàêî îáúåêò òîò æå ñàìûé, áîëåå òîãî çàïîëíÿåòñÿ îíà òîé æå ñàìîé ïðîöåäóðîé, èñïîëüçóþùåé ïåðâóþ è ïîñëåäíþþ ñòðîêó (ïåðâûé è ïîñëåäíèé ñòîëáåö). Ãëàâíîå â òàáëèöàõ íå çíàêè, ãëàâíîå – ïðîöåññ çàïîëíåíèÿ. Îò÷¸òëèâåå âñåãî ýòî âèäíî íà òàáëèöå óìíîæåíèÿ êâàòåðíèîíîâ
+h +i +j +k +00 +01 +10 +11
+i –h +k –j +01 –00 +11 –10
+j –k –h +i +10 –11 –00 +01
+k +j –i –h èëè +11 +10 –01 –00.
Äåéñòâèòåëüíî, ìû æå ìîæåì âìåñòî èì¸í: +h, +i, +j, +k, – âçÿòü èìåíà: +h, –i, +j, +k.  ðåçóëüòàòå, çíàêè â ñòîëáöå "i" è ñòðîêå "i" ñìåíÿòñÿ íà ïðîòèâîïîëîæíûå
+h –i +j +k
–i –h –k +j
+j +k –h +i
+k –j –i –h
îäíàêî â òàáëèöå óìíîæåíèÿ âñå èìåíà äîëæíû èìåòü çíàê "+" (ñîãëàñíî Ïðàâèë çàïèñè òàáëèö óìíîæåíèÿ), ïîýòîìó
+h +i +j +k
+i –h –k +j
+j +k –h +i
+k –j –i –h – ñóòü òàáëèöà óìíîæåíèÿ èçîìîðôíîãî êâàòåðíèîíà (ñîõðàíÿåòñÿ ãëàâíîå ñâîéñòâî êâàòåðíèîíà – ïîñëåäíÿÿ ñòðîêà òàáëèöû óìíîæåíèÿ ÿâëÿåòñÿ çåðêàëüíûì îòðàæåíèåì ïîñëåäíåãî ñòîëáöà ñî ñìåíîé çíàêà íà ïðîòèâîïîëîæíûé.
Îò÷¸òëèâî âèäíî, ÷òî òàáëèöà óìíîæåíèÿ îêòîíèîíîâ ñîñòîèò èç ÷åòûð¸õ êâàòåðíèîíîâ
+0 +1 +2 +3 +4 +5 +6 +7
+1 –0 +3 –2 +5 –4 –7 +6
+2 –3 –0 +1 +6 +7 –4 –5
+3 +2 –1 –0 +7 –6 +5 –4
+4 –5 –6 –7 –0 +1 +2 +3
+5 +4 –7 +6 –1 –0 –3 +2
+6 +7 +4 –5 –2 +3 –0 –1
+7 –6 +5 +4 –3 –2 +1 –0
Çäåñü ìû ñòàëêèâàåìñÿ ñ îñíîâíûì ôåíîìåíîì òåîðèè ìíîæåñòâ, ëåæàùåé â îñíîâàíèÿõ ìàòåìàòèêè: ãëàâíîå íå òî, êàê ýòè ñèñòåìû áûëè ïîëó÷åíû âïåðâûå, à ãëàâíîå òî, ÷òî îíè ñàìè î ñåáå "ãîâîðÿò". Áóäü òî þëèàíñêèé êàëåíäàðü, èëè Âèôëååìñêàÿ çâåçäà èëè òàáëèöà óìíîæåíèÿ ãèïåðêîìïëåêñíûõ åäèíèö, à îíà "ãîâîðèò", ÷òî ïîñëåäíèå äâå äâîè÷íûå öèôðû îïðåäåëÿþò èìåíà êâàòåðíèîíà, òîãäà êàê âñå îñòàëüíîå - ñóòü çíàêè è "äîâåñêè" ê èìåíàì. Äåéñòâèòåëüíî
+0 +1 +2 +3 +0 +1 +2 +3
+1 –0 +3 –2 +1 –0 –3 +2
+2 –3 –0 +1 +2 +3 –0 –1
+3 +2 –1 –0 +3 –2 +1 –0 ("äîâåñîê" ê èìåíàì âòîðîé òàáëèöå ðàâåí 4, à çíàêè èìåí âíå ãëàâíîé äèàãîíàëè çàìåíåíû íà ïðîòèâîïîëîæíûå)
+0 –1 –2 –3 –0 +1 +2 +3
+1 +0 –3 +2 –1 –0 –3 +2
+2 +3 +0 –1 –2 +3 –0 –1
+3 –2 +1 +0 –3 –2 +1 –0 (äîâåñîê ê ïåðâîé òàáëèöå ðàâåí 4, à çíàêè èìåí â ïîñëåäíèõ òð¸õ ñòîëáöàõ çàìåíåíû íà ïðîòèâîïîëîæíûå; äîâåñîê êî âòîðîé òàáëèöå ðàâåí 0, à çíàêè èìåí âíå äèàãîíàëè â ïîñëåäíèõ òð¸õ ñòðîêàõ çàìåíåíû íà ïðîòèâîïîëîæíûå).
Ôàêòè÷åñêè ìû îáíàðóæèëè áîëåå ýôôåêòèâíûé àëãîðèòì "óäâîåíèÿ" òàáëèöû, ðàáîòà êîòîðîãî áóäåò ïðîèëëþñòðèðîâàíà íà "óäâîåíèè îêòîíèîíà Êýëè". Îòïðàâëÿåìñÿ îò îêòîíèîíà Êýëè
+0 +1 +2 +3 +4 +5 +6 +7
+1 –0 +3 –2 +5 –4 –7 +6
+2 –3 –0 +1 +6 +7 –4 –5
+3 +2 –1 –0 +7 –6 +5 –4
+4 –5 –6 –7 –0 +1 +2 +3
+5 +4 –7 +6 –1 –0 –3 +2
+6 +7 +4 –5 –2 +3 –0 –1
+7 –6 +5 +4 –3 –2 +1 –0. Ïîä íèì ðàçìåùàåì åãî æå, óìíîæàÿ âñå ñòîëáöû íà –1, êðîìå ïåðâîãî
+0 –1 –2 –3 –4 –5 –6 –7 +0
+1 +0 –3 +2 –5 +4 +7 –6 –1 –0
+2 +3 +0 –1 –6 –7 +4 +5 –2 +3 –0
+3 –2 +1 +0 –7 +6 –5 +4 –3 –2 +1 –0
+4 +5 +6 +7 +0 –1 –2 –3 –4 +5 +6 +7 –0
+5 –4 +7 –6 +1 +0 +3 –2 –5 –4 +7 –6 +1 –0
+6 –7 –4 +5 +2 –3 +0 +1 –6 –7 –4 +5 +2 –3 –0
+7 +6 –5 –4 +3 +2 –1 +0 –7 +6 –5 –4 +3 +2 –1 –0. Ïîñëå ÷åãî âñ¸, ÷òî ëåæèò íèæå ãëàâíîé äèàãîíàëè âåðõíåé òàáëèöû, ïåðåíîñèì âïðàâî–âíèç ñ ïðîòèâîïîëîæíûì çíàêîì, çàïîëíèâ íèæíþþ ïîëîâèíó óäâîåííîãî îáúåêòà, îòðàæàåì åãî îò ãëàâíîé äèàãîíàëè ñî ñìåíîé çíàêà íà ïðîòèâîïîëîæíûé. Îñòàëîñü ïðèáàâèòü ê èìåíàì âòîðîé è òðåòüåé ÷åòâåðòè óäâîåííîé òàáëèöû "÷èñëî 8".
Îñíîâíûì ðàáî÷èì ýëåìåíòîì ýòîãî ýòàïà ÿâëÿåòñÿ "êâàòåðíèîí Ãàìèëüòîíà", ïîýòîìó ýòîò ýòàï èññëåäîâàíèÿ ãèïåðêîìïëåêñíûõ ñèñòåì ìîæíî îõàðàêòåðèçîâàòü, êàê "êâàòåðíèçàöèÿ".
Ñâèäåòåëüñòâî î ïóáëèêàöèè ¹223062901615