Global Joural of Sciece Frotier Research Mathematics ad Decisio Scieces Volume 12 Issue 11 Versio 10 Year 2012 Type : Double Blid Peer Reviewed Iteratioal Research Joural Publisher: Global Jourals Ic (USA Olie ISSN: 2249-42 & Prit ISSN: 0975-589 By Floreti Smaradache Uiversity of New Mexico, USA Abstract - I this paper we show how to usig the extesio trasformatio i I hig i order to trasformig a hexagram to aother oe Each biary hexagram (ad similarly the previous trigram has a degree of Yag ad a degree of Yi As i eutrosophic logic ad set, for each hexagram <H> there is correspodig a opposite hexagram <atih>, while i betwee them all other hexagrams are eutralities deoted by <euth>; a eutrality has a degree of <H> ad a degree of <atih> A geeralizatio of the trigram (which has three stacked horizotal lies ad hexagram (which has six stacked horizotal lies to -gram (which has stacked horizotal lies is provided Istead of stacked horizotal lies oe ca cosider stacked vertical lies - without chagig the compositio of the trigram/hexagram/-gram Afterwards, circular represetatios of the hexagrams ad of the -grams are give GJSFR-F lassificatio : MS 2010: 97G50 Strictly as per the compliace ad regulatios of : 2012 Floreti Smaradache This is a research/review paper, distributed uder the terms of the reative ommos Attributio-Nocommercial 30 Uported Licese http://creativecommosorg/liceses/by-c/30/, permittig all o commercial use, distributio, ad reproductio i ay medium, provided the origial work is properly cited
Floreti Smaradache 2012 Abstract - I this paper we show how to usig the extesio trasformatio i I hig i order to trasformig a hexagram to aother oe Each biary hexagram (ad similarly the previous trigram has a degree of Yag ad a degree of Yi As i eutrosophic logic ad set, for each hexagram <H> there is correspodig a opposite hexagram <atih>, while i betwee them all other hexagrams are eutralities deoted by <euth>; a eutrality has a degree of <H> ad a degree of <atih> A geeralizatio of the trigram (which has three stacked horizotal lies ad hexagram (which has six stacked horizotal lies to -gram (which has stacked horizotal lies is provided Istead of stacked horizotal lies oe ca cosider stacked vertical lies - without chagig the compositio of the trigram/hexagram/gram Afterwards, circular represetatios of the hexagrams ad of the -grams are give I Itroductio Year 1Global Joural of Sciece Frotier Research Volume XII Issue ersio I V XI I hig, which meas The Book of hages, is oe of the oldest classical hiese texts It is formed of 4 hexagrams I hig is part of the hiese culture, philosophy ad diviizatio Accordig to I hig everythig is i a cotiuous chage At the begiig, betwee 2800-2737 B, origiatig with the culture hero Fu Xi, there have bee 8 trigrams, ad withi the time of the legedary Yu (2194-2149 B the trigrams were expaded ito 4 hexagrams Each trigram was formed by three stacked horizotal lies The two trigrams formed a hexagram Therefore a hexagram is formed by six stacked horizotal lies; ad each stacked horizotal lie is either ubroke lie (, called Yag, or broke lie (, called Yi Yag is associated with MALE, positive, givig, creatio, digit 1, ad Yi is associated with FEMALE, egative, receivig, receptio, digit 0 i the Taoist philosophy I Taoism, Yag ad Yi complemet each other, like i the taijitu symbol: F Figure 1 Author : Uiversity of New Mexico, Mathematics ad Sciece Departmet 705 Gurey Ave Gallup, NM 87301, USA E-mail : smarad@umedu 2012 Global Jourals Ic (US
The umber of all possible trigrams formed with ubroke or broke lies is 2 3 = 8 Ad the umber of all possible hexagrams also formed with ubroke or broke lies is 2 = 4 A hexagram is formed by two trigrams: the first trigram (first three lies is called lower trigram ad represets the ier aspect of the chage, while the secod trigram (last three lies is called upper trigram ad represets the outer aspect of the chage II Aalyzig the Hexagrams Ref 22012 Year Global Joural of Sciece Frotier Research F Volume XII Issue XI V ersio I As i eutrosophy (which is a philosophy that studies the ature of etities, their opposites, ad the eutralities i betwee them, we have the followig for the I hig hexagrams: - To each hexagram <H> a ati-hexagram <atih> is correspodig, ad 2 eutral hexagrams <euth> are i betwee <H> ad <atih> - Each <euth> has a degree of <H> ad a degree of <atih> The degrees are amog the umbers 1/, 2/, 3/, 4/, 5/ ad the sum of the degree of <H> ad degree of <atih> is 1 - Let s ote the 2 eutral hexagrams by <euth 1 >, <euth 2 >,, <euth 2 > For each eutral hexagram <euth i > there is a eutral hexagram <euth j >, with i j, which is the opposite of it - For each stacked horizotal lie the extesio trasformatio is the followig: ad T( x T: {Yag, Yi} {Yag, Yi} _ x, where x _ is the opposite of x, ie T(Yag = Yi or T( = T(Yi = Yag or T( = To trasform a hexagram ito aother hexagram oe uses this extesio trasformatio oce, twice, three times, four times, five, or six times The maximum umber of extesio trasformatios used (six occurs whe we trasform a hexagram ito its opposite hexagram III Hexagram Table The below Hexagram Table is take from Iteret ([1] ad [2]; istead of stacked horizotal lies oe cosiders stacked vertical lies - without affectig the results of this article I this table oe shows the moder iterpretatio of each hexagram, which is a retraslatio of Richard Wilhelm s traslatio Hexagram Hexagram Table Moder Iterpretatio 01 Force ( 乾 qiá Possessig reative Power & Skill 02 Field ( 坤 kū Needig Kowledge & Skill; Do ot force matters ad go with the flow 1 Wilhelm (tras, Richard; ary Bayes (tras, "The I hig or Book of hages", from Iteret 2012 Global Jourals Ic (US
03 Sproutig ( 屯 zhū Sproutig 04 Evelopig ( 蒙 még Detaied, Eveloped ad Iexperieced 05 Attedig ( 需 xū Uivolvemet (Wait for ow, Nourishmet 0 Arguig ( 訟 sòg Egagemet i oflict 07 Leadig ( 師 shī Brigig Together, Teamwork 08 Groupig ( 比 bǐ Uio 09 Small Accumulatig ( 小畜 xiǎo chù Accumulatig Resources 10 Treadig ( 履 lǚ otiuig with Alertess 11 Pervadig ( 泰 tài Pervadig 12 Obstructio ( 否 pǐ Stagatio 13 ocordig People ( 同人 tóg ré Fellowship, Partership 14 Great Possessig ( 大有 dà yǒu Idepedece, Freedom 15 Humblig ( 謙 qiā Beig Reserved, Refraiig 1 Providig-For ( 豫 yù Iducemet, New Stimulus 17 Followig ( 隨 suí Followig 18 orruptig ( 蠱 gǔ Repairig 19 Nearig ( 臨 lí Approachig Goal, Arrivig 20 Viewig ( 觀 guā The Withholdig 21 Gawig Bite ( 噬嗑 shì kè Decidig 22 Adorig ( 賁 bì Embellishig 23 Strippig ( 剝 bō Strippig, Flayig 24 Returig ( 復 fù Returig 25 Without Embroilig ( 無妄 wú Without Rashess wàg 2 Great Accumulatig ( 大畜 dà chù Accumulatig Wisdom 27 Swallowig ( 頤 yí Seekig Nourishmet 28 Great Exceedig ( 大過 dà guò Great Surpassig 29 Gorge ( 坎 kǎ Darkess, Gorge 30 Radiace ( 離 lí ligig, Attachmet 31 ojoiig ( 咸 xiá Attractio 32 Perseverig ( 恆 hég Perseverace Joural of Sciece Frotier Research F Volume XII Issue XI V ersio I 3Global Year 2012 2012 Global Jourals Ic (US
2012 Year 4 Global Joural of Sciece Frotier Research F Volume XII Issue XI V ersio I Hexagram Moder Iterpretatio 33 Retirig ( 遯 dù Withdrawig 34 Great Ivigoratig ( 大壯 dà zhuàg Great Boldess 35 Prosperig ( 晉 jì Expasio, Promotio 3 Brightess Hidig ( 明夷 míg yí Brilliace Ijured 37 Dwellig People ( 家人 jiā ré Family 38 Polarisig ( 睽 kuí Divisio, Divergece 39 Limpig ( 蹇 jiǎ Haltig, Hardship 40 Takig-Apart ( 解 xiè Liberatio, Solutio 41 Dimiishig ( 損 sǔ Decrease 42 Augmetig ( 益 yì Icrease 43 Partig ( 夬 guài Separatio 44 ouplig ( 姤 gòu Ecouterig 45 lusterig ( 萃 cuì Associatio, ompaioship 4 Ascedig ( 升 shēg Growig Upward 47 ofiig ( 困 kù Exhaustio 48 Wellig ( 井 jǐg Repleishig, Reewal 49 Skiig ( 革 gé Abolishig the Old 50 Holdig ( 鼎 dǐg Establishig the New 51 Shake ( 震 zhè Mobilizig 52 Boud ( 艮 gè Immobility 53 Ifiltratig ( 漸 jià Auspicious Outlook, Ifiltratio 54 overtig The Maide ( 歸妹 guī mèi Marryig 55 Aboudig ( 豐 fēg Goal Reached, Ambitio Achieved 5 Sojourig ( 旅 lǚ Travel 57 Groud ( 巽 xù Subtle Ifluece 58 Ope ( 兌 duì Overt Ifluece 59 Dispersig ( 渙 huà Dispersal 0 Articulatig ( 節 jié Disciplie 1 etre ofirmig ( 中孚 zhōg fú Stayig Focused, Avoid Misrepresetatio 2 Small Exceedig ( 小過 xiǎo guò Small Surpassig 2012 Global Jourals Ic (US
3 Already Fordig ( 既濟 jì jì ompletio 4 Not-Yet Fordig ( 未濟 wèi jì Icompletio IV Examples of Extesio Trasformatios Used for Hexagrams Ref 3 ai We Extesio Set ad No-ompatible Problems [J] Joural of Scietific Exploratio, 1983, (1: 83-97 As a example of studyig the above Hexagram Table, let s take the first hexagram ad deote it by <H> = The its opposite diagram happeed to be its secod hexagram: <atih> = Their moder iterpretatio is cosistet with them, sice <H> meas Possessig reative Power & Skill, while <atih> meas the opposite, ie Needig Kowledge & Skill (because <atih> does t have kowledge ad skills Hexagram <H> is kow as Force, while <atih> as Field, or the Force works the Field As i Exteics fouded ad developed by ai We [3, 4], to trasform <H> ito <atih> oe uses the extesio trasformatio T(Yag=Yi six times (for each stacked vertical lie The other 2 hexagrams have a percetage of <H> ad a percetage of <atih> There are: 0 = 1 hexagram that has / = 100% percetage of <H> ad 0/ = 0% percetage of <atih>; 1 = hexagrams that have 5/ percetage of <H> ad 1/ percetage of <atih>; 2 = 15 hexagrams that have 4/ percetage of <H> ad 2/ percetage of <atih>; 3 = 20 hexagrams that have 3/ percetage of <H> ad 3/ percetage of <atih>; 4 = 15 hexagrams that have 2/ percetage of <H> ad 4/ percetage of <atih>; 5 = hexagrams that have 1/ percetage of <H> ad 5/ percetage of <atih>; = 1 hexagram that has 0/ = 0% percetage of <H> ad / = 100% percetage of <atih> The total umber of hexagrams is: k (1 1 1 15 20 15 1 4 k 0 Joural of Sciece Frotier Research F Volume XII Issue XI V ersio I 5Global Year 2012 For the followig eutral hexagram ( Gorge <euth 29 > = 2012 Global Jourals Ic (US
its opposite is aother eutral hexagram ( Radiace <euth 30 > = <euth 29 > ca be obtaied from the hexagram <H> by usig four times the extesio trasformatio T(Yag = Yi for the first, third, fourth, ad sixth stacked vertical lies 2012 Year Hexagram <euth 29 > is 2/ = 33% <H> ad 4/ = 7% <atih> <euth 30 > ca be obtaied from the hexagram <H> by usig two times the extesio trasformatio T(Yag = Yi for the secod, ad fifth stacked vertical lies Hexagram <euth 30 > is 4/ = 7% <H> ad 2/ = 33% <atih> Global Joural of Sciece Frotier Research F Volume XII Issue XI V ersio I V ircular Represetatio of the Hexagrams Shao Yug i the 11 th cetury has displayed the hexagrams i the formats of a circle ad of a rectagle We represet the hexagrams i the format of a circle, but such that each hexagram <H i > is diametrically opposed to its opposite hexagram <atih i > We may start with ay hexagram <H 0 > as the mai oe: <H 0 > <H 1 > <atih 31 > <H 2 > O O <atih 2 > <atih 1 > <H 31 > <atih 0 > Figure 2 2012 Global Jourals Ic (US
VI Geeralizatio of Hexa-Grams to N-Grams The 3-gram (or trigram ad the -gram (or hexagram ca be geeralized to a -gram, where is a iteger greater tha 1 We defie the -gram as formed by stacked horizotal lies; ad each stacked horizotal lie is either ubroke lie (, called Yag, or broke lie (, called Yi Therefore we talk about biary -grams The umber of all possible biary -grams is equal to 2 Similarly to hexagrams we have: - To each -gram <G> a ati--gram <atig> is correspodig, ad 2-2 eutral - grams <eutg> are i betwee <G> ad <atig> - Each <eutg> has a degree of <G> ad a degree of <atig> The degrees are amog the umbers 1/, 2/,, (-1/ ad the sum of the degree of <G> ad degree of <atig> is 1 - Let s ote the 2-2 eutral -grams by <eutg 1 >, <eutg 2 >,, eutg 2 For each 1 eutral -gram <eutg i > there is a eutral -gram <eutg j >, with i j, which is the opposite of it - For each stacked horizotal lie the extesio trasformatio is the same: Joural of Sciece Frotier Research Volume XII Issue XI V ersio I 7Global Year 2012 T: {Yag, Yi} {Yag, Yi} ad T( x _ x, where x _ is the opposite of x, ie T(Yag = Yi or T( = T(Yi = Yag or T( = F To trasform a -gram ito aother -gram oe uses this extesio trasformatio oce, twice, three times, ad so forth up to 2 2 times The maximum umber of extesio trasformatios used (2-2 occurs whe we trasform a -gram ito its opposite -gram To trasform a -gram <G> ito its opposite <atig> oe uses the extesio trasformatio T(Yag=Yi 2 times (for each stacked vertical lie The other 2 2 -grams have a percetage of <G> ad a percetage of <atig> There are: 0 = 1 -gram that have / = 100% percetage of <G> ad 0/ = 0% percetage of <atig>; 1 = -grams that have (-1/ percetage of <G> ad 1/ percetage of <atig>; 2 = (-1/2 -grams that have (-2/ percetage of <G> ad 2/ percetage of <atig>; 2012 Global Jourals Ic (US
k! k! k! <atig>; -grams that have (-k/ percetage of <G> ad k/ percetage of 2012 Year 8 = 1 -gram that has 0/ = 0% percetage of <G> ad / = 100% percetage of <atig> The total umber of -grams is: k (1 1 1 ( 1 / 2 2 k 0 Global Joural of Sciece Frotier Research F Volume XII Issue XI V ersio I VII ircular Represetatio of the N-Grams We represet the -grams i the format of a circle, but such that each -gram <G i > is diametrically opposed to its opposite -gram <atig i > We may start with ay -gram <G 0 > as the mai oe: <G 0 > <G 1 > eutg 1 2 1 <G 2 > O O <atig 2 > <atig 0 > G 1 2 1 <atig 1 > Figure 3 2012 Global Jourals Ic (US
VIII oclusio I this article the coectio betwee I hig (The Book of hage, Exteics, ad eutrosophics has bee made The a geeralizatio from aciet trigrams ad hexagrams to - grams, 1, was preseted at the ed, together with the geometric iterpretatios of hexagrams ad -grams A extesio trasformatio is used to chage from a hexagram to aother oe, ad i geeral from a -gram to aother -gram refereces référeces referecias 1 Wilhelm (tras, Richard; ary Bayes (tras, "The I hig or Book of hages", from Iteret 2 I hig, http://ewikipediaorg/wiki/i_hig 3 ai We Extesio Set ad No-ompatible Problems [J] Joural of Scietific Exploratio, 1983, (1: 83-97 4 Yag huya, ai We Extesio Egieerig [M] Beijig: Public Library of Sciece, 2007 5 F Smaradache, Geeralizatios of the Distace ad Depedet Fuctio i Exteics to 2D, 3D, ad -D, vixraorg, http://vixraorg/abs/1200014 ad http://vixraorg/pdf/1200014v1pdf, 2012 F Smaradache, V Vlădăreau, Applicatios of Exteics to 2D-Space ad 3D-Space, vixraorg, http://vixraorg/abs/1200043 ad http://vixraorg/pdf/1200043v2pdf, 2012 Joural of Sciece Frotier Research Volume XII Issue XI V ersio I 9Global Year 2012 F 2012 Global Jourals Ic (US