iTranslated by AI

The content below is an AI-generated translation. This is an experimental feature, and may contain errors. View original article
🌀

Calculating Pi Using B13 (BASE = 3120): Public Notes

に公開
Python
円周率
数論
b13
tech

The 720 That Appears When Wrapping π with B13 (BASE = 3120) - Public Note -

1. Introduction

In my previous article, "Where Do Circles Come From? - Light Cone, Fullerene, Ramanujan Formula - Public Note -", I presented several approaches to the circle. This time, I have verified the circumference ratio π using B13.

What can we see when we read π not in base-10, but in the base BASE = 3120 of the B13 framework?

In B13, BASE = 3120 = 13 × 240 = 60 × 52 is confirmed as the fundamental lattice for angles and geometry. The CMB first acoustic peak ℓ* = 302 = 13 × 23 + 3, the total angular deficit of the C60 fullerene at 720°, and the electron shell capacities of 2/8/18/32 are all explained on the same lattice.

What emerges when we read the circumference ratio on this lattice? This paper documents that implementation and observation.

2. Methodology — 3120 Balanced Expansion

We express π using the following series:

\pi = \sum_{n=1}^{\infty} \frac{a_n}{3120^n}

Each coefficient a_n is an integer, determined uniquely by the following procedure:

1. Initialize residual r with π
2. Multiply r by 3120         (Scale the circle by 3120)
3. Take the nearest integer a_n   (Choose the closest point on the 3120-point lattice)
4. r ← r - a_n              (Carry the residual to the next stage)
5. Advance to the next stage and return to 2

Only addition and nearest-integer rounding are required. Concepts such as factorials, powers, square roots, trigonometric functions, or infinite sums are unnecessary.

This is equivalent to repeating the operation "Which point on a circle drawn with 3120 points is this closest to?" on a circle scaled up by 3120 at each step.

Implementation

from decimal import Decimal, getcontext

BASE = 3120

def evaluate_pi(n_terms=15, prec_digits=200):
    getcontext().prec = prec_digits
    pi = _pi_decimal(prec_digits)
    base = Decimal(BASE)

    residual = pi
    coefficients = []
    for _ in range(n_terms):
        shifted = residual * base
        a_n = _nearest_int(shifted)
        coefficients.append(a_n)
        residual = shifted - Decimal(a_n)
    return coefficients

The core calculation consists of only these three lines:

shifted = residual * base       # Scale
a_n = _nearest_int(shifted)     # Select nearest point
residual = shifted - Decimal(a_n)  # Update residual

3. Results

The following table shows the results expanded to 15 stages:

Stage n Coefficient a_n Factorization Cumulative Error
1 9802 2 × 13² × 29 10⁻⁴·¹³
2 −720 −2⁴ × 3² × 5 10⁻⁷·³¹
3 −1475 −5² × 59 10⁻¹⁰·⁸⁴
4 −1354 −2 × 677 10⁻¹⁴·⁹¹
5 −367 Prime 10⁻¹⁷·⁸⁶
6 −1280 −2⁸ × 5 10⁻²²·²⁶
7 −159 −3 × 53 10⁻²⁴·⁷⁹
8 1456 2⁴ × 7 × 13 10⁻³¹·¹¹
9 −2 −2 10⁻³²·²⁰
10 −556 −2² × 139 10⁻³⁵·³²
11 1292 2² × 17 × 19 10⁻³⁹·⁰⁷
12 727 Prime 10⁻⁴²·²⁸
13 −1382 −2 × 691 10⁻⁴⁶·⁶²
14 197 Prime 10⁻⁴⁹·³⁴
15 1190 2 × 5 × 7 × 17 10⁻⁵²·⁸⁰

Precision improves by approximately 3.48 digits per stage on average. Since the theoretical upper limit is log₁₀(3120) ≈ 3.494 digits/stage, π is being wrapped with near-maximum efficiency.

Two Notable Stages

a₁ = 9802 = 2 × 13² × 29 = 26 × F₁₄

Here, F₁₄ = 377 = 13 × 29 is the 14th term of the Fibonacci sequence. In the first stage, π appears as an "integer ratio having 13² × F₁₄ in the numerator".

\pi \approx \frac{9802}{3120} = \frac{2 \times 13^2 \times 29}{60 \times 52} = \frac{26 \times F_{14}}{3120}

a₂ = −720

The second-stage coefficient is ±720. This matches the total angular deficit of 720° for the C60 fullerene (in B13 units, 2 × 13).

C60 is a truncated icosahedron consisting of 12 pentagons and 20 hexagons. Each pentagon has an angular deficit of 60° from a regular polygon, and the 12 faces together result in a total deficit of 720° = 2 × 13 B13 units (confirmed in documentation v114).

I will not provide interpretations for a₃ onwards in this paper. This is a crucial stance. I am drawing a line here to avoid fabricating correlations.

4. Comparative Experiment — Is 720 Specific to π or BASE?

A question arises:

Does a₂ = −720 appear due to the nature of π, or the nature of BASE = 3120 itself?

Since BASE = 3120 has 720 as a divisor (3120 / 720 = 13/3), if BASE had a property of "automatically engraving 13-related integers into the coefficient sequence," one would expect 720 and multiples of 13 to appear frequently in early stages for constants other than π.

To isolate this, I applied the same procedure to the following mathematical constants:

  • e (Napier's constant)
  • φ (Golden ratio)
  • √2, √3, √5
  • ln 2

Comparison of Early-Stage Coefficients

Coefficients from the 2nd to 6th stages obtained by expanding each constant in a 3120 equilibrium representation:

Constant a₂ a₃ a₄ a₅ a₆
π −720 −1475 −1354 −367 −1280
e 123 −1151 −1343 −1498 460
φ 830 187 1471 −1051 273
√2 1081 −1555 −369 −26 123
√3 −5 1189 1058 1483 −1122
√5 −1460 375 −179 1018 545
ln 2 −1188 −267 191 1019 770

Stages Where "720" Appears

Results after expanding 7 constants up to 30 stages:

Constant Stage where absolute value 720 appears
π a₂(Early)
√5 a₂₈(Deep)
Other 5 constants None

Only in π did 720 appear in the early stage a₂. Since a₂₈ for √5 is a deep-layer stage, and sporadic matches are possible given the probability of a value hitting each stage (2 × |a_n|/BASE ≈ 1/4.3 viewed in finite stages), this cannot be called unique.

I checked whether the set of C60-related integers (12, 20, 30, 60, 90, 120, 180, 360, 720, 1560, 3120) appears in early stages where n ≤ 5.

Constant C60 integers in early stages
π a₂ = −720
Other 6 constants None (all appearances are in deep stages where n ≥ 18)

π is the only constant with C60-related integers in the early stages.

Appearance Rate of Multiples of 13

To examine the perspective that "since BASE = 3120 contains 13, multiples of 13 appear frequently in the coefficient sequence," I tabulated the proportion of coefficients divisible by 13 from the second stage onwards for 29 stages (the random expectation is 1/13 ≈ 7.7%).

Constant Number of multiples of 13 / 29 stages Appearance Rate Binomial test p-value
π 1 3.45% 0.52
e 2 6.90% 1.00
φ 2 6.90% 1.00
√2 1 3.45% 0.52
√3 0 0.00% 0.17
√5 1 3.45% 0.52
ln 2 3 10.34% 0.72

For all constants, p > 0.16. The appearance of multiples of 13 is indistinguishable from a random distribution.

5. Interpretation

From these comparisons, the following can be read:

(A) 720 is not automatically generated by BASE

The appearance rate of multiples of 13 is consistent with random expectation for all constants. The view that BASE = 3120 itself automatically engraves a 13-structure into the coefficient sequence is not supported.

(B) 720 may have appeared as a feature of π

Among the 7 constants, only π had C60-related integers in the early stages (n ≤ 5). This may reflect the relationship between π and circles, and between circles and C60 (the discretized geometry of spheres).

(C) However, it cannot be called "unique"

There is a limit to statistical rigor with 7 constants. One should remain at describing the fact that "early appearance was observed only for π," and it is premature to conclude that it is a "property unique to π."

6. Conclusion

When π is re-read using the equilibrium expansion of BASE = 3120, 13² × 29 = 169 × 29 appears in the first stage, and −720, which numerically matches the C60 total angular deficit, appears in the second stage. When the same expansion is applied to the other 6 basic constants, C60-related integers do not appear in the early stages.

This is a promising observation for reading π on the B13 grid.

Discussion — On the Necessity of Proof

This expansion is not a mathematical discovery. When the value of π is given, I am merely re-reading it using BASE 3120.

For any real number x,

x = \sum_{n=1}^{\infty} \frac{a_n}{3120^n}

holds by definition. Each coefficient a_n is uniquely determined by rounding to the nearest integer, and convergence is self-evident because the residual shrinks to 1/3120 or less at each stage. This is an operation at the same level as "writing π in decimal as 3.14159..." which requires no proof.

Closed-form series like Ramanujan's formula require proof that a specific series truly converges to π (using modular equations or elliptic integrals). On the other hand, the expansion in this paper requires no such proof. This is because I am simply reading the value of π in a different base.

Therefore, a₁ = 26 × F₁₄ and a₂ = −720 observed in this paper are not proven theorems, but facts that appear when π is read in BASE = 3120. The observation that these facts numerically match the C60 total angular deficit may reflect the nature of the value of π itself.

Three-Layer Classification

[Confirmed] I implemented the BASE=3120 equilibrium expansion of π. The coefficients are a₁=9802=2×13²×29, a₂=−720, a₃=−1475, a₄=−1354, a₅=−367…. a₂=−720 numerically matches the C60 total angular deficit. As a comparative experiment, applying the same expansion to e, φ, √2, √3, √5, ln2 showed that C60-related integers appear in early stages (n≤5) only for π. On the other hand, the appearance rate of 13 multiples was within the range of random expectation, and the view that BASE=3120 automatically engraves a 13-structure into the coefficient sequence was not supported.

[Observation] The structure of a₁ and a₂ may be a feature of π.

[Hypothesis] The B13 equilibrium expansion of π suggests a structural connection with C60 geometry. However, statistical rigor is weak at this stage.

Future Challenges

  • Extend comparative constants to 20 or more (γ, ζ(3), Catalan constant, ζ(5), Khinchin constant, etc.)
  • Rigorous modeling of the probability of C60 integers appearing in early stages
  • Attempt analytical derivation of a₁, a₂ (Can one derive 9802/3120 - 720/3120² + ... from the closed form of π?)
  • Re-verification with other continuous quantities (e^π instead of e, π², etc.)

Implementation
The full implementation is included in the following repository:

b13_pi.py — B13 (BASE=3120) Equilibrium Expansion of π

b13_constants_compare.py: Comparative experiment with multiple constants

Both modules share BASE = 3120 and are consistent with the b13phase package.

Discussion