---
© 2015-2026 Wesley Long & Daisy Hope. All rights reserved.
Synergy Research — FairMind DNA
License: CC BY-SA 4.0
Originality: 100% — Original π derivation methods from Synergy Research (2018–2024)
---

# SSM π Derivation Methods
## Eight Independent Approaches to Deriving π from Geometry

**Wesley Long — Synergy Research**

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## Overview

Between 2018 and 2024, Synergy Research produced eight independent methods for deriving π from geometric first principles. Each method uses a different algebraic route but draws on the same underlying structures: the unit square, point structures, the Golden Ratio, and the Quadrian Components.

All eight are implemented in `js/ssm.pi.rank.js` and ranked against true π alongside historical values, observed measurements, and rational fractions.

---

## 1. Rational Pi (2018)

$$\pi_R = \frac{28}{9} + \frac{1}{28} - \frac{1}{189}$$

**Value:** 3.1415343915343916 (6 matching digits)

**Form:** Sum of three unit fractions. Uses 28 = 4×7, 9 = 3², 189 = 27×7 = 3³×7.

**Significance:** Earliest SSM approach. Purely rational — no irrational numbers involved.

---

## 2. Turtle Pi (2019)

$$\pi_T = \frac{r \times 6 + \frac{r}{2} \times \frac{1+1/5}{2+1/10}}{d}$$

with $d = 1$, $r = d/2$.

Simplifies to:

$$\pi_T = 3 + \frac{1}{2} \times \frac{6/5}{21/10} = 3 + \frac{1}{7} = \frac{22}{7}$$

**Value:** 3.142857142857143 (4 matching digits)

**Form:** Geometric construction from a circle's diameter. Rediscovers Archimedes' 22/7 from a different construction path.

**Significance:** Shows that 22/7 is not just a historical approximation — it emerges naturally from geometric subdivision.

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## 3. Syπ — The Gradient Equation (2018)

$$\Pi(n) = \frac{3940245000000}{2217131n + 1253859750000}$$

**Value at n=162:** 3.141592684309533 (9 matching digits)

**Form:** Linear fractional (Möbius) transformation. See `SYPI_NOTATION.md` for full algebra.

**Significance:** The central SSM π equation. π is not a constant but a gradient — a function of position. Every physical equation that uses π implicitly selects a gradient position.

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## 4. Easy123 Pi (2021)

$$\pi_{123}(n) = \frac{2^1 \times 3^2 \times (3^2+1)}{R(n)}$$

where $R(n)$ is expressed entirely in powers of 1, 2, and 3:

$$R(n) = \frac{(3^2+1)^3((3^2+1)^3 \times 2^{-3}+1)}{2^3 \times 3^3 \times (3^2+1)+2} - \left(1-(3^3+1)n \times (3^2+1)^{-6} \times (2^{-1} \times 3^{-4} \times (3^2+1)^2+3)\right)$$

**Value at n=1:** 3.142487054628346 (4 matching digits — identical to Syπ(1))

**Form:** Rational function using only {1, 2, 3} as coefficients. This is the Syπ equation rewritten to make explicit that it reduces to powers of 2 and 3.

**Significance:** Proves claim G-04 constructively. The Syπ equation contains no "hidden" constants.

---

## 5. Eye Pi (2023)

$$\pi_E = \frac{U}{R}$$

where $U = 3 \times 2 \times (3 \times 10) = 180$ and $R$ is built from a recursive decimal series:

$$d = 2 + \sum_{j=1}^{10} \frac{1}{(10j)^j}$$

$$R = d \times (3^3 + 1/2^2) + \frac{1}{3^2(6^2 + 2^2) - \frac{1}{5 - 1/250}}$$

**Value:** 3.141529884073451 (6 matching digits)

**Form:** Complex rational composition. Uses only {1, 2, 3, 10} as inputs.

**Significance:** Achieves 6-digit accuracy through recursive series construction rather than algebraic manipulation.

---

## 6. Bubble Pi (2023)

$$\pi_B = \frac{Q}{4} - Z \times p$$

where:
- $d = 1$, $r = 1/2$, $H = 1/6$ (Hexagon), $p = 13/2$ (Radial half)
- $Z = 1/(2 \times 4 \times 8 \times 7 \times 5) = 1/2240$ (Doubling Circuit reciprocal)
- $X = (2d/H)(15/8) = 12 \times 15/8 = 22.5$
- $q = \sqrt{d^2 + r^2} = \sqrt{5}/2$ (Quadrian Ratio)
- $Q = X/(2d/q) = 22.5 \times q/2$

**Value:** 3.1415688076447936 (6 matching digits)

**Form:** Geometric derivation from the hexagonal subdivision of the unit circle, corrected by the Doubling Circuit.

**Significance:** Directly uses the SSM point structures (6, 8, 13, 15) and the Doubling Circuit product (2240). Connects π to the same geometry that produces particle masses.

---

## 7. Phi Pi (2023)

$$\pi_\Phi = \frac{6}{5} \times \Phi^2$$

where $\Phi = (1+\sqrt{5})/2$ is the Golden Ratio.

**Value:** 3.141640786499874 (5 matching digits)

**Form:** Scaled square of the Golden Ratio.

**Significance:** The simplest SSM π — just two Quadrian Components (5 and 6) plus the Golden Ratio. Shows that π and Φ are structurally adjacent.

**Note:** $\Phi^2 = \Phi + 1 \approx 2.618$, so $\pi_\Phi = 6(Φ+1)/5 = 6Φ/5 + 6/5$.

---

## 8. GEP:163 — Ramanujan Pi (2024)

$$\pi_G = \frac{\ln(262537412640768744)}{\sqrt{163}}$$

**Value:** 3.141592653589793 (17 matching digits — ties Math.PI at float64)

**Form:** Logarithmic ratio involving the Ramanujan constant $e^{\pi\sqrt{163}}$.

**Significance:** This is the most precise SSM π. It is not new mathematics — the near-integer property $e^{\pi\sqrt{163}} \approx 262537412640768743.99999999999925...$ is Ramanujan's classical result. What the SSM adds is the observation that 163 is also a Heegner number (class number 1), and that this π lives at Syπ position $n \approx 162.006$ — indistinguishable from the Syπ(162) position.

The connection to absolute zero: $\Pi(-273150) = 6.078...$, and $-273.15°\text{C}$ is absolute zero.

---

## Summary Table

| # | Method | Year | Value | Match | Form |
|---|---|---|---|---|---|
| 1 | Rational Pi | 2018 | 3.14153439... | 6 | Unit fractions |
| 2 | Turtle Pi | 2019 | 3.14285714... | 4 | Geometric (= 22/7) |
| 3 | Syπ(162) | 2018 | 3.14159268... | 9 | Möbius transformation |
| 4 | Easy123(1) | 2021 | 3.14248705... | 4 | Powers of {1,2,3} |
| 5 | Eye Pi | 2023 | 3.14152988... | 6 | Recursive series |
| 6 | Bubble Pi | 2023 | 3.14156880... | 6 | Hexagonal geometry |
| 7 | Phi Pi | 2023 | 3.14164078... | 5 | Golden Ratio |
| 8 | GEP:163 | 2024 | 3.14159265... | 17 | Ramanujan constant |

---

## Classification by Approach

| Class | Methods | What they share |
|---|---|---|
| **Algebraic** | Rational, Easy123 | Pure integer arithmetic, no irrationals |
| **Geometric** | Turtle, Bubble | Physical construction from circles/hexagons |
| **Golden** | Phi, GEP:163 | Golden Ratio / Φ as primary ingredient |
| **Gradient** | Syπ | π as a function of position |
| **Series** | Eye | Recursive decimal expansion |

---

## Open Question G-1

> Are all 8 methods derivations of the same underlying structure, or are they independent?

Partial answer: Easy123 is provably the same as Syπ (claim G-04). Bubble Pi uses the same point structures (6, 8, 15) as the Bubble Mass equation. Phi Pi uses the Golden Ratio which underlies the Quadrian Ratio. GEP:163 connects to Syπ through position correspondence. The others may be independent routes to the same geometric truth, or may reveal deeper connections not yet understood.

---

## How to Run

```
node js/ssm.pi.rank.js 200        # from pub/
python py/ssm.pi.rank.py 200      # Python counterpart
```

All 8 methods are included in the ranking alongside historical values, observed measurements, rational fractions, and the Syπ gradient sweep.

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*Synergy Standard Model v2.0 — © 2015–2026 Synergy Research. All rights reserved.*