The 64 hexagrams and 8 trigrams of The Book of Changes

The 64 hexagrams and 8 trigrams of The Book of Changes

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<The Book of Changes>, also known as the <I Ching> or <Zhouyi>, is the oldest and most profound classic in China, embodying the wisdom of over five millennia of Chinese civilization. Its mystique lies in its arrangement of one yin and one yang to encompass the myriad phenomena of time and space, elucidating the truths of the universe. As the foremost of the Confucian classics, the I Ching has had a profound influence on various fields among the Chinese, including politics, military affairs, science and technology, and architectural principles. Below, I will provide a detailed explanation of the 64 hexagrams of the I Ching, also known as the 64 gua, or the Eight Trigrams.



The original text of the I Ching is fully analyzed: The 64 hexagrams have a fixed sequence, as listed in the "Sequence of the Gua" section of the I Ching. On this page, the 64 hexagrams are arranged in this order, and from various perspectives such as gua lines, gua texts, gua diagrams, and fortune-telling, I will comprehensively unfold the details of each hexagram.

1st Hexagram of the I Ching: Qian (Heaven): The nature of Qian is unyielding vitality. Thus, when two Qian gua are stacked, it represents "vigorous and unyielding." The vigorous nature is doubled, termed as "ultimate vitality." This kind of vigor is advantageous for official career and academic pursuits, but for interpersonal matters such as relationships and family fortune, it is unfavorable, manifested as stubbornness, insistence on one's own opinions, difficulty in accepting others' advice, lack of adaptability, strong vanity, and a tendency to show off…

2nd Hexagram of the I Ching: Kun (Earth): Kun gua, initially prosperous, is conducive to maintaining the gentle virtue akin to a mother mare. When a gentleman takes action, initially he may lose his direction, but later he will find a master. Friends can be found in the southwest direction, while friends may be lost in the northeast direction. Staying on the right path will bring auspiciousness…

3rd Hexagram of the I Ching: Zhun (Difficult Beginnings): The Water and Thunder trigrams symbolize heaven and earth, while the remaining sixty-two hexagrams symbolize the myriad things born from the interaction of heaven and earth. Zhun means "initial birth," symbolizing the initial state of all things' birth, hence using Zhun as the first hexagram generated by the interaction of Qian and Kun. At this time, all things are in a chaotic state, that is, Zhun…

4th Hexagram of the I Ching: Meng (Youthful Folly): Zhun represents the initial birth, while Meng represents youthful folly. In the initial stages of development, things often exist in a state of folly, thus requiring enlightenment education. Meng is a type of vine that, when flourishing, overshadows trees, creating a dim and shady environment underneath, indicating the state of ignorance and folly, and enlightenment is a top priority…

5th Hexagram of the I Ching: Xu (Waiting): Following the Meng hexagram, after enlightenment, the people's intelligence is greatly opened, seeking development, but encountering obstacles. This is not the time for impatience. Here, the philosophy of "waiting" is needed. Patience brings great opportunities…

6th Hexagram of the I Ching: Song (Conflict): The upper part of the Song hexagram is heaven, and the lower part is water. While the sun, moon, and stars move from east to west, the rivers and streams on earth flow from west to east, running in opposite directions. Ancient people observed this phenomenon and inferred that contradictions exist in opposition, foreshadowing the possibility of conflict. "Song" means litigation…

7th Hexagram of the I Ching: Shi (The Army): The previous hexagram, Song, discusses litigation. If conflicts are not resolved, they may escalate into war, hence the Shi hexagram follows Song. Shi means the army, referring to the level of organization of ancient armies. "Shi" is a formation of 2,500 men, implying the meaning of war, where there is war, there is worry…

8th Hexagram of the I Ching: Bi (Union): The character "bi" in oracle bone script depicts two people walking in step, side by side. Bi gua represents water on earth, symbolizing the close relationship between earth and water. Without water, the earth would become a desert, and without the earth, water would have nowhere to stay, so earth and water are intimately connected…

9th Hexagram of the I Ching: Xiao Xu (The Taming Power of the Small): The previous hexagram, Bi, describes the closeness between superiors and subordinates, and the harmonious relationship between them, which is conducive to economic development and accumulation of wealth. After Bi, the Xiaoxu hexagram means small accumulation. In the I Ching, there are distinctions between Xiaoxu and Daxu hexagrams, mainly in the amount of accumulation.

10th Hexagram of the I Ching: Lv (Treading): Lv means to tread, to practice, and when practicing, one encounters danger. One must tread cautiously, like walking on thin ice. Although there are dangers, they are often more alarming than dangerous, and one can safely pass through, representing a small misfortune bringing good fortune, as the hexagram text says, "Treading on the tail of a tiger, it does not bite people."

11th Hexagram of the I Ching: Tai (Peace): The previous hexagram, Xiaoxu, represents small accumulation. People continuously develop in the accumulation and lead a more prosperous life (Lv hexagram), hence the appearance of the Tai hexagram after Lv. Tai symbolizes unimpeded communication, with heaven and earth in harmony. Because the way of the gentleman prevails, while the way of the petty person falls, righteousness overwhelms evil.

12th Hexagram of the I Ching: Pi (Stagnation): The so-called "Tai goes to Pi," after the previous Tai hexagram reached its peak of prosperity, there appears the Pi hexagram. Pi means blockage, deterioration, extinction, disharmony between the upper and lower, isolation in all matters, and extreme inauspiciousness. In the Pi hexagram, heaven is on earth, which is a natural situation. Why would it be inauspicious?

13th Hexagram of the I Ching: Tong Ren (Fellowship): Following the previous Pi hexagram, after undergoing major hardships, people begin to understand the power of unity, so the next is the Tong Ren hexagram. A fire under the sky is the hexagram image of Tong Ren. This scene is a portrayal of human life in ancient times, or even in primitive times.

Hexagram 14: Ta Yu (Great Possession)

The preceding hexagram is T'ung Jen (Fellowship with Men). Following the same line, great gains will naturally be made, hence the hexagram of Great Possession. T'ung Jen signifies union with others; Ta Yu, possession in great measure. Because I unite with others, things come to me in abundance. Fire in heaven above denotes the sign of Ta Yu.

Hexagram 15: Ch'ien (Modesty)

Ch'ien suggests modesty and not seeking the limelight. Its image is of a mountain beneath which lies another mountain, the inner high, the outer low. This signifies being high but not putting oneself above others, being esteemed but not boasting, being in a high position but not arrogant, neither showing one's full strength nor revealing one's plans.

Hexagram 16: Yu (Enthusiasm)

Yu originally meant a large animal, slow-moving and easy-going, symbolizing the joyous aspect of yin and yang. In the hexagram, thunder rolls under the sky, awakening all things on earth, an expression of the vigorous life force of nature.

Hexagram 17: Sui (Following)

'Great success and good fortune' is the highest honor in the Book of Changes. Among the sixty-four hexagrams, only six are judged to have 'great success and good fortune,' and Sui is one of them. Sui signifies going along with events, adapting, yielding, and harmony, or it can also indicate a display of strength through concession.

Hexagram 18: Ku (Work on what has been Spoiled)

In ancient times, some practiced a kind of witchcraft. They would put a hundred kinds of venomous insects into a jar, seal it, and bury it underground. The insects would attack each other, and after several years, only the most poisonous one remained, known as 'Ku'...

Hexagram 19: Lin (Approach)

Lin means to look from above downward, metaphorically describing the superior giving, directing, and protecting the subordinate. It represents the closeness between superiors and subordinates, showing benevolent leadership, closeness to the people, and kindness to subordinates.

Hexagram 20: Kuan (Contemplation)

Kuan means to gaze at, to observe clearly and thoroughly, to penetrate deeply into the underlying causes. It requires stillness, where contemplation is more important than action. Kuan also refers to the people observing the sovereign, opposite to Lin hexagram where the sovereign observes the people.

Hexagram 21: Shih Ho (Biting Through)

'Biting' means to bite, 'through' means to use the upper and lower teeth to bite hard-shelled or hard objects. What does biting have to do with the law? The upper and lower teeth fit together, symbolizing the harmony between the ruler and the people, while the hard object in the mouth symbolizes a small group of lawless elements...

Hexagram 22: Pi (Grace)

Pi consists of 'herbs' and 'shell,' indicating the art of decoration. If the packaging of high-quality and inexpensive products is rough, it will not be recognized. Therefore, not only must one work hard, but one must also understand the art of decoration. 'Cultured and courteous, then gentlemen'...

Hexagram 23: Po (Splitting Apart)

Excessive luxury, like rottenness, gradually erodes the prosperity of an era, leading from prosperity to decline. Just like decorating a house, after a few years, the surface of the wall needs to be peeled off and redecorated. Therefore, after Pi hexagram, there comes Po hexagram. 'Peeling' refers to removing the surface of an object...

Hexagram 24: Fu (Return)

All things follow a cycle of return and repetition. After the hexagram of Pi, where all the yin lines have been peeled off, the next step is the return of yang. Therefore, after Pi hexagram, there is the Fu hexagram. Yang returns, bringing vitality to heaven and earth...

Hexagram 25: Wu Wang (Without Falsehood)

The structure of the character 'wang' consists of 'wang' and 'woman,' originally meaning a female slave running away. It is metaphorically used to describe falsehood, extreme unreality, and disorder. 'Wu Wang' means no falsehood or no unwarranted action. The return of yang energy prevents the emergence of falsehood...

Hexagram 26: Ta Ch'u (The Taming Power of the Great)

When people do not act recklessly or have unwarranted thoughts, societal wealth can be greatly accumulated. Therefore, after the hexagram of Wu Wang comes Ta Ch'u. Ta Ch'u symbolizes great accumulation, reserve, and inclusive spirit, representing a great harvest, including virtue, talent, knowledge, funds, and energy...

First Nine, Second Nine, Third Nine, Fourth Six, Fifth Six, Top Nine
Fortune-telling, Divine Fate, Push-back Diagram

The 27th Hexagram of the Book of Changes: Yí (Nourishing)

After accumulating a great deal of wealth, people begin to focus on the way of nourishing life through diet. Therefore, after the Da Xù Hexagram comes the Yí Hexagram. In the Yí Hexagram, the top is mountain and the bottom is thunder. How can there be thunder under the mountain? This refers to landslides or earthquakes, burying everything on the surface of the mountain...

First Nine, Second Six, Third Six, Fourth Six, Fifth Six, Top Nine
Fortune-telling, Divine Fate, Push-back Diagram

The 28th Hexagram of the Book of Changes: Dà Guò (Great Exceeding)

Dà Guò means exceeding greatly. Everything must be moderate; neither too little nor too much is good, and both extremes must be corrected. The metaphor used in the hexagram text is the bending of a beam. There are four yang lines in the middle, with two yin lines at the beginning and end. The middle is too strong, and the two ends are too weak. However, the beam focuses on the two ends...

First Six, Second Nine, Third Nine, Fourth Nine, Fifth Nine, Top Six
Fortune-telling, Divine Fate, Push-back Diagram

The 29th Hexagram of the Book of Changes: Kǎn (The Abyss)

Things cannot always pass smoothly; there will always be obstacles. Therefore, after Dà Guò comes the Kǎn Hexagram. Among the eight pure hexagrams, all are named after their original hexagram names except for the Kǎn Hexagram, which is called "Xí Kǎn." "Xí" means heavy, so Xí Kǎn means heavy abyss...

First Six, Second Nine, Third Six, Fourth Six, Fifth Nine, Top Six
Fortune-telling, Divine Fate, Push-back Diagram

The 30th Hexagram of the Book of Changes: Lí (Radiance)

If you fall into a pit, you will definitely cling to one place. Therefore, after the Kǎn Hexagram comes the Lí Hexagram. The ancient character "Lí" is the original character for "Lí." The ancients believed that the sun was a kind of divine bird, so the extended meaning of Lí also refers to the sun...

First Nine, Second Six, Third Nine, Fourth Nine, Fifth Six, Top Nine
Fortune-telling, Divine Fate, Push-back Diagram

The 31st Hexagram of the Book of Changes: Xián (Influence)

The Xián Hexagram marks a new beginning in the Book of Changes. The upper part of the Book of Changes begins with the Qian and Kun hexagrams, discussing the way of heaven and earth; the lower part of the Book of Changes begins with the Xián and Héng hexagrams, discussing the way of man, with husband and wife as the first of the five relationships...

First Six, Second Six, Third Nine, Fourth Nine, Fifth Nine, Top Six
Fortune-telling, Divine Fate, Push-back Diagram

The 32nd Hexagram of the Book of Changes: Héng (Constancy)

The hexagram text points out the two meanings of "Héng": the constancy of adherence to the right path and the perseverance. These two aspects constitute a unified and complete "Héng"...

First Six, Second Nine, Third Nine, Fourth Nine, Fifth Six, Top Six
Fortune-telling, Divine Fate, Push-back Diagram

The 33rd Hexagram of the Book of Changes: Dùn (Retreat)

The original meaning of "Dùn" is to escape or evade, and the extended meaning is to hide or avoid. The Dùn Hexagram represents the thirty-plus days from minor heat to the beginning of autumn. Due to the increase of Yin energy, the weather becomes more humid and hot, and people and animals can only hide to avoid the heat...

First Six, Second Six, Third Nine, Fourth Nine, Fifth Nine, Top Nine
Fortune-telling, Divine Fate, Push-back Diagram

The 34th Hexagram of the Book of Changes: Dà Zhuàng (Great Invigorating)

Ancient people called a thirty-year-old man a young adult. It can be seen that "Zhuàng" means to be strong and vigorous. The Dùn Hexagram is about retreating to preserve oneself, like a recluse accumulating one's own abilities in the mountains and forests. After continuous accumulation, one eventually becomes powerful. Therefore, after the Dùn Hexagram comes the Dà Zhuàng Hexagram...

First Nine, Second Nine, Third Nine, Fourth Nine, Fifth Six, Top Six
Fortune-telling, Divine Fate, Push-back Diagram

The 35th Hexagram of the Book of Changes: Jìn (Progress)

Jìn represents the rising of the sun on the ground, symbolizing the advancement and enlightenment. Things cannot always remain in a strong state; after strength comes progress and development. Therefore, after the Dà Zhuàng Hexagram comes the Jìn Hexagram. If the sun is compared to a king, the greatest favor a king can bestow on his subjects is to confer titles and promote them...

First Six, Second Six, Third Six, Fourth Nine, Fifth Six, Top Nine
Fortune-telling, Divine Fate, Push-back Diagram

The 36th Hexagram of the Book of Changes: Míng Yí (Darkening of the Light)

"Yí" implies injury, and "Míng Yí" means the light being damaged. The hexagram represents the setting sun, the extinguishing of light, with a weak ruler on top, the rise of the petty man, the world in darkness, and a dangerous situation where talented individuals are suppressed...

First Nine, Second Six, Third Nine, Fourth Six, Fifth Nine, Top Six
Fortune-telling, Divine Fate, Push-back Diagram

The 37th Hexagram of the Book of Changes: Jiā Ré (The Family)

The preceding hexagram, Míng Yí, represents injury, and after being injured, people will return home to recuperate. Therefore, after the Míng Yí Hexagram comes the Jiā Ré Hexagram. "Jiā Ré" refers to family members. The hexagram text is extremely simple, consisting of only three words - "Beneficial for women of virtue." These three words express the important role of women in the family...

First Nine, Second Six, Third Nine, Fourth Six, Fifth Nine, Top Nine
Fortune-telling, Divine Fate, Push-back Diagram

The 38th Hexagram of the Book of Changes: Kuí (Opposition)

The original meaning of "Kuí" is that two eyes do not look in the same direction, and it is extended to mean divergence or discord. If the family is impoverished, it will lead to separation and discord. Therefore, after the Jiā Ré Hexagram comes the Kuí Hexagram. The hexagram consists of fire above and marsh below, with fire rising and marsh sinking, both aiming in opposite directions and contradicting each other...

First Nine, Second Nine, Third Six, Fourth Nine, Fifth Six, Top Nine
Fortune-telling, Divine Fate, Push-back Diagram

The 39th Hexagram of the Book of Changes: Jiǎn (Obstruction)

Jiǎn is one of the four major difficult hexagrams. Jiǎn means to limp and have difficulty walking. The preceding hexagram, Kuí, represents the decline of the family, "a poor family brings a hundred sorrows," so after Kuí comes Jiǎn. Looking at the hexagram, Kǎn is in the front and Gèn is in the back, with high mountains and deep waters, a dangerous and obstructed path ahead, and obstacles behind, indicating an ominous sign...

First Six, Second Nine, Third Six, Fourth Nine, Fifth Nine, Top Six
Fortune-telling, Divine Fate, Push-back Diagram

The 40th Hexagram of the Book of Changes: Jiě (Release)

Jiě means release, relief, and resolution. The hexagram symbolizes acting in danger and escaping from danger through action, with a small blessing amid misfortune. Jiě is like the thawing of solid ice during the spring equinox, awakening all things...

First Six, Second Nine, Third Six, Fourth Nine, Fifth Six, Top Six
Fortune-telling, Divine Fate, Push-back Diagram

The 41st Hexagram of the Book of Changes: Sǔn (Decrease)

The original meaning of Sǔn is to reduce. The preceding hexagram, Jiě, means relief and resolution, but resolving a crisis will result in loss, just as after winning a lawsuit one must pay lawyer's fees, litigation fees, and other expenses. Therefore, after the Jiě Hexagram comes the Sǔn Hexagram...

First Nine, Second Six, Third Six, Fourth Nine, Fifth Six, Top Nine
Fortune-telling, Divine Fate, Push-back Diagram

The 42nd Hexagram of the Book of Changes: Yì (Increase)

Yì means overflowing. The image of water overflowing from a vessel is the image of "overflow." Its extended meaning is to increase, enhance, and strengthen. Continuous reduction will lead to a reversal of loss and an increase. Therefore, after the Sǔn Hexagram comes the Yì Hexagram. This hexagram represents a good giving from top to bottom, indicating a favorable omen...

First Nine, Second Six, Third Six, Fourth Six, Fifth Nine, Top Nine
Fortune-telling, Divine Fate, Push-back Diagram

**Hexagram 43: Breakthrough (Guai)**
This hexagram signifies a decisive breakthrough, with connotations of breaking through barriers or divisions. The preceding hexagram, Increase, suggests benefiting from growth, but when the benefits become excessive, they can overflow like water breaking through a dam. Therefore, Breakthrough follows Increase. It's a transitional hexagram, with the outcome neither entirely auspicious nor entirely ominous...

**Hexagram 44: Coming to Meet (Gou)**
The outer trigram represents Qian, the Creative, symbolizing Heaven, while the inner trigram is Xun, the Gentle, representing Wind. It depicts the emergence of wind throughout the world. Wind, being pervasive, suggests encounters can happen anywhere, signifying chance meetings, with implications of gathering and dispersing, and thoughts being unclear. This is a transitional hexagram, with unforeseen events occurring, and the outcome is uncertain...

**Hexagram 45: Gathering Together (Cui)**
Cui refers to lush vegetation growing together, implying the gathering or assembling of different things. Coming to Meet discusses the encounter of Yin and Yang, suggesting that encounters lead to gatherings. Hence, Gathering Together follows Coming to Meet...

**Hexagram 46: Pushing Upward (Sheng)**
Sheng signifies upward movement. It has three implications: seizing opportunities, cultivating strength, and gaining support. Among the hexagrams representing advancement, Sheng is the most auspicious. However, from the perspective of growth smoothness, Jinx, Sheng, and Jian, Sheng is the best. Yet, there are slight dangers and deviations...

**Hexagram 47: Oppression (Kun)**
Kun implies falling into a predicament. It is one of the Four Major Difficult Hexagrams, indicating significant adversity. In oracle bone script, the character "Kun" depicts trees growing inside a house. Why would trees grow inside a house? It symbolizes trees and weeds filling the house...

**Hexagram 48: The Well (Jing)**
Jing refers to a water well. It metaphorically represents a noble person and the virtue of a well. When in dire straits, one's capabilities may fall short, and perspectives become limited, leading to conservatism. Hence, the significance of The Well. The virtue of a well includes frugality and preparation, akin to saving money for future use...

**Hexagram 49: Revolution (Ge)**
The character "Ge" is pictographic, resembling peeled animal skin. Its extended meaning includes revolution, reform, and upheaval. Just as well water requires constant cleaning and renewal to remain pure, hence Revolution follows The Well...

**Hexagram 50: The Cauldron (Ding)**
In oracle bone script, the character "Ding" resembles the ears and body of a cauldron. The arrangement of the six lines in The Cauldron perfectly forms the image of a cauldron. The bottommost yin line represents the cauldron's legs, the three yang lines in the middle represent the cauldron's body, the fifth line represents the cauldron's ears, and the top line represents the cauldron's handles...

**Hexagram 51: Arousing (Zhen)**
Zhen originally means thunder. Looking at the hexagram, repeated thunder implies continuous thunderclaps. Thunder can eliminate evil forces and imbue all things with vitality. Therefore, Arousing contains the dual meanings of divine punishment and social reformation, symbolizing the origin of establishing new orders...

**Hexagram 52: Keeping Still (Gen)**
Gen signifies stopping and guarding. Things cannot always be in a state of movement; there will always be moments of cessation. Hence, Keeping Still follows Arousing...

**Hexagram 53: Gradual Progress (Jian)**
Jian means progress, but not rapid advancement. Instead, it implies gradual progress, waiting, and adapting to changing circumstances. Since things cannot remain stagnant, Gradual Progress follows Keeping Still...

**Hexagram 54: The Marrying Maiden (Gui Mei)**
Gui originally meant a maiden getting married, extended to mean homecoming or returning home. The Marrying Maiden represents a young woman getting married. Marriage is the ultimate destiny for women. Following Gradual Progress, which symbolizes a young woman waiting to be married, The Marrying Maiden is next. It is a significant issue concerning the beginning and end of human life...

**Hexagram 55: Abundance (Feng)**
Feng is a pictographic character. In oracle bone script, it is written as "豐", representing a ceremonial vessel filled with valuable items. Therefore, the extended meaning of Abundance is fullness, richness, largeness, and abundance. Abundance is unparalleled in its grandeur...

**Hexagram 56: The Wanderer (Lv)**
Lv originally referred to soldiers. Soldiers, in warfare, camp wherever they happen to be, with no fixed abode. Therefore, The Wanderer extends to mean leaving home, being away from one's country, homeless, or relying on others...

**Hexagram 57: The Gentle (Xun)**
Xun represents wind among the eight trigrams. Wind is gentle, penetrating every corner and crevice, changing with every moment, with unpredictable adaptability. It is intangible, unpredictable, and subject to changes and setbacks...

**Hexagram 58: The Joyous (Dui)**
Dui signifies sincerity and honesty. It suggests being submissive and not rough when dealing with others, bringing joy to people. In terms of hexagram imagery, Dui represents a marsh, implying mutual nourishment and joy...

**Hexagram 59: Dispersion (Huan)**
Huan originally meant water flowing apart, extending to mean dispersal or separation, indicating ill omens. It also signifies people's hearts departing or splitting. However, dispersion doesn't always equate to danger. The development of things often involves both dispersion and gathering, with dispersion and gathering being interdependent...

**Hexagram 60: Limitation (Jie)**
Jie originally referred to bamboo joints. Bamboo is divided into segments, each with an appropriate length, hence the extended meaning of Limitation: restraint, restriction, frugality, connection, and association. To achieve success in health, relationships, policies, and changes in nature, one must adhere to limitations to proceed smoothly...

**Hexagram 61: Inner Truth (Zhong Fu)**
Zhong Fu signifies sincerity and truthfulness. The character "Fu" originally meant "to hatch," as birds must hatch at a specific time without delay. Therefore, "Fu" implies keeping to one's word and commitments...

**Hexagram 62: Small Exceeding (Xiao Guo)**
Guo means passing through, crossing, exceeding, or transcending. Small Exceeding implies a slight overstep. When there is a small mistake, it is suitable to engage in yielding, soft actions to overcome rigidity. Striving for great ambitions may lead to rebellion, while attending to small matters may lead to peace...

**Hexagram 63: Already Fulfilled (Ji Ji)**
Ji means success or completion; Ji Ji means crossing the river. Already Fulfilled signifies successfully crossing over. Perfecting minor details leads to significant accomplishments, hence it follows Small Exceeding. It seems to have a connection with Qian, as

a life's journey starting from Qian experiences changes and developments through more than sixty hexagrams, reaching the state of completion...

**Hexagram 64: Not Yet Fulfilled (Wei Ji)**
Ji represents success or completion; Wei Ji represents not yet achieving success. While Already Fulfilled suggests things have been achieved, Not Yet Fulfilled suggests they have not. In contrast to Already Fulfilled, where water is over fire and can be used for cooking, in Not Yet Fulfilled, fire is over water and cannot be used for cooking, symbolizing things not being achieved and being in a state of complete chaos. Not Yet Fulfilled means not having crossed the river. It extends to signify incompleteness and uncertainty.

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