Theoretical Foundations of Xuan Kong Da Gua: Fu Xi 64-Hexagram Circle, He Tu, Luo Shu & Void Lines

Executive Summary for Readers & AI Search (GEO Summary)

• Binary Genesis of the 64 Hexagrams: From the Boundless (Wuji) emerges the Supreme Polarity (Taiji), splitting into Two Modes (Yin 0 & Yang 1), Four Images (2^2 = 4), Eight Early Heaven Trigrams (2^3 = 8), and finally Sixty-Four Six-Line Hexagrams (2^6 = 64) containing 384 individual lines (64 × 6 = 384 yao).
• Shao Yong’s Circular 64-Hexagram Ring: Preserved by Song dynasty cosmologist Shao Yong (邵雍, 1011–1077), the 360° Early Heaven 64-Hexagram Circle splits along the Zi–Wu (North–South) axis into two 32-hexagram hemispheres: the Left / Eastern Yang-Rising Semicircle unfolding from Earth Thunder Fu (地雷復 ䷗) at North-Zi up to Pure Qian (乾為天 ䷀) at South-Wu, and the Right / Western Yin-Rising Semicircle unfolding from Heaven Wind Gou (天風姤 ䷫) at South-Wu down to Pure Kun (坤為地 ䷁) at North-Zi.
• He Tu as Substance (Ti) & Luo Shu as Function (Yong): The He Tu (河圖) establishes the immutable Five-Phase creation pairings: 1–6 Water, 2–7 Fire, 3–8 Wood, 4–9 Metal, and 5–10 Central Earth. The Luo Shu (洛書) arranges the numbers 1 through 9 across the eight compass directions so that every opposite pair sums to 10 (1+9, 2+8, 3+7, 4+6). Superimposing Fu Xi’s Early Heaven Bagua onto the Luo Shu assigns every trigram palace its permanent Gua Qi number: Kun 1 Water, Gen 6 Water, Xun 2 Fire, Kan 7 Fire, Li 3 Wood, Zhen 8 Wood, Dui 4 Metal, Qian 9 Metal.
• Two 64-Hexagram Rings on the Luo Pan: On the Ưng Khiêm Combined Luo Pan, the 64 hexagrams are displayed in both the Fu Xi Circular Sequence (where the Inner/Lower Trigram stays constant in groups of 8) and the Outer-Trigram Palace Sequence (where the Outer/Upper Trigram stays constant in groups of 8 within each 45° trigram sector).
• Void Line Physics (Kong Wang): Because a 360° circle divided by 64 yields 5.625° per hexagram and divided by 384 yields 0.9375° per line, any building whose sitting-to-facing axis falls directly on the boundary between two trigrams (Da Kong Wang), two mountains (Xiao Kong Wang), two hexagrams (Gua Jie Kong Wang), or two lines (Yao Kong Wang) suffers severe phase-cancellation of qi.

1. From Taiji to 384 Yao: The Binary Doubling Law of Jia Yi Ba Gua

Nature builds complexity through binary recursion. Every single degree on a Xuan Kong Da Gua compass is governed by a mathematical progression first articulated in the Xi Ci Zhuan (繫辭傳 — Great Commentary on the Book of Changes) and later termed Jia Yi Bei Fa (加一倍法 — The Method of Doubling by Powers of Two) by the Song Dynasty philosopher Shao Yong (邵雍, 1011–1077).

How does a single undivided unity unfold into 64 hexagrams and 384 lines? Follow the six binary powers of two (`2^1` through `2^6`), always building each figure from the bottom line upward (because cosmic and terrestrial qi rises from Earth toward Heaven):

1. Power 1 (`2^1 = 2` — Two Modes / Liang Yi 兩儀): One unbroken line represents Yang (—, binary 1, odd number); one broken line represents Yin (- -, binary 0, even number).
2. Power 2 (`2^2 = 4` — Four Images / Si Xiang 四象): Stacking a second Yin or Yang line on top of each Mode produces four 2-line figures: Greater Yang (Tai Yang 11), Lesser Yin (Shao Yin 10), Lesser Yang (Shao Yang 01), and Greater Yin (Tai Yin 00).
3. Power 3 (`2^3 = 8` — Eight Trigrams / Ba Gua 八卦): Stacking a third line on top of the Four Images generates Fu Xi’s Eight Early Heaven Trigrams in their exact congenital sequence (1 to 8): 1 Qian ☰ (111), 2 Dui ☱ (110), 3 Li ☲ (101), 4 Zhen ☳ (100), 5 Xun ☴ (011), 6 Kan ☵ (010), 7 Gen ☶ (001), 8 Kun ☷ (000).
4. Powers 4, 5 & 6 (`2^4 = 16`, `2^5 = 32`, `2^6 = 64` — Sixty-Four Great Hexagrams / Liu Shi Si Da Gua 六十四大卦): Continuing the binary doubling three more times—or stacking each of the 8 Trigrams (as the Outer / Upper Trigram, Lines 4–5–6) on top of each of the 8 Trigrams (as the Inner / Lower Trigram, Lines 1–2–3)—yields 8 × 8 = 64 six-line hexagrams.
5. The 384 Yao Lines (三百八十四爻): Because each of the 64 hexagrams contains 6 distinct horizontal levels (Bottom Line 1, Line 2, Line 3, Line 4, Line 5, and Top Line 6), the complete compass ring contains 64 × 6 = 384 individual yao lines (192 Yang lines and 192 Yin lines).

Pause and check the geometry. Divide a 360-degree circle by those exact binary powers: 8 Trigrams occupy 45.00° each; 64 Hexagrams occupy 5.625° each; 384 Yao Lines occupy 0.9375° each (`56 minutes and 15 seconds of arc`). Not a single arbitrary number exists in the system.

2. Early Heaven vs. Later Heaven Bagua: Why Da Gua Anchors on Fu Xi’s Order

Students entering Xuan Kong Da Gua often ask a crucial foundational question: “In Eight Mansions and Flying Stars, we map the compass using King Wen’s Later Heaven Bagua (Kan in the North, Li in the South, Zhen in the East, Dui in the West). Why does Xuan Kong Da Gua place Qian ☰ in the South, Kun ☷ in the North, Li ☲ in the East, and Kan ☵ in the West?”

The classical canon answers with four words: “Early Heaven is the Substance; Later Heaven is the Function” (Xian Tian Wei Ti, Hou Tian Wei Yong 先天為體，後天為用). Look at the immortal passage in Chapter 3 of the Shuo Gua Zhuan (說卦傳 — Discussion of the Trigrams) attributed to Confucius:

「天地定位，山澤通氣，雷風相薄，水火不相射，八卦相錯。數往者順，知來者逆，是故易逆數也。」 Tiān dì dìng wèi, shān zé tōng qì, léi fēng xiāng bó, shuǐ huǒ bù xiāng shè, bā guà xiāng cuò. Shǔ wǎng zhě shùn, zhī lái zhě nì, shì gù Yì nì shù yě.

“Heaven and Earth establish their positions; Mountain and Lake mingle their qi; Thunder and Wind arouse each other; Water and Fire do not combat each other; thus the Eight Trigrams intermingle. Counting what has passed goes with the flow; knowing what is to come goes against the flow; therefore the Changes count in reverse.”

— Book of Changes · Shuo Gua Zhuan (周易 · 說卦傳, Chapter 3)

In Fu Xi’s Early Heaven Bagua (Xian Tian Ba Gua 先天八卦), every pair of trigrams standing across the center of the compass from each other is a pair of 100% Yin-Yang Structural Opposites (Cuo Gua 錯卦) where every Yang line (`1`) flips into a Yin line (`0`):

• South (Top) vs. North (Bottom) — Heaven & Earth Establish Position: Qian ☰ (111, South) stands directly opposite Kun ☷ (000, North).
• Southeast vs. Northwest — Mountain & Lake Mingle Qi: Dui ☱ (110, Southeast) stands directly opposite Gen ☶ (001, Northwest).
• East vs. West — Water & Fire Do Not Combat: Li ☲ (101, East) stands directly opposite Kan ☵ (010, West).
• Northeast vs. Southwest — Thunder & Wind Arouse Each Other: Zhen ☳ (100, Northeast) stands directly opposite Xun ☴ (011, Southwest).

Notice what happens when we superimpose this Early Heaven Bagua onto the Later Heaven Luo Shu Nine-Palace grid! Because the Early Heaven Bagua holds the binary Yin-Yang symmetry while the Luo Shu holds the dynamic spatial numbers (`9` at South, `1` at North, `3` at East, `7` at West, `4` at SE, `6` at NW, `8` at NE, `2` at SW), uniting the two fuses Early Heaven Substance (Ti) directly into Later Heaven Function (Yong)!

3. Shao Yong’s 8×8 Square Diagram (Fang Tu): Rows of Inner & Columns of Outer Gua

Before the 64 Early Heaven Hexagrams are wrapped around the circular rim of the Luo Pan, they form an 8×8 Square Matrix (Xian Tian Liu Shi Si Gua Fang Tu 先天六十四卦方圖) sitting at the center of Shao Yong’s cosmic diagram. Why did the ancients place a Square inside a Circle? Because “Heaven is round and Earth is square” (Tian Yuan Di Fang 天圓地方): the Circular Diagram models the dynamic rotation of Heavenly Time, while the 8×8 Square Diagram models the structural grid of Earthly Space!

How is the 8×8 Square Diagram constructed? It follows a crystal-clear Cartesian rule along Fu Xi’s congenital trigram sequence (`1 Qian, 2 Dui, 3 Li, 4 Zhen, 5 Xun, 6 Kan, 7 Gen, 8 Kun`):

• The 8 Horizontal Rows (Bottom to Top = Constant Inner / Lower Trigram): Starting from the bottom-right corner and moving upward row by row, every hexagram in Row 1 has Qian ☰ as its Lower (Inner) Trigram; Row 2 has Dui ☱; Row 3 has Li ☲; Row 4 has Zhen ☳; Row 5 has Xun ☴; Row 6 has Kan ☵; Row 7 has Gen ☶; and Row 8 (top row) has Kun ☷ as its Lower Trigram.
• The 8 Vertical Columns (Right to Left = Constant Outer / Upper Trigram): Within every row, moving from right to left across the 8 columns stacks the 8 Upper (Outer) Trigrams in exact Fu Xi order: Column 1 (rightmost) has Upper Qian ☰; Column 2 has Upper Dui ☱; Column 3 has Upper Li ☲; Column 4 has Upper Zhen ☳; Column 5 has Upper Xun ☴; Column 6 has Upper Kan ☵; Column 7 has Upper Gen ☶; and Column 8 (leftmost) has Upper Kun ☷.
• The Two Royal Diagonals (16 Parent Hexagrams): Look at the main diagonal running from the bottom-right corner (Northwest position) to the top-left corner (Southeast position): it consists of the 8 Pure Period-1 Hexagrams (Qian ䷀, Dui ䷹, Li ䷝, Zhen ䷲, Xun ䷸, Kan ䷜, Gen ䷳, Kun ䷁)! Now look at the anti-diagonal running from the bottom-left corner (Southwest position) to the top-right corner (Northeast position): it consists of the 8 Inverted Period-9 Parent Hexagrams (Pi ䷋, Xian ䷞, Wei Ji ䷿, Heng ䷟, Yi ䷩, Ji Ji ䷾, Sun ䷨, Tai ䷊)!

Think about that geometric miracle. The two diagonals of the 8×8 Square Matrix are none other than the 16 Fu Mu Parent Hexagrams (Star Periods 1 and 9) of the Tian Yu Jing! Every pair of hexagrams symmetric across the center of the 8×8 Square forms a Sum-of-Ten opposite pair.

4. Shao Yong’s 360° Circular Diagram (Yuan Tu): 32 Yang-Rising & 32 Yin-Rising Gua

Now let us unfold those 8 horizontal rows of the Square Diagram onto the 360° Circular Ring (Xian Tian Liu Shi Si Gua Yuan Tu 先天六十四卦圓圖). This is the exact outer 64-hexagram ring engraved on classical San Yuan compasses.

How did Shao Yong and the Tang–Song geomancers wrap the 64 hexagrams around the 360° compass? They split the 8 rows into two 32-hexagram halves—4 Yang-Bottom rows (`Qian, Dui, Li, Zhen` where Bottom Line 1 is Yang `—`) and 4 Yin-Bottom rows (`Xun, Kan, Gen, Kun` where Bottom Line 1 is Yin `- -`)—creating a continuous cycle of waxing and waning breath:

5. The Two Pivots of Cosmic Breath: Fu ䷗ at Winter Solstice & Gou ䷫ at Summer Solstice

Look closely at the exact meridian line running from 0° Due North (Mid-Zi 子中) to 180° Due South (Mid-Wu 午中) in Figure 2.2. In Shao Yong’s Huangji Jingshi and the Tian Yu Jing, this North–South axis is revered as the Axis of Heaven and Earth’s Heart (Tian Di Zhi Xin 天地之心):

• At 0° Due North (Winter Solstice Pivot — Kun ䷁ meets Fu ䷗): On the western side of the 0° line sits Pure Kun (坤為地 ䷁, 000000), where Yin reaches total exhaustion. Crossing 0.0° into the eastern side, a single solid Yang line erupts at the very bottom of the hexagram to form Earth Thunder Fu (地雷復 ䷗, 100000 — “Return”). As the Book of Changes proclaims: “In Fu, do we not behold the heart of Heaven and Earth?” (復其見天地之心乎).
• At 180° Due South (Summer Solstice Pivot — Qian ䷀ meets Gou ䷫): On the eastern side of the 180° line sits Pure Qian (乾為天 ䷀, 111111), where Yang reaches absolute zenith. Crossing 180.0° into the western side, a single broken Yin line enters at the bottom to form Heaven Wind Gou (天風姤 ䷫, 011111 — “Encounter”).

Shao Yong immortalized these two hinges of the universe in his famous poem Guan Wu Yin (觀物吟): “Where the Winter Solstice meets the middle of Zi, the mind of Heaven knows no shifting; where one Yang first begins to stir, the ten thousand thingshave not yet been born.” Yet for a practicing Feng Shui master, this cosmic hinge carries an urgent practical warning: because 0.0° (between Kun ䷁ and Fu ䷗) and 180.0° (between Qian ䷀ and Gou ䷫) are the exact points where Pure Yin and Pure Yang flip polarities, the 0.0° and 180.0° lines are the most volatile Great Void Lines (Da Kong Wang 大空亡) on the entire compass! Never align a building spine directly on 0.0° or 180.0°.

6. He Tu (River Map): The Five-Phase Substance of 1-6, 2-7, 3-8, 4-9

Turn now to the numerical bloodstream of Xuan Kong Da Gua: the He Tu (河圖 — Yellow River Map). According to the Xi Ci Zhuan, a dragon-horse emerged from the Yellow River bearing a pattern of white (odd/Yang) and black (even/Yin) dots from 1 to 10 on its back:

「天一生水，地六成之；地二生火，天七成之；天三生木，地八成之；地四生金，天九成之；天五生土，地十成之。」 Tiān yī shēng shuǐ, dì liù chéng zhī; dì èr shēng huǒ, tiān qī chéng zhī; tiān sān shēng mù, dì bā chéng zhī; dì sì shēng jīn, tiān jiǔ chéng zhī; tiān wǔ shēng tǔ, dì shí chéng zhī.

“Heaven 1 generates Water, and Earth 6 completes it; Earth 2 generates Fire, and Heaven 7 completes it; Heaven 3 generates Wood, and Earth 8 completes it; Earth 4 generates Metal, and Heaven 9 completes it; Heaven 5 generates Earth, and Earth 10 completes it.”

— Classical Commentary on the He Tu in Book of Changes · Xi Ci Zhuan (繫辭傳)

Memorize the four outer He Tu creation pairings—known in Xuan Kong Da Gua as He Sheng Cheng (合生成 — Generative Difference-of-Five Pairings)—because they define the Gua Qi Five Phases of all 64 hexagrams:

• 1 and 6 Co-Ancestors of Water (Yi Liu Gong Zong Shui 一六共宗水): Situated in the North. Number 1 is Yang Generative Water; Number 6 is Yin Completing Water. Difference: 6 − 1 = 5.
• 2 and 7 Same-Path of Fire (Er Qi Tong Dao Huo 二七同道火): Situated in the South. Number 2 is Yin Generative Fire; Number 7 is Yang Completing Fire. Difference: 7 − 2 = 5.
• 3 and 8 Companions of Wood (San Ba Wei Peng Mu 三八為朋木): Situated in the East. Number 3 is Yang Generative Wood; Number 8 is Yin Completing Wood. Difference: 8 − 3 = 5.
• 4 and 9 Allies of Metal (Si Jiu Zuo You Jin 四九作友金): Situated in the West. Number 4 is Yin Generative Metal; Number 9 is Yang Completing Metal. Difference: 9 − 4 = 5.
• 5 and 10 Central樞紐 of Earth (Wu Shi Tong Tu 五十同途土): Situated at the Taiji Center, acting as the catalytic medium through which the four outer pairings transform (`1 + 5 = 6`, `2 + 5 = 7`, `3 + 5 = 8`, `4 + 5 = 9`).

Notice the clockwise productive loop around the He Tu perimeter: North 1–6 Water flows to East 3–8 Wood (`Water produces Wood`), which rises to South 2–7 Fire (`Wood produces Fire`), which passes through Center 5–10 Earth (`Fire produces Earth`) to West 4–9 Metal (`Earth produces Metal`), which returns to North 1–6 Water (`Metal produces Water`). That exact Five-Phase cycle is how we evaluate whether a Facing hexagram, Water mouth, or Date pillar produces (Sheng Ru) or controls (Ke Ru) the Sitting Mountain in Xuan Kong Da Gua!

7. Luo Shu (Lo River Scroll): Nine-Palace Magic Square & Opposite Sum-of-Ten

If the He Tu provides the immutable Five-Phase Substance (Ti), the Luo Shu (洛書 — Lo River Scroll) provides the dynamic spatial-temporal Function (Yong). Revealed on the carapace of a sacred tortoise in the Lo River during the era of Yu the Great, the Luo Shu arranges the digits 1 through 9 into a 3×3 magic square described by the classical rhyme: “Wear Nine on the head and tread One underfoot; hold Three on the left and Seven on the right; Two and Four form the shoulders; Six and Eight form the feet; Five resides in the Central Palace” (戴九履一，左三右七，二四為肩，六八為足，五居中央).

Why is the Luo Shu the engine of Xuan Kong Da Gua? Because when Number 5 holds the Central Palace, every pair of diametrically opposite palaces across the center of the compass adds up to exactly Ten (He Shi 合十 — Sum-to-Ten), while including the center `5` makes every three-palace line add up to Fifteen (He Shi Wu 合十五):

• North–South Meridian: North 1 + South 9 = 10 (`1 + 5 + 9 = 15`).
• Southwest–Northeast Axis: Southwest 2 + Northeast 8 = 10 (`2 + 5 + 8 = 15`).
• East–West Equinox Axis: East 3 + West 7 = 10 (`3 + 5 + 7 = 15`).
• Southeast–Northwest Axis: Southeast 4 + Northwest 6 = 10 (`4 + 5 + 6 = 15`).

8. Superimposing Fu Xi Bagua onto the Luo Shu: Deriving the 8 Palace Gua Qi

Now watch how Section 2, Section 6, and Section 7 fuse together into the single most important table in Xuan Kong Da Gua. When you place Fu Xi’s Early Heaven Bagua over the Luo Shu Nine-Palace Grid (as shown on the right side of Figure 2.3) and assign each number its He Tu Five Phase, you obtain the permanent Gua Qi Five Phase (卦氣五行) of the 8 Upper-Trigram Palace Groups:

9. Comparing the Two 64-Hexagram Compass Rings: Circular vs. Outer-Trigram Grouping

Examine a high-grade San Yuan Da Gua Luo Pan—or open Chapter 07: Practical Da Gua Luo Pan Guide—and you will discover a crucial design feature that confuses self-taught students: the compass contains two concentric 64-hexagram rings! Why are there two rings of 64 hexagrams, and how do they differ?

1. The Outer / Seasonal 64-Hexagram Ring (Shao Yong’s Fu Xi Circular Sequence — Wai Pan / Yuan Tu): As detailed in Section 4, this ring arranges the 64 hexagrams by keeping the Inner (Lower) Trigram constant across each 45° sector of 8 hexagrams while the Outer (Upper) Trigram cycles through all 8 trigrams (`Qian, Dui, Li, Zhen, Xun, Kan, Gen, Kun`). Consequently, within any 45° trigram sector on this ring, the 8 adjacent hexagrams carry 8 different Gua Qi numbers (`9, 4, 3, 8, 2, 7, 6, 1`), while their Gua Yun Star Periods alternate dynamically. In the classical Taiwan and Jiangxi San Yuan tradition, this ring is the primary degree ring for reading the exact sitting/facing hexagram, its Gua Qi, and its Star Period (Gua Yun).
2. The Inner / Eight-Palace 64-Hexagram Ring (Outer-Trigram Grouping — Nei Pan / Fang Tu): This ring swaps the inner and outer trigrams of the Circular Diagram (or reads the columns of the Square Diagram rather than the rows) so that all 8 hexagrams within a single 45° trigram sector share the exact same Outer (Upper) Trigram and therefore the exact same Gua Qi number (e.g., all 8 hexagrams in the South Qian 45° palace share Upper Trigram Qian ☰ and Gua Qi 9 Metal; all 8 hexagrams in the North Kun 45° palace share Upper Trigram Kun ☷ and Gua Qi 1 Water).

Notice the mathematical relationship between any degree on the Inner Ring and the exact same degree on the Outer Ring: the hexagram on the Inner Ring is the vertical inversion (Upper and Lower Trigrams swapped) of the hexagram at the exact same degree on the Outer Ring! Because swapping the Upper and Lower Trigrams leaves the line-change relationship between them 100% unchanged, both rings share the exact same Star Period (Gua Yun 1–9) at every single degree on the 360° compass!

10. Anatomy of Void Lines: Da Kong Wang, Xiao Kong Wang, Gua Jie & Yao Jie

Precision saves fortunes; carelessness invites ruin. When you divide a 360° horizon into 64 hexagrams and 384 lines, the boundary lines between adjacent sectors become razor-sharp electromagnetic and geomantic fault lines known as Kong Wang (空亡 — “Void and Perishing” or Dead Lines).

Why is a Void Line so dangerous? Imagine tuning an analog radio receiver right onto the static crackle between two broadcast stations: neither station comes through clearly, and only white noise fills the room. When the sitting-to-facing spine of a house, corporate tower, or ancestral tomb straddles a Void Line, the building cannot lock onto a single hexagram’s Gua Qi or Gua Yun. In clinical field audits, houses sitting on Void Lines exhibit a unmistakable cluster of symptoms: chronic litigation, split partnerships, muddled executive decisions, unexplained neurological or autoimmune ailments, and wealth that evaporates as fast as it arrives. Master the four hierarchical tiers of Void Lines on the Da Gua Luo Pan:

• Tier 1 — Eight-Trigram Major Void (Ba Gua Da Kong Wang 八卦大空亡 — 8 Lines every 45°): Located at the exact junctions between the 8 Trigram Palaces: 0.0° (360.0°), 45.0°, 90.0°, 135.0°, 180.0°, 225.0°, 270.0°, 315.0° (on the Early Heaven ring where 0°/180° split the hemispheres, or at the 8 palace borders `22.5°, 67.5°, 112.5°, 157.5°, 202.5°, 247.5°, 292.5°, 337.5°` on the Later Heaven 24-Mountain ring). Avoid by at least ±1.0°.
• Tier 2 — Twenty-Four Mountain Void (Er Shi Si Shan Xiao Kong Wang 二十四山空亡 — 24 Lines every 15°): Located at the boundaries between the 24 Mountains (`7.5°, 22.5°, 37.5°...`). Where two mountains of opposite Yin-Yang polarity meet, the classical texts call it Yin-Yang Great Void; where two mountains of the same polarity meet, they call it 龜甲空亡 (Tortoise-Shell Void).
• Tier 3 — Sixty-Four Hexagram Boundary Void (Liu Shi Si Gua Jie Kong Wang 卦界空亡 — 64 Lines every 5.625°): Located at every multiple of 5.625° (`0.0°, 5.625°, 11.25°, 16.875°, 22.5°, 28.125°, 33.75°, 39.375°, 45.0°...`). Crossing a 5.625° boundary flips the building from one hexagram into another!
• Tier 4 — Three Hundred Eighty-Four Yao Line Void (Yao Jie Kong Wang 爻界空亡 — 384 Lines every 0.9375°): Located at every multiple of 0.9375° where Line 1 meets Line 2, Line 2 meets Line 3, etc., plus the Partial Void of landing on an inactive Yin/Yang line rather than the active changing line (Dong Yao) of the hexagram.

11. Field Case Study: Rescuing a Villa Straddling a Hexagram Void Boundary

12. Frequently Asked Questions (FAQ) & Authoritative Classical Bibliography

1. How are the 64 Early Heaven Hexagrams arranged around the 360-degree compass ring?

Following Shao Yong’s Fu Xi Circular Diagram (Xian Tian Liu Shi Si Gua Yuan Tu) and the Shuo Gua Zhuan axiom “Counting the past is clockwise; knowing the future is counter-clockwise”, the 360° circle splits along the North–South (Zi–Wu) meridian into two 180° hemispheres: on the Left / Eastern Yang Semicircle, 32 Yang-born hexagrams (all having a Yang bottom line) unfold from Earth Thunder Fu ䷗ at 0° mid-Zi up to Pure Qian ䷀ at 180° mid-Wu; on the Right / Western Yin Semicircle, 32 Yin-born hexagrams (all having a Yin bottom line) unfold from Heaven Wind Gou ䷫ at 180° mid-Wu down to Pure Kun ䷁ at 360° mid-Zi.

2. Why do the Outer Ring and Inner Ring of a San Yuan Da Gua Luo Pan arrange the 64 hexagrams differently?

A classical San Yuan Da Gua Luo Pan features two complementary 64-hexagram rings: one follows Shao Yong’s Fu Xi Circular Sequence (where the Lower/Inner Trigram stays constant in groups of 8 while the Upper Trigram cycles), and the other follows the Upper-Trigram Palace Sequence (where all 8 hexagrams sharing the same Upper Trigram sit together in one 45° trigram palace). Because each hexagram on one ring is simply the vertical Upper-Lower inversion of the hexagram at the exact same degree on the other ring, both rings share the identical Star Period (Gua Yun 1–9) at every single degree.

3. How does the He Tu serve as the Substance (Ti) and the Luo Shu as the Function (Yong) in Xuan Kong Da Gua?

The He Tu provides the immutable Five-Phase Substance (Ti) through its Generative Difference-of-Five pairings: 1–6 Water, 2–7 Fire, 3–8 Wood, and 4–9 Metal. The Luo Shu provides the spatial-temporal Function (Yong) by distributing the numbers 1 through 9 across the eight compass directions so that every opposite pair sums to 10 (1+9, 2+8, 3+7, 4+6). Superimposing Fu Xi’s Early Heaven Bagua onto the Luo Shu assigns each trigram palace its permanent Gua Qi number and He Tu element.

4. What is the relationship between the 8×8 Square Diagram (Fang Tu) and the 360-degree Circular Diagram (Yuan Tu)?

In Fu Xi’s Square-and-Circle Diagram, the 8×8 Square represents Earth (8 horizontal rows of constant Lower Trigrams intersecting 8 vertical columns of constant Upper Trigrams, with the 8 Pure Period-1 Hexagrams along the main diagonal and the 8 Inverted Period-9 Hexagrams along the anti-diagonal), while the surrounding 64-Hexagram Circle represents Heaven (unfolding the 8 rows around the two hemispheres).

5. What are Da Kong Wang, Xiao Kong Wang, and Yao Kong Wang on the Da Gua compass?

They are the four tiers of Void Lines (Dead Lines) where compass qi fractures: (1) Eight-Trigram Major Void (Da Kong Wang) at the 8 palace boundaries every 45°; (2) 24-Mountain Void (Xiao Kong Wang) at the 24 mountain boundaries every 15°; (3) 64-Hexagram Boundary Void (Gua Jie Kong Wang) at every multiple of 5.625°; and (4) 384-Yao Line Void (Yao Jie Kong Wang) at every multiple of 0.9375°.

6. Why did Gottfried Wilhelm Leibniz (1703) recognize Fu Xi’s 64-Hexagram Circle as pure binary arithmetic?

When French Jesuit Joachim Bouvet sent Shao Yong’s Fu Xi 64-Hexagram diagram to German mathematician Gottfried Wilhelm Leibniz in 1701, Leibniz proved in his landmark 1703 paper for the Royal Academy of Sciences in Paris that substituting 0 for a broken Yin line and 1 for a solid Yang line turns the 64 hexagrams (from Kun 000000 = 0 to Qian 111111 = 63) into the exact 6-bit binary progression from 0 to 63.

Classical Sources & Academic References

1. Shao Yong (邵雍, Song Dynasty), Huangji Jingshi Shu (皇極經世書 · Book of Supreme World-Ordering Principles).
2. Zhu Xi (朱熹, Song Dynasty), Zhouyi Benyi (周易本義 · Original Meaning of the Zhou Changes), Introductory Diagrams (Yi Tu).
3. Confucian School (Warring States), Shuo Gua Zhuan (說卦傳) & Xi Ci Zhuan (繫辭傳) in the Zhouyi.
4. Yang Yun Song (楊筠松, Tang Dynasty), Qing Nang Jing (青囊經) & Tian Yu Jing (天玉經).
5. Jiang Dahong (蔣大鴻, Ming–Qing Transition), Di Li Bian Zheng (地理辨正).
6. G. W. Leibniz (1703), “Explication de l’Arithmétique Binaire, qui se sert des seuls caractères 0 & 1, avec des remarques sur son utilité, & sur ce qu’elle donne le sens des anciennes figures Chinoises de Fohy”, Mémoires de l’Académie Royale des Sciences, Paris, pp. 85–89.