Just Intonation Intervals at A4 = 432 Hz
Frequencies are computed with: f_interval = 432 * ratio. These are pure (just) ratios. In 12-TET the actual frequencies differ slightly so you can play in all keys.
| Interval | Ratio | Frequency (Hz) |
|---|---|---|
| Unison | 1:1 | 432.00 |
| Octave | 2:1 | 864.00 |
| Perfect Fifth | 3:2 | 648.00 |
| Perfect Fourth | 4:3 | 576.00 |
| Major Third | 5:4 | 540.00 |
| Minor Third | 6:5 | 518.40 |
| Major Sixth | 5:3 | 720.00 |
| Minor Sixth | 8:5 | 691.20 |
| Major Second | 9:8 | 486.00 |
| Minor Second | 16:15 | 460.80 |
| Major Seventh | 15:8 | 810.00 |
| Minor Seventh | 9:5 | 777.60 |
| Minor Seventh (alt.) | 16:9 | 768.00 |
Note: Values are rounded to 2 decimal places for readability.
- Maya Long Count Calendar: 5,125.36 years × 5 = 25,626.8 ≈ Precession of the Equinoxes (4 sides x 9 steps x 3 parts per side x 4 = 432 (add x 2 (layers top) = 864)
- Kali Yuga = 432,000 years
- Dvapara Yuga = 864,000 years
- Treta Yuga = 1,296,000 years
- Satya Yuga = 1,728,000 years
- Total Maha Yuga = 4,320,000 years
Ancient Egypt / Great Pyramid
- Base perimeter ≈ 3,024 ft = 4 × 756 ft; Height aligns with 51.84° and harmonic ratios.
- Great Pyramid of Giza & Earth ratios: The pyramid’s height and base perimeter have been claimed to scale to the Earth at a ratio of ~1 : 43,200 — meaning if the pyramid’s dimensions are multiplied by 43,200, they approximate the Earth’s radius or circumference.
- Because a day has 86,400 seconds, 1/43,200 of a day is exactly 2 seconds — hence the notion that “every 2 seconds” of Earth’s rotation corresponds to that same ratio.
- The number 43,200 itself factors into 72 × 600 and is linked to both precession cycles (72 yrs per degree) and time measurement (43,200 = 60 × 60 × 12) in modern claims.
- Note: These claims are contested by mainstream scholarship; they remain speculative and symbolic rather than universally accepted as empirical evidence.
- The 36 decans were star groups the ancient Egyptians used to divide the night sky and the year into equal parts. Each decan rose on the horizon every 10 days, marking a “week” in their 360-day year (36×10 = 360 days + 5 festival days). At night, each decan’s rising signaled a new hour, so priests could tell time by the stars. The system appears on coffin lids and temple ceilings (like Seti I’s tomb). Later, the Greeks adopted the 36 decans into astrology, making them 10-degree divisions of each zodiac sign, each with its own deity or planetary ruler. In short: → Egyptian decans = 36 star clocks for time + 36 calendar weeks, later becoming the foundation of zodiac astrology.
-
Gematria
- Jesus = 888
- Christ = 1480
- Lord = 800
- Holy Ghost = 1080
- 432 Hz: Resonates with the universe's natural harmonic scale
- Base 60 (Sumerian) used for time, angle, and celestial calculations — harmonically linked
- 72 years per degree of the Great Year (25,920 / 360 = 72)
- 12 tribes of Israel — the root of 144 (12 × 12 = 144), symbolizing completeness and divine order. The number 12 recurs across ancient cosmologies: 12 months, 12 zodiac signs, 12 hours of day and night, and 12 tones in the musical octave. In sacred geometry, it frames cosmic balance and spiritual structure.
- Qur'an (Kufa Tradition) — The Holy Qur'an contains 6,236 verses (Kufa/Hafs recension). Symbolically, 6×2×3×6 = 216 → digital root 9, reflecting the same 9-based harmonic cycle found across sacred systems.
Base‑20 (Vigesimal) System: The Maya used a base‑20 count, reflecting the natural count of fingers and toes — a “hands and feet” mathematics. Unlike a strict 20×20×20 structure, the third level (Tun) is adjusted to 18×20 = 360 to align with the solar year, showing harmony between celestial cycles and human measure.
Cycle Hierarchy:
1 Kin = 1 day
1 Uinal = 20 Kin = 20 days
1 Tun = 18 Uinal = 360 days
1 Katun = 20 Tun = 7 200 days ≈ 19.7 years
1 Baktun = 20 Katun = 144 000 days ≈ 394.26 years
13 Baktun = 1 Great Cycle = 1 872 000 days ≈ 5 125.36 years
Precessional Harmony: Five Maya Great Cycles ≈ 25 626 years — nearly equal to Earth’s precessional year (~ 25 920 years). This resonance unites Maya, Hindu, and Sumerian cosmologies through shared numeric constants: 3 000, 6 000, 9 000, 12 000, 24 000, 36 000, 54 000, 72 000, 108 000, 144 000, 432 000, 72, 216, 8640, 43 200. Each expresses a fractal scaling of time and consciousness.
Numerical Synthesis:
– 360 days (Tun) ↔ 360° circle ↔ solar year
– 144 000 days (Baktun) ↔ Revelation / pyramid ratio key
– 72 yrs per degree ↔ precession constant
– 432 Hz ↔ cosmic frequency base tone
– 864 000 miles (Sun diameter) ↔ Maya and Hindu scales in mirrored ratio
Fractal Time Concept: The Maya viewed time as recursive — each unit a seed of the next. Kin : Uinal : Tun : Katun : Baktun form a living sequence of micro‑to‑macro harmony, echoing the same base‑6, 12, 60 resonances found in other sacred systems.
| Maya Unit | Count (days) | Equivalent Years | Harmonic Relation |
|---|---|---|---|
| Kin | 1 | 1 day | Unit of life / heartbeat |
| Uinal | 20 | 0.0548 yr | 20 digits — base of creation |
| Tun | 360 | 0.986 yr | Circle of degrees / solar year |
| Katun | 7 200 | 19.7 yr | ≈ Lunar nodes cycle |
| Baktun | 144 000 | 394.26 yr | 144 = 12² — completion |
| 13 Baktun | 1 872 000 | 5 125.36 yr | 1 Great Cycle / Mayan Age |
| 5 × 13 Baktun | 9 360 000 | 25 626.8 yr | ≈ Precession of the Equinoxes |
Many ancient systems modeled the year as 360 days. Interpreting the Maya Long Count through that lens makes several harmonics exact and easier to see.
| Maya Unit | Count (days) | Years @ 365.2425 | Years @ 360 | Harmonic Notes |
|---|---|---|---|---|
| Kin | 1 | 0.00274 | 0.00278 | Seed unit |
| Uinal | 20 | 0.0548 | 0.0556 | Base‑20 digit cycle |
| Tun | 360 | 0.986 | 1.000 | Exactly 1 “ancient” year |
| Katun | 7,200 | 19.73 | 20.00 | 20 “ancient” years |
| Baktun | 144,000 | 394.26 | 400.00 | 20 Katun = 400 “ancient” years |
| 13 Baktun | 1,872,000 | 5,125.36 | 5,200.00 | 1 Great Cycle = 13×400 yrs |
| 5 × 13 Baktun | 9,360,000 | 25,626.8 | 26,000.0 | ≈ Precession (ancient rounding) |
Takeaway: With a 360‑day year, Tun = 1 year, Katun = 20 years, Baktun = 400 years, Great Cycle = 5,200 years, and Five Great Cycles = 26,000 years — making the precessional harmony clean in the “ancient time” framework.
Hindu Yugas
| Yuga / Phase | Divine Years |
Human Years |
Human / 360 = Divine | Total Divine (Cycle) |
Total Human (Cycle) |
|---|---|---|---|---|---|
| KALI YUGA | |||||
| Sandhya | 100 | 36,000 | 100 | ||
| Kali | 1,000 | 360,000 | 1,000 | 1,200 | 432,000 |
| Sandhyamsa | 100 | 36,000 | 100 | ||
| DWAPARA YUGA | |||||
| Sandhya | 200 | 72,000 | 200 | ||
| Dwapara | 2,000 | 720,000 | 2,000 | 2,400 | 864,000 |
| Sandhyamsa | 200 | 72,000 | 200 | ||
| TRETA YUGA | |||||
| Sandhya | 300 | 108,000 | 300 | ||
| Treta | 3,000 | 1,080,000 | 3,000 | 3,600 | 1,296,000 |
| Sandhyamsa | 300 | 108,000 | 300 | ||
| SATYA YUGA | |||||
| Sandhya | 400 | 144,000 | 400 | ||
| Satya | 4,000 | 1,440,000 | 4,000 | 4,800 | 1,728,000 |
| Sandhyamsa | 400 | 144,000 | 400 | ||
| TOTAL MAHAYUGA | |||||
| TOTAL | 12,000 | 4,320,000 | 12,000 | 12,000 | 4,320,000 |
HEBREW CUBIT (COMMON / SHORT) ~ 0.4572 m = 18 in
440 HC -> 201.168 m -> 660.000 ft (exact 660 ft 0 in) -> 7920.0 in (base-6: 100400, base-60: 2:12, root 9: 7+9+2+0=18 -> 1+8=9)
280 HC -> 128.016 m -> 420.000 ft (exact 420 ft 0 in) -> 5040.0 in (base-6: 35200, base-60: 1:24, root 9: 5+0+4+0=9)
NOTE:
The meter is defined by fixing the speed of light c = 299 792 458 m/s
and defining the second via the cesium-133 hyperfine transition
frequency ("atomic vibration") of 9 192 631 770 Hz.
So the meter is tied to light speed and that atomic frequency --
decidedly not anatomical.
Harmonic Notes
- 1080 → "Holy Ghost" (Greek: Pneuma Hagion)
- 432 → Tuning frequency / "Heaven" harmonic / 12 × 36
- 666 → "Calculation Key", resonance distortion or mirror code
- 144000 → Book of Revelation, harmonic completion
- 888 → "Jesus" in Greek
- 72 → Divine names in Kabbalah / 1° Precession shift
- 3 → Avraham, Yitzhak, Yaakov — Patriarchal triad, foundation of faith
- 6 → Sheshet Yemei Bereshit — Six days of Creation
- 9 → Emet (אמת) = 441 → 4 + 4 + 1 = 9 — “Truth,” the seal of God
- 12 → Shivtei Yisrael — Twelve tribes of Israel
- 24 → Books of the Tanakh (תנ״ך) — 24 canonical writings
- 36 → Lamed-Vav Tzadikim — 36 hidden righteous souls
- 54 → Sidrei Torah Parashot — 54 weekly Torah portions
- 72 → Shem Ha-Meforash — 72-fold Divine Name (Exodus 14:19–21)
- 108 → Elohim (אלהים) ≈ 108 symbolic completion of divine order
- 216 → Gevurah (גבורה) = 216 — Strength, one of God’s emanations
- 360 → HaGalgal (הגלגל) — Celestial wheel, completeness of cycle
- 432 → David (דוד) = 14 → 432/3 = 144 — Royal lineage harmonic
- 540 → Yisrael (ישראל) = 541 ≈ 540 — Symbolic unity of the people
- 720 → Keter + Malkhut — Crown and Kingdom, upper and lower union
- 1080 → Ruach Ha-Kodesh — Holy Spirit (aligns with Greek Holy Ghost)
- 2160 → Light of the Moon — Lunar sphere ≈ 2160 mi radius
- 4320 → Yovel Cycle — 120 × 36 = 4320 symbol of renewal and release
- 8640 → Ohr Ha-Olam (אור העולם) — “Light of the World,” doubled 4320
Hebrew Gematria (Short Intro + Table)
Gematria is a simple system where each Hebrew letter equals a number. Add the letter values of a word to get its total. In Torah study, this does not replace the plain meaning; it highlights patterns and supports ideas found in the text.
| Name | Hebrew | Value | Name | Hebrew | Value | Name | Hebrew | Value |
|---|---|---|---|---|---|---|---|---|
| Alef | א | 1 | Yod | י | 10 | Qof | ק | 100 |
| Bet | ב | 2 | Kaf | כ | 20 | Resh | ר | 200 |
| Gimel | ג | 3 | Lamed | ל | 30 | Shin | ש | 300 |
| Dalet | ד | 4 | Mem | מ | 40 | Tav | ת | 400 |
| He | ה | 5 | Nun | נ | 50 | |||
| Vav | ו | 6 | Samekh | ס | 60 | |||
| Zayin | ז | 7 | Ayin | ע | 70 | |||
| Chet | ח | 8 | Pe | פ | 80 | |||
| Tet | ט | 9 | Tsadi | צ | 90 |
- Standard values run from 1 to 400 (Alef..Tav).
- Mispar Katan: reduce a total by summing its digits to 1..9 (example: 611 => 6+1+1 = 8).
- To compute a word: add each letter value. Example: Torah = Tav(400)+Vav(6)+Resh(200)+He(5) = 611.
Counting In Different Bases (Base 10, Base 6, Base 60)
A base is the number of unique digits you use before you roll over to a new place. Place values grow by powers of the base: ones (base^0), base (base^1), base^2, base^3, and so on. To see the conversion, fill in a number and press your button above.
- Base 10 (0..9): what most of us use. After 9 comes 10 (one ten and zero ones).
- Base 6 (0..5): after 5 comes 10_6 (one six and zero ones). Examples below.
- Base 60 (0..59): after 59 comes 10_60 (one sixty and zero ones). Think clocks: 60 seconds = 1 minute.
| Idea | Base 10 | Base 6 | Base 60 |
|---|---|---|---|
| Digits used | 0..9 | 0..5 | 0..59 |
| First rollover | 9 -> 10 | 5 -> 10_6 | 59 -> 10_60 |
| Place values | 1, 10, 100, 1000, ... | 1, 6, 36, 216, ... | 1, 60, 3600, 216000, ... |
Tiny Counting Examples
Base 10 (0..12):
0, 1, 2, 3, 4, 5,
6, 7, 8, 9, 10, 11, 12
Base 6 (0..14_6):
0, 1, 2, 3, 4, 5,
10_6, 11_6, 12_6,
13_6, 14_6
Base 60 (0..11_60):
0, 1, 2, ... 58, 59,
10_60, 11_60
How to Convert (conceptual)
- Place-value view: write the number as sums of powers of the base.
- Division view: divide by the base, record the remainder, repeat with the quotient; read remainders backward.
- Try it: enter a number and press your button above to see its form in base 6 or base 60.
This section shows the classic binary doubling chain 1 → 2 → 4 → 8 → 16 → 32 → … and compares each value’s digital root (sum of digits reduced to 1–9). The key observation: the digital roots of powers of two cycle as 2, 4, 8, 7, 5, 1 and never hit 3, 6, or 9. That is the “3‑6‑9 doesn’t appear in binary roots” idea.
Examples you mentioned
- 18 → 1 + 8 = 9
- 36 → 3 + 6 = 9
Contrast: for powers of two, the digital-root cycle stays on the six‑step wheel [2, 4, 8, 7, 5, 1]. No 3, 6, or 9 appear.
The pentagram inscribes a regular pentagon whose interior angle is 108°. Five such corners give 5 × 108° = 540° (digital root 5+4+0 = 9). These “holy” counts connect to zodiac/precession cycles long cited in esoteric math: 72, 432, 2160, 25,920. We contrast the classical schema (25,920 years = 360° × 72 y/°) with a modern astronomical estimate (~25,772 years).
Platonic Set & Angle Links
Key Numbers & Digital Roots
All Five Platonic Solids — Angle & Digital-Root Map
DR = 9 Contexts Across Platonic Solids
Metatron’s Cube is the line network formed by connecting the centers of the 13 circles (Flower/Seed of Life extension). Within that network, the edge-graphs of all five Platonic solids can be embedded by appropriate selections of lines. This section maps those solids, their dualities, and their characteristic angle sums into the same 3‑6‑9 digital-root lens you’ve used elsewhere.
Dualities & Star Forms
Counts Inside the 13‑Circle Frame (conceptual)
The Hebrew Bible / KJV gives physical and temporal measurements in clean integer blocks. We treat cubits as 18 inches = 1.5 feet, and regnal years as 360‑day years the way ancient calendars are often modeled.
Below: Solomon’s Temple (1 Kings 6), the Ark of the Covenant (Exodus 25:10), and Jehoram timelines (2 Kings 3:1 KJV). For each value we also show the digital root: add the digits until you get one number.
Solomon’s Temple — Dimensions
| Zone | Biblical Size (cubits) | Size in Inches (1 cubit = 18″) |
Root of Inches (digit sum → 1 digit) |
Same in Feet (12″ = 1 ft) |
Notes |
|---|---|---|---|---|---|
| Main House (“the temple of the LORD”) |
60 cubits long 20 cubits wide 30 cubits high 1 Kings 6:2 |
Length: 60×18″ = 1080″ Width: 20×18″ = 360″ Height: 30×18″ = 540″ Floor area: 1080″ × 360″ = 388,800 in² |
1080 → 1+0+8+0 = 9 360 → 3+6+0 = 9 540 → 5+4+0 = 9 388,800 → 3+8+8+8+0+0 = 27 → 2+7 = 9 |
Length: 90.0 ft Width: 30.0 ft Height: 45.0 ft Floor area: 2,700 ft² Volume: 121,500 ft³ |
Long narrow hall lined with cedar and gold. |
| Holy Place / Nave |
40 cubits long 20 cubits wide 30 cubits high 1 Kings 6:17‑18 |
Length: 40×18″ = 720″ Width: 20×18″ = 360″ Height: 30×18″ = 540″ |
720 → 7+2+0 = 9 360 → 3+6+0 = 9 540 → 5+4+0 = 9 |
Length: 60.0 ft Width: 30.0 ft Height: 45.0 ft Floor area: 1,800 ft² Volume: 81,000 ft³ |
Main working space for priestly ritual. |
| Most Holy Place / Inner Sanctuary (Debir / “Holy of Holies”) |
20 cubits × 20 cubits × 20 cubits 1 Kings 6:20 |
Edge: 20×18″ = 360″ per side (= 360″ × 360″ floor) |
360 → 3+6+0 = 9 |
Edge: 30.0 ft per side Floor area: 900 ft² Volume: 27,000 ft³ |
Perfect cube, overlaid with gold. Restricted / presence zone. |
| Porch / Vestibule (Entrance) |
10 cubits deep 20 cubits wide ~30 cubits high* 1 Kings 6:3 *Height ~30 cubits assumed to match the house in 6:2; 2 Chronicles 3:4 reads 120 cubits, often seen as a copy inflation. |
Depth: 10×18″ = 180″ Width: 20×18″ = 360″ Height: 30×18″ = 540″ |
180 → 1+8+0 = 9 360 → 3+6+0 = 9 540 → 5+4+0 = 9 |
Depth: 15.0 ft Width: 30.0 ft Height: 45.0 ft Floor area: 450 ft² Volume: 20,250 ft³ |
Entry hall / porch in front of the main door. |
Ratio / harmony note
The core block of the Temple is 60 : 20 : 30 cubits (1 Kings 6:2 KJV). Reduce that and you’re looking at clean integers (3 : 1 : 1.5 if you divide through by 20), then a 20‑cubit cube for the Most Holy Place (1 Kings 6:20 KJV).
Plain language: the Temple body is about 1080″ long × 360″ wide × 540″ high (≈ 90 × 30 × 45 ft), and the inner gold room is a ~360″ cube (≈ 30 ft cube). All inch values collapse to 9.
Ark of the Covenant — Dimensions
“And they shall make an ark of shittim wood:
two cubits and a half shall be the length thereof,
and a cubit and a half the breadth thereof,
and a cubit and a half the height thereof.”
Exodus 25:10 KJV
Conversion:
1 cubit = 18 inches = 1.5 feet.
| Object | Biblical Size (cubits) | Size in Inches (1 cubit = 18″) |
Root of Inches (digit sum → 1 digit) |
Same in Feet (12″ = 1 ft) |
Notes |
|---|---|---|---|---|---|
|
Ark of the Covenant Exodus 25:10 KJV |
Length: 2.5 cubits Width: 1.5 cubits Height: 1.5 cubits |
Length: 2.5×18″ = 45″ Width: 1.5×18″ = 27″ Height: 1.5×18″ = 27″ |
45 → 4+5 = 9 27 → 2+7 = 9 27 → 2+7 = 9 |
Length: 45″ ÷ 12 = 3.75 ft (≈3′9″) Width: 27″ ÷ 12 = 2.25 ft (≈2′3″) Height: 27″ ÷ 12 = 2.25 ft (≈2′3″) Volume (outer box): ~18.98 ft³ |
Gold-covered chest with rings and poles; carried by hand. |
Top surface / lid area:
45″ × 27″ = 1,215 in².
Root: 1+2+1+5 = 9.
(Same in feet: 3.75 ft × 2.25 ft ≈ 8.44 ft²)
Outer box volume (rectangular bound):
45″ × 27″ × 27″ = 32,805 in³.
Root: 3+2+8+0+5 = 18 → 1+8 = 9.
(~18.98 ft³)
Summary: about 45″ × 27″ × 27″ (≈ 3.75 ft × 2.25 ft × 2.25 ft). Inch values resolve to 9.
Jehoram (Joram) king of Israel — Time Math
“Now Jehoram the son of Ahab began to reign over Israel in Samaria the eighteenth year of Jehoshaphat king of Judah, and reigned twelve years.” 2 Kings 3:1 KJV
We model an ancient / prophetic year as 360 days.
| Span | Years | Days @ 360‑day year | Root of Days (digit sum → 1 digit) |
|---|---|---|---|
| Eighteenth year marker | 18 years | 18 × 360 = 6480 days | 6480 → 6+4+8+0 = 18 → 1+8 = 9 |
| Reign length | 12 years | 12 × 360 = 4320 days | 4320 → 4+3+2+0 = 9 |
| Total window | 18 + 12 = 30 years | 6480 + 4320 = 10800 days | 10800 → 1+0+8+0+0 = 9 |
The regnal math that frames when “the word of the prophet”
reaches the king (2 Kings 3) collapses to 9,
the Ark dimensions collapse to 9,
and the Temple inch-counts collapse to 9
Assumptions: 1 cubit = 18 inches = 1.5 ft; prophetic year = 360 days (30‑day months). Base‑6 uses digits 0–5. “Root” = digital root (sum of digits to 1–9; 0→0).
Why base‑6? Your sacred‑number set (e.g., 36, 72, 108, 144, 216, 432, 2,592…) compresses cleanly in senary: 2,592 → 20000, 1,080 → 5000, 540 → 2300, 108 → 300, etc.
Verse
Text (KJV excerpt)
Inches
Root
Feet
Base‑6
of inches / daysExpanded Time
days→hours→minutes→secondsNotes
Rev 21:17
“...an hundred and forty and four cubits...”
2,592
9
216
20000
—
Wall of New Jerusalem; 144 ↔ 216 (6³) resonance.
Rev 21:16
“...twelve thousand furlongs. The length and the breadth and the height of it are equal.”
—
—
—
—
—
Included for the 12 ↔ 144 pairing with v.17 (not cubits).
Dan 3:1
“...height was threescore cubits, and the breadth thereof six cubits...”
1,080 × 108
9 × 9
90 × 9
5000 × 300
—
Nebuchadnezzar’s image; neat 108/1080 patterning.
Gen 6:15
“...length three hundred cubits, breadth fifty cubits, height thirty cubits.”
5,400 × 900 × 540
9 × 9 × 9
450 × 75 × 45
41000 × 4100 × 2300
—
Noah’s Ark dimensions; recurring /9 digital roots.
1 Kgs 7:23
“...ten cubits from the one brim to the other... height five cubits... a line of thirty cubits...”
Ø180; h=90; C=540
9 / 9 / 9
Ø15; h=7.5; C=45
Ø500; h=230; C=2300
—
“Molten Sea” in the Temple court.
Ezek 40:5
“...a measuring reed of six cubits long by the cubit and an hand breadth...”
108
9
9
300
—
“Long” cubit noted, harmony still at 6×.
Prophetic Times (360‑day year)
Rev 11:2
“...tread under foot forty and two months.”
1,260
9
—
110 (42 in base‑6)
1,260 d → 30,240 h → 1,814,400 m → 108,864,000 s
42×30 = 1,260 days.
Rev 11:3; 12:6
“...a thousand two hundred and threescore days.”
1,260
9
—
5500
1,260 d → 30,240 h → 1,814,400 m → 108,864,000 s
Equivalent to 42 months at 30 days/month.
Rev 12:14
“...for a time, and times, and half a time...”
1,260
9
—
5500
1,260 d → 30,240 h → 1,814,400 m → 108,864,000 s
Same span as above (3.5×360).
Gen 7:24
“And the waters prevailed... an hundred and fifty days.”
150
6
—
410
150 d → 3,600 h → 216,000 m → 12,960,000 s
5 months × 30‑day months in Flood chronology.
🔢 Harmonic Base Calculator with Harmonic Math Key
We show three ultra-simple processes that race toward φ ≈ 1.6180339887…:
- Fibonacci ratios: Fn+1/Fn → φ.
- Reciprocal map: xn+1 = 1 + 1/xn (start at 1) → φ.
- Sum-multiplicative ladder: starting from a,b > 0, update (a,b) ← (b, a+b). Ratios b/a → φ.
Extending φ = 1.618 033 988 749 894 … through 100 decimal truncations, the digital root (sum of digits → 1 to 9) returns 9 at specific intervals: 10, 18, 26, 30, 32, 34, 46, 47, 50, 54, 57, 79, 84, 87, 94, 95.
These positions mirror the same harmonic numbers that resonate through the Base Kalk framework — 432, 369, 396, 234, 144, 108, 54 — repeating as fractal reflections of Root 9 within φ itself.
Each point where φ prefix → Root 9 marks a resonant closure in its decimal spiral, echoing the geometry of precession and the same Mayan–Platonic harmonic structure.
A constant in φ — Root 9 reveals its own cosmic timing.
Table 1 — Root 9 Intervals within φ · Reflections of the Harmonic Base Kalk Sequence
Position
φ (truncated)
Digital Root
Resonant Harmonic
10 1.6180339887 9 432 18 1.6180339887498948 9 369 26 1.6180339887498948482 9 396 30 1.6180339887498948482045 9 234 32 1.618033988749894848204586 9 144 34 1.61803398874989484820458683 9 108 46 1.6180339887498948482045868343656 9 54
| DR=9 hit (digits after comma) | Total Sum up to that digit | Digital Root | φ fractional prefix (truncated) |
|---|---|---|---|
| 5 | 18 | 9 | .61803 |
| 17 | 99 | 9 | .61803398874989484 |
| 23 | 126 | 9 | .61803398874989484820458 |
| 35 | 180 | 9 | .61803398874989484820458683436563811 |
| 44 | 216 | 9 | .61803398874989484820458683436563811772030917 |
| 45 | 225 | 9 | .618033988749894848204586834365638117720309179 |
| 52 | 261 | 9 | .6180339887498948482045868343656381177203091798057628 |
| 55 | 270 | 9 | .6180339887498948482045868343656381177203091798057628621 |
| 68 | 324 | 9 | .61803398874989484820458683436563811772030917980576286213544862270526 |
| 69 | 324 | 9 | .618033988749894848204586834365638117720309179805762862135448622705260 |
| 96 | 441 | 9 | .618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391 |
Table 2 — Constant in φ · Root 9 Intervals (Excluding the 1) · After‑comma cumulative sums where DR = 9.
| Digit Position (after comma) | Total Sum up to that digit | Digital Root |
|---|---|---|
| 5 | 18 | 9 |
| 17 | 99 | 9 |
| 23 | 126 | 9 |
| 35 | 180 | 9 |
| 44 | 216 | 9 |
| 45 | 225 | 9 |
| 52 | 261 | 9 |
| 55 | 270 | 9 |
| 68 | 324 | 9 |
| 69 | 324 | 9 |
| 96 | 441 | 9 |
| 102 | 468 | 9 |
| 124 | 576 | 9 |
| 139 | 639 | 9 |
| 140 | 648 | 9 |
| 146 | 666 | 9 |
| 147 | 666 | 9 |
| 165 | 756 | 9 |
| 172 | 801 | 9 |
| 184 | 855 | 9 |
| 187 | 864 | 9 |
| 189 | 873 | 9 |
| 193 | 891 | 9 |
| 194 | 900 | 9 |
| 198 | 918 | 9 |
| 199 | 927 | 9 |
| 210 | 981 | 9 |
| 214 | 999 | 9 |
| 222 | 1035 | 9 |
| 228 | 1062 | 9 |
| 229 | 1071 | 9 |
| 239 | 1125 | 9 |
| 240 | 1134 | 9 |
| 249 | 1179 | 9 |
| 257 | 1215 | 9 |
| 258 | 1224 | 9 |
| 263 | 1248 | 9 |
| 264 | 1257 | 9 |
| 269 | 1281 | 9 |
| 277 | 1317 | 9 |
| 287 | 1362 | 9 |
| 288 | 1371 | 9 |
| 298 | 1425 | 9 |
| 299 | 1434 | 9 |
| 303 | 1452 | 9 |
| 304 | 1461 | 9 |
| 309 | 1485 | 9 |
| 316 | 1512 | 9 |
| 317 | 1521 | 9 |
| 332 | 1593 | 9 |
| 338 | 1620 | 9 |
| 339 | 1629 | 9 |
| 348 | 1674 | 9 |
| 358 | 1728 | 9 |
| 367 | 1773 | 9 |
| 371 | 1791 | 9 |
| 384 | 1854 | 9 |
| 385 | 1863 | 9 |
| 491 | 2349 | 9 |
Table 3 — First 500 fractional digits of φ · Digital‑Root 9 occurrences and cumulative sums (excluding the 1).
When summing only the digits after the comma in φ = 1.618 033 988 749 894 …, the digital root again cycles to 9 at specific harmonic positions: 18, 126, 180, 216, 270, 324.
These intervals resonate with the same 3 – 6 – 9 structure and mirror the sacred time divisions of 432, 144, 108 and 216, forming a secondary wave of Root 9 within φ’s infinite unfolding — the after‑comma harmonics.
Root 9 repeats — whether the 1 is counted or not.
The Golden Ratio - "As Above, So Below"
What the ratio expresses
Take a whole unity, say 1. Divide it into two parts: a larger part and a smaller part, such that the whole : larger = larger : smaller. That special value is the golden ratio (phi) ~ 1.618.
In symbols: let the whole = 1, the larger part = a, the smaller part = b. Then 1 : a = a : b -> which yields a/b = a/(1 - a) = phi ~ 1.618.
Self-similarity => "micro contains macro / macro contains micro"
Because when you divide in that ratio, you get a structure that embeds itself: the whole (1) relates to the larger part (~ 0.618 of 1) in the same way the larger part relates to the smaller part.
So the larger part is a scaled-down version of the whole in proportions. And the smaller part likewise is a further scaled version of that. Thus you get a nested sequence of proportions each echoing the same pattern - the macro contains the micro, and the micro contains another micro-pattern - "as above, so below."
Why nature might use it
In growth, reproduction, and pattern-making (plants, shells, branching, phyllotaxis, etc.), a division or multiplication in the golden ratio lets new growth mirror the proportions of existing structure. Because of the self-similar relation, adding or subdividing in that ratio preserves a harmonic proportion across scale.
For example: start with size 1. Multiply by phi (~ 1.618) -> ~ 1.618. That new size can then be subdivided so the larger part is ~ 1 and the smaller ~ 0.618, and the proportions map the same way. Each "generation" of growth remains proportionally consistent, preserving form through scale.
A simple numeric illustration
Start with 1 -> multiply by phi ~ 1.618 -> whole ~ 1.618.
Divide in golden ratio: larger ~ 1.000, smaller ~ 0.618.
Check: larger : smaller ~ 1.000 : 0.618 ~ 1.618 (same ratio).
Divide the larger part again: larger ~ 0.618, smaller ~ 0.382, and so on.
Each level replicates the ratio pattern. Because each "part" carries the same proportion as the "whole", the micro-level carries the macro-pattern.
Plain metaphor
Imagine a plant branching: the main stem is the "whole", the major branch the "larger part", the minor branch the "smaller part". If the growth splits in golden ratio, the branch-structure keeps the same proportion at each level. Each branch (micro) reflects the form of the trunk (macro). Thus: 1 : 1.618 acts like a "growth-division formula" by which the total remains in the same proportions as it was and will be ("past-present-future in one"). The pattern reproduces itself.
Summing all digital roots across the full alphabet (A = 1 … Z = 26) and their harmonic extensions (×6, ×9) gives:
| Root Column | Σ |
|---|---|
| Root(Value) | 126 |
| Root(6×n) | 153 |
| Root(9×n) | 234 |
| Total | 513 |
The combined total of 513 forms a direct harmonic link to the triadic constant 153 — the 17th triangular number (1 + 2 + … + 17 = 153), known since antiquity as the “Fisherman’s Number.”
When balanced by its mirror: 513 + 153 = 666, it reveals the same Root 9 closure found throughout Base Kalk’s harmonic geometry — the constant resonance of 3 – 6 – 9 within the Phi field.
In essence: 153 completes 666 — the harmonic seal of Root 9.
The Sacred Key - Live Relationships click to open
Explore live harmonic relationships among the sacred constants: 3, 6, 9, 12, 24, 36, 54, 72, 108, 144, 216, 432, 720, 1080, 2160, 4320, and 8640.
Live Sacred Relationships (based on input)
Vesica Piscis — Sacred Intersection of Light and Form
153
Vesica Piscis - Sacred Intersection of Light and Form
1) Pythagoras in one line
For any right-angled triangle: a2 + b2 = c2
a and b are the two short sides (the legs).
c is the longest side (the hypotenuse).
Kid version: "Square the legs, add them, then take the square root to get the slanted side."
2) Three tiny recipes for square roots
√2 - the square's diagonal
Picture a 1x1 square and draw the diagonal.
Legs: a = 1, b = 1
Compute: 12 + 12 = 1 + 1 = 2 → c = √2 (approx. 1.414)
Kid talk: "One step across and one step up makes a root-two slanted side."
√3 - the Vesica Piscis height
Make two unit circles (radius = 1) so each center lies on the other circle. The almond-shaped overlap is the Vesica Piscis.
Inside it is an equilateral triangle of side 1. Its half-height is:
height = sqrt(12 - (1/2)2) = sqrt(1 - 1/4) = sqrt(3/4) = √3 / 2
The full top-to-bottom line in the vesica stacks two of these: (√3 / 2) + (√3 / 2) = √3 (approx. 1.732)
Kid talk: "In the magic almond, the tall middle is root-three when circles have size 1."
√5 - the 1x2 rectangle's diagonal
Draw a 1x2 rectangle and its diagonal.
Legs: a = 1, b = 2
Compute: 12 + 22 = 1 + 4 = 5 → c = √5 (approx. 2.236)
Kid talk: "Go one square across and two squares up; the shortcut is root-five long."
3) Why this is Ratio 1 magic
Use unit size (call it "1") as your measuring stick:
- A 1x1 square gives √2 as its diagonal by a2 + b2 = c2.
- The Vesica Piscis (two unit circles) hides a side-1 equilateral triangle, so its full vertical span is √3.
- A simple 1x2 rectangle gives √5 as its diagonal by the same rule.
In short: divide space with unit shapes and use a2 + b2 = c2, and the square roots appear naturally:
12 + 12 = 2 → √2 |
1 - (1/2)2 → √3 / 2 → √3 |
12 + 22 = 5 → √5
4) Do-it-yourself steps (child-friendly)
- Choose unit 1. Draw with side or radius = 1.
- Make a right triangle (square diagonal, vesica height triangle, or a 1x2 rectangle).
- Square, add, root. Write the two leg lengths, square each, add them, then take the square root to get the slanted side or full height.
Historical note on credit
Pythagoras (6th century BCE) is widely credited with the theorem a2 + b2 = c2, but the idea is older and appears in several ancient cultures:
- Mesopotamia (Old Babylonian period, about 1800–1600 BCE): clay tablets (for example, Plimpton 322) list integer triples like 3–4–5 that satisfy the rule.
- Ancient Egypt (about 1650 BCE): rope-stretchers used the 3–4–5 triangle for right angles in land surveying (Rhind Mathematical Papyrus).
- Ancient India (about 800–500 BCE): the Sulba Sutras (for example, Baudhayana) state the right-triangle relation and list many triples.
- Ancient China (compiled about 1st century BCE): the Zhoubi Suanjing explains the Gougu (legs-hypotenuse) theorem, equivalent to a2 + b2 = c2.
Simple takeaway for kids: Pythagoras made it famous, but builders and scribes across the ancient world already used this right-corner rule to measure land and make things square and true.
| Formula (exact) | Notes | Formula (limit) |
|---|---|---|
| φ = (1 + √5) / 2 | Closed form (golden ratio) | NaN |
| φ = 2 · cos(36°) | From pentagon/decagon geometry | NaN |
| φ^2 = φ + 1 | Characteristic quadratic; implies φ = (1 + √5) / 2 | NaN |
| 1/φ = φ − 1 | Self‑reciprocal property | NaN |
| NaN | Ratio of consecutive Fibonacci numbers | φ = lim_{n→∞} F_{n+1} / F_n |
| Expression | Hours | Minutes | Seconds |
|---|---|---|---|
| 1 day | 24.000000 | 1440 | 86400.0 |
| 1 hour | 1.000000 | 60 | 60.0 |
| 1 minute | 0.016667 | 1 | 60.0 |
| 3 days | 72.000000 | 4320 | 259200.0 |
| 36 hours | 36.000000 | 2160 | 129600.0 |
| 72 hours | 72.000000 | 4320 | 259200.0 |
| 1440 minutes | 24.000000 | 1440 | 86400.0 |
| Target | Derivations (examples) |
|---|---|
| 3 | base element of the sequence |
| 6 | 3 + 3 ; 3 × 2 |
| 9 | 3 × 3 ; 6 + 3 |
| 12 | 6 + 6 ; 3 × 4 ; 24 ÷ 2 |
| 24 | 12 × 2 ; 3 × 8 ; 6 × 4 |
| 36 | 6 × 6 ; 12 × 3 ; 72 ÷ 2 |
| 54 | 6 × 9 ; 108 ÷ 2 |
| 72 | 36 × 2 ; 216 ÷ 3 |
| 108 | 36 × 3 ; 9 × 12 ; 216 ÷ 2 |
| 216 | 108 × 2 ; 36 × 6 ; 24 × 9 |
| 360 | 1080 ÷ 3 ; 2160 ÷ 6 ; 4320 ÷ 12 ; 8640 ÷ 24 ; 25920 ÷ 72 |
| 432 | 216 × 2 ; 12 × 36 |
| 864 | 432 × 2 ; 24 × 36 |
| 1080 | 2160 ÷ 2 |
| 2160 | 1080 × 2 ; 4320 ÷ 2 |
| 4320 | 2160 × 2 ; 8640 ÷ 2 |
| 8640 | 4320 × 2 ; 25920 ÷ 3 |
| 25920 | 4320 × 6 ; 2160 × 12 ; 8640 × 3 ; 1080 × 24 |
Four self-similar growth patterns that keep showing up in nature, geometry, sound, and cosmic scale:
Spiral of Theodorus (right-triangle root steps),
Phyllotaxis / Vogel model (sunflower packing at the golden angle),
Logarithmic Spiral (r = a·ebθ — shells & galaxies),
and a Cymatic / Rose Spiral (r = a·sin(kθ) standing-wave petals).
1. Spiral of Theodorus
√1 → √10
Radius after n steps = √n. We stop at √10.
2. Phyllotaxis / Vogel
Golden-angle packing
3. Logarithmic Spiral
r = a·ebθ
4. Cymatic / Rose Spiral
r = a·sin(kθ)
Pre-flood (pre-deluvian) rulers and how long each one was said to reign. The total adds up to 432,000 years, which locks straight into the 432 / 3‑6‑9 harmonic timelines.
| Name of King | Years of Reign | Digital Root of Years |
|---|---|---|
| Aloros | 36,000 | 9 |
| Alaporos | 10,800 | 9 |
| Amelon | 46,800 | 9 |
| Ammenon | 43,200 | 9 |
| Megalaros | 64,800 | 9 |
| Daonos | 36,000 | 9 |
| Euedoroches | 64,800 | 9 |
| Amempsinos | 36,000 | 9 |
| Opartes | 28,800 | 9 |
| XIsuhtros | 64,800 | 9 |
| TOTAL YEARS | 432,000 | 9 |
Every reign length in the list reduces to digital root 9 (sum of digits → 9). The grand total, 432,000 years, is itself a core “432” harmonic number used in cosmology lore, Vedic/Yuga cycle talk, and sacred time-base systems. This is the same 432 resonance we’re already tracking in frequency tuning and precessional timing blocks above.
The Earth is not a perfect sphere — the equator bulges.
• Equatorial diameter ≈ 7,926 miles
• Polar diameter ≈ 7,900 miles
• Difference ≈ 26 miles
• A diagonal / tropic-to-tropic style measure ~7,920 miles
• A perfect sphere with the same surface area as Earth would have diameter ≈ 7,920 miles, which is often treated as a “sacred number.”
Sakwala Chakraya - Wheel of the Universe
The Sakwala Chakraya is a shallow rock carving often described as a cosmological map. It is located in Sri Lanka, in the ancient city of Anuradhapura, inside the Ranmasu Uyana gardens. The site sits a short walk from the Isurumuniya temple, near the Tissa Wewa reservoir.
- Country: Sri Lanka
- City/Area: Anuradhapura
- Place: Ranmasu Uyana (near Isurumuniya and Tissa Wewa)
Core idea
- The stone encodes time-as-geometry in a sexagesimal (base-60) spirit: vesica-piscis gates, 3-6-9 ladders, and harmonic partitions that resolve into the Great Year = 25,920 sacred years (360-day year framing).
- Numbers like 432, 216, 144, 108, 36 act as phase nodes of that larger cycle.
Symbol map & harmonic arithmetic
- 20 border symbols × 22 × 6 = 2640
- 2640 − 2160 = 480 (with 2160 echoing the Moon’s diameter motif)
- 25,920 / 480 = 54 → 54 × 2 = 108 (solar/lunar distance lore)
- 25,920 / 20 = 1296 → cascade: 1296/3 = 432, /6 = 216, /9 = 144, /36 = 36
- Balances: 72 + 36 = 108, 480 − 360 = 120 (360° baseline)
Vesica layout
- 9 vesica piscis → nine “portals” (1:√3 union geometry).
- Vertical symmetry count: 4 up + 1 center + 4 down = 9.
Total symbol counts
- 33 inner (3×3 = 9; 3+3 = 6)
- 20 outer + 1 center → 33 + 20 + 1 = 54 (→ 108 when doubled)
- Optional reading: 32 (exclude center ring) × 20 = 640, 640 × 9 = 5760; 8640 − 5760 = 2880 → 1440 (folds into the 144-family; 144 cubits / 144,000 motif).
Precessional time (sacred-year lens)
- Great Year = 25,920 sacred years (360-day).
- Phase nodes:
- 480 years = 1/54 of 25,920 (critical step)
- 1440 years = 1/18 of 25,920 = 2 decans ≈ 20% of ecliptic (Egyptian 36-decan clock)
- Time mirrors: 480 hours, 28,800 minutes (sexagesimal echoes)
440 and the base systems
- 20 × 22 = 440 in base-10.
- Converting 44010 to base‑6 gives 20126 (how the “2012” figure arises from 20×22): repeated division by 6 yields remainders 2,1,0,2 → reversed as 20126.
- In base‑60, 44010 = 7;2060 (i.e., 7×60 + 20).
- Keeping 2012 as a symbolic phase marker aligns with the mirror-node motif, while the literal sexagesimal form is 7;20.
Base‑10 ↔ Base‑60: form vs flow
- Base‑10 = form (measures, matter, commerce).
- Base‑60 = flow (time, angles, sky‑temple geometry).
- Their resonant crossings show up at 432, 216, 144, 108, 36—numbers the slab foregrounds to fuse geometry, music, and calendrics.
Reading the stone as a “time‑gate mandala”
- 9 vesicas = nine gates / memory layers.
- 54 points (33 + 20 + 1) → 108 when doubled (solar/lunar distance lore).
- The 1296 fan‑out (432 / 216 / 144 / 36) maps fractal phases nested inside the 25,920‑year envelope.
One‑breath summary
Base‑10 speaks the shape; Base‑60 conducts the motion; frequency (e.g., 440 Hz → 20126, 7;2060, and the sacred set 432/144/108) bridges them. Sakwala Chakraya makes the Great Year legible as geometry and audible as harmonic number.
2012 in Base-60 is A Tesseract Moment in Time not the End...
- Base plan: perfect square, 403 ft 6.5 in × 403 ft 6.5 in (≈ 4,842.5 in per side).
- Overall form: 9 stacked levels — 6 square terraces (with 4 circumambulatory galleries) + 3 circular rings, crowned by a central stupa.
- Height: about 115 ft today; original stack (incl. “hidden foot”) often cited near 138 ft.
- Circular rings: 3 concentric rings with 72 perforated stupas arranged 32 + 24 + 16 around the great central stupa.
- Statues & reliefs: 504 seated Buddhas; 2,672 narrative relief panels.
- Geometric language: concentric square→circle mandala, strict axial symmetry, common proportional reading 4 : 6 : 9.
- Sacred numerics: 72 → 7+2 = 9 and 504 → 5+0+4 = 9 (digital-root / base-9 resonance).
Golden Ratio & Human Proportions
Definition (Golden Section)
A segment is divided in the golden ratio when the ratio of the whole to the larger part equals the ratio of the larger part to the smaller part.
(a+b)/b = b/a = φ⇒ a^2 + ab − b^2 = 0⇒ (a/b)^2 + (a/b) − 1 = 0⇒ a/b = φ⁻¹ = (√5 − 1)/2 ≈ 0.618 034⇒ φ = (1 + √5)/2 ≈ 1.618 034
Worked Example
Take a person of height 192.5 cm. Golden-ratio partitions of the height fall at the following levels:
| Level | Centimeters |
|---|---|
| Total height (H) | 192.50 |
| Navel level (H/φ) | 118.97 |
| Solar/diaphragm region (H/φ²) | 73.53 |
| Knee region (H/φ³) | 45.44 |
| Mid-shin/ankle region (H/φ⁴) | 28.09 |
These are guidelines; individual anatomy varies.
Personal Proportions Tool
Enter your total height and get the corresponding landmarks based on the Rasart φ‑scheme (navel ≈ H/φ, knee ≈ H/φ², fingertips (arm raised) ≈ 2·H/φ). Values are shown in centimeters/millimeters and feet–inches.
| Landmark | Exact (cm) | Nearest whole cm | Δ (mm) | Feet–Inches |
|---|---|---|---|---|
| Total height (H) | — | — | — | — |
| Navel height (H/φ) | — | — | — | — |
| Knee height (H/φ²) | — | — | — | — |
| Fingertips, arm raised (2·H/φ) | — | — | — | — |
φ = (1 + √5) / 2 ≈ 1.618 033 988. These are canonical guides; real anatomy varies.
Golden Ratio Cascade (×φ / ÷φ)
The table shows the geometric progression around H by powers of φ. Positive n multiplies by φ, negative n divides by φ.
| n | Value (cm) | Feet–Inches |
|---|---|---|
| +8 | 9043.40 | 296′ 8.39″ |
| +7 | 5589.13 | 183′ 4.44″ |
| +6 | 3454.27 | 113′ 3.95″ |
| +5 | 2134.86 | 70′ 0.50″ |
| +4 | 1319.41 | 43′ 3.45″ |
| +3 | 815.44 | 26′ 9.04″ |
| +2 | 503.97 | 16′ 6.41″ |
| +1 | 311.47 | 10′ 2.63″ |
| +0 | 192.50 | 6′ 3.79″ |
| -1 | 118.97 | 3′ 10.84″ |
| -2 | 73.53 | 2′ 4.95″ |
| -3 | 45.44 | 1′ 5.89″ |
| -4 | 28.09 | 0′ 11.06″ |
| -5 | 17.36 | 0′ 6.83″ |
| -6 | 10.73 | 0′ 4.22″ |
| -7 | 6.63 | 0′ 2.61″ |
| -8 | 4.10 | 0′ 1.61″ |
Vitruvian Man — Hidden Triangle
The fact that the difference between the positive and the negative root is −1 has a curious meaning. For a rectangle with length 1.618… and width 1: if you place a square along the length with a side equal to the width, the remaining rectangle has the same proportions (length 1.618… and width 1), repeating ad infinitum.
- Boundary identification (one‑point compactification): $$ (0,\infty)^{*}=(0,\infty)\cup\{\omega\},\quad \lim_{r\to 0^{+}} r=\lim_{r\to\infty} r=\omega,\quad \boxed{0 \equiv \infty}. $$
- Equivalence‑class statement (quotient = circle of scales): $$ [0,\infty]/(0\sim\infty)\cong S^{1},\qquad \boxed{0 =_{\text{quot}} \infty}. $$
- Inversion duality (extended line): $$ x\mapsto \frac{1}{x}\ \text{on}\ \overline{\mathbb{R}}\ \Rightarrow\ \boxed{0 \leftrightarrow \infty}. $$
“Infinitely small” (No‑thing) and “infinitely large” (All) coincide by the chosen identification on scale.
Precession & Sunrises (Northern Hemisphere)
Precession (big picture): Earth’s spin axis slowly traces a cone because the Sun and Moon torque the equatorial bulge. Rate ≈ 50.3″/year → one full cycle in ~25,772 years (often rounded to 25,920 ≈ 1° per 72 years). The equinox points drift westward along the ecliptic, so the background zodiac constellations behind the equinoxes/solstices change over millennia.
- Vernal (March) equinox currently lies in Pisces (moving toward Aquarius).
- September equinox is in Virgo.
- Solstitial longitudes are in/near Gemini (June) and Sagittarius/Ophiuchus (December).
Sunrise directions through the year
Let latitude be φ; Earth’s axial tilt (obliquity) ≈ 23.44°.
| Event | Solar declination δ | Sunrise (N hemi) | Day length | Notes |
|---|---|---|---|---|
| March & September Equinox | 0° | Due East; sets due West | ≈ 12 h / 12 h | “Equal night” (small variations from refraction/latitude) |
| June (Summer) Solstice | +23.44° | North–East; sets North–West | Longest day | Tropic of Cancer overhead Sun; midnight sun above Arctic Circle |
| December (Winter) Solstice | −23.44° | South–East; sets South–West | Shortest day | Polar night above Arctic Circle |
Azimuth A measured from North (East = 90°). At δ = 0° (equinox), A = 90° (due East). Positive δ (summer) shifts sunrise north of East; negative δ (winter) shifts it south of East.
Sunrise Azimuths by Latitude (Northern Hemisphere)
Azimuth is measured from North, clockwise. East = 90°. Dates shown are for 2025.
| Event | Date (2025) | Solar declination | Sunrise azimuth | Comment |
|---|---|---|---|---|
| June (Summer) Solstice | Jun 21 | +23.44° | -- | -- |
| March Equinox | Mar 20 | 0° | -- | -- |
| September Equinox | Sep 22 | 0° | -- | -- |
| December (Winter) Solstice | Dec 21 | −23.44° | -- | -- |
Method
We use cos A = sin δ / cos φ, where A is azimuth from North, δ is solar declination, and φ is latitude. If |sin δ/cos φ| > 1, sunrise does not occur (midnight sun or polar night).
Earth Orbit Diagram
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Glossary of Terms
Short explanations of the most important words used on this page.
- 432 Hz
- A musical tuning where the note A is set to 432 vibrations per second.
- Root-9 / Digital Root
- Reducing a number to one digit by adding its digits. Example: 432 -> 4+3+2 = 9.
- Base-10
- The normal number system we use every day (0-9).
- Base-60
- Older counting system that goes 0-59. Used in time (60 seconds, 60 minutes) and angles (360 degrees).
- Precession
- Earth's slow wobble. One full cycle takes about 26,000 years (often rounded to 25,920).
- Equinox
- The 2 days in the year when day and night are equal length everywhere on Earth.
- Solstice
- The longest and shortest days of the year.
- Phi / Golden Ratio
- A special proportion written as 1.618... found in shells, art, and nature.
- Platonic Solids
- 5 perfect 3D shapes: cube, tetrahedron, octahedron, dodecahedron, icosahedron.
- Vesica Piscis
- Shape made by overlapping 2 circles so their centers touch.
- Seed of Life
- Pattern made from 7 circles. Used in many symbols and mandalas.
- Flower of Life
- Repeating geometric pattern based on the Seed of Life.
- Metatron's Cube
- Made from 13 circles and connecting lines. Holds all 5 Platonic solids inside it.
- Harmonics
- Frequencies that are simple multiples of each other, like 432, 864, 216.
- Frequency
- How many times something repeats in 1 second.
- Cycle
- Something that repeats, like seasons, waves, breath, music, orbits.
- Ratio / Proportion
- A comparison of values, like 2:1 or 3:2.
- Sacred Geometry
- The study of shapes and patterns believed to reflect natural order.
- Yuga Cycle
- Very long repeating ages of time described in ancient India.
- Decan
- One of 36 divisions of the zodiac sky, each 10 degrees wide.
- Base Conversion
- Changing a number from one base (like 10) to another (like 6).
- Resonance
- When two vibrating things match in frequency and amplify each other.
