Paper 1-3: Architecture of Three Major Transformation Channels Based on Multiple Expressions of π

Author: Zhang Suhang (Heluo Mathematical School)

Abstract

The core innovation of the \mathcal{\Pi} operator lies in the fact that different mathematical expressions of π are not merely equivalent numerical descriptions, but act as channel signatures corresponding to distinct logics of dimensional transformation. According to the mathematical structures of these expressions, this paper classifies the \mathcal{\Pi} operator into three primary channels. Channel I (geometric π) corresponds to rigid body rotation transformation; Channel II (series-type π) corresponds to periodic infinitesimal transformation; Channel III (integral and complex-variable π) corresponds to spatial field mapping. We systematically elaborate the functional positioning, transformation rules, applicable scenarios and interrelationships of each channel, establish criteria for channel selection, and construct a unified operational framework for subsequent geometric verification and higher-dimensional extension.

Keywords: \mathcal{\Pi} operator; three major channels; geometric π; series-type π; integral-type π; dimensional transformation

1. Introduction

1.1 Problem Statement

Paper 1-1 defines the symbolic system of the \mathcal{\Pi} operator, and Paper 1-2 establishes the curl-preservation axiom. Nevertheless, a core problem remains unresolved:

Given a specific two-dimensional object, what "path" can transform it into a three-dimensional object? How do different expressions of π guide such a selection?

The key insight of this paper is presented as follows:

Each mathematical expression of π essentially serves as an eigenvalue or generating function of the \mathcal{\Pi} operator under a specific transformation logic. The choice of expression determines the adopted transformation channel.

1.2 Origin of the Channel Concept

The term "channel" is inspired by urban interchange systems:

- Different ramps lead to different destinations.

- All ramps share the same overpass (the \mathcal{\Pi} operator).

- Travelers (two-dimensional objects) select appropriate ramps according to their destinations (transformation targets).

Mathematically, the correspondence is defined as:

- Channel I: π defined as the ratio of circumference to diameter of a circle → generation of rotating solids

- Channel II: π expressed as the sum of an infinite series → superposition of periodic structures

- Channel III: π derived from Gaussian integral → mapping of field distributions

1.3 Paper Organization

Section 2 presents an overview and classification criteria for the three channels. Sections 3 to 5 elaborate on each channel respectively. Section 6 discusses the collaboration and conversion between channels. Section 7 provides the decision-making process for channel selection. Section 8 draws the conclusion.

2. Overview of the Three Major Channels

2.1 Classification Criteria

The three channels are classified based on the mathematical structures of π expressions and their implied transformation logics:

- Channel I (Geometric π): Representative formula \displaystyle \pi = \lim_{n \to \infty} n\sin\frac{\pi}{n}; Transformation logic: rigid body rotation; Output features: rotating solids (sphere, cylinder, cone).

- Channel II (Series-type π): Representative formula \displaystyle \frac{\pi}{4} = 1-\frac{1}{3}+\frac{1}{5}-\cdots; Transformation logic: periodic superposition; Output features: periodic surfaces (helix, ripple).

- Channel III (Integral & Complex-variable π): Representative formula \displaystyle \int_{-\infty}^{\infty} e^{-x^2}dx = \sqrt{\pi}; Transformation logic: field mapping; Output features: spatial fields (Gaussian field, quantum field).

2.2 Unified Notation

The \mathcal{\Pi} operators for respective channels are denoted as:

\mathcal{\Pi}^{(I)}, \quad \mathcal{\Pi}^{(II)}, \quad \mathcal{\Pi}^{(III)}

For default dimensional elevation (2→3), the notation is simplified to \mathcal{\Pi}^{(k)}. The full notation \mathcal{\Pi}^{(k)}_{m \leftarrow n} is adopted when specifying the source and target dimensions.

2.3 Schematic of Channel Workflow

A three-dimensional output is obtained via three independent paths. A two-dimensional input is converted into spheres, cylinders or cones through Channel I (geometric π, rigid body rotation); into helices or ripples through Channel II (series-type π, periodic superposition); and into Gaussian fields or wave fields through Channel III (integral & complex-variable π, field mapping).

3. Channel I: Geometric π → Rigid Body Rotation Transformation

3.1 Basis of Expressions

Channel I adopts the fundamental geometric definitions of π:

\pi = \frac{C}{2r} = \frac{A}{r^2}

along with its limit form:

\pi = \lim_{n \to \infty} n \cdot \sin\frac{\pi}{n}

and integral form:

\pi = 2\int_{-1}^{1}\sqrt{1-x^2}\,dx

3.2 Transformation Rules

Core operation: Rotate a two-dimensional figure by 2\pi radians around its axis of symmetry or a specified axis.

Formal definition:

\mathcal{\Pi}^{(I)}(G_2, L) = \left\{ (x,y,z) \,\bigg|\, \big(\sqrt{y^2+z^2},\, x\big) \in G_2,\ \text{where axis } L \text{ is the } x\text{-axis} \right\}

where G_2 is represented in the (r,x) coordinate system, and r denotes the distance to the rotation axis.

Curl preservation (Axiom 1 in Paper 1-2):

\mathcal{\Pi}^{-1}\big(\mathcal{\Pi}^{(I)}(G_2)\big) = G_2

3.3 Typical Transformation Examples

Common two-dimensional inputs and their corresponding three-dimensional outputs together with volume formulas are listed below:

- Circle (radius r) → Sphere: \displaystyle V = \frac{4}{3}\pi r^3

- Rectangle (height h, half-width r) → Cylinder: \displaystyle V = \pi r^2 h

- Right triangle (base r, height h) → Cone: \displaystyle V = \frac{1}{3}\pi r^2 h

- Semicircle (radius r) → Hemisphere: \displaystyle V = \frac{2}{3}\pi r^3

- Ellipse (semi-axes a, b) → Ellipsoid (rotated about the major axis): \displaystyle V = \frac{4}{3}\pi a b^2

3.4 Operator Expression for Volume Transformation

For a solid of revolution, the general formula is given by:

V_3 = \mathcal{\Pi}^{(I)}(A_2) = 2\pi \int x \cdot y(x)\,dx

where y(x) is the generatrix function. The formula degenerates to that of a cylinder when y(x) is constant, and to that of a sphere when y(x) describes a circular arc.

3.5 Applicable Scenarios

Applicable cases: Objects with a well-defined axis of rotational symmetry; generatrix functions of simple forms (straight line, circular arc, elliptical arc); tasks aiming to generate closed rotating solids.

Inapplicable cases: Objects without rotational symmetry; objects with overly complex generatrix functions (Channel II or Channel III is recommended instead).

4. Channel II: Series-type π → Periodic Infinitesimal Transformation

 4.1 Basis of Expressions

Channel II employs series expansions of π:

Leibniz series:

\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots = \sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1}

Euler series:

\frac{\pi^2}{6} = \sum_{n=1}^{\infty} \frac{1}{n^2}

Wallis product:

\frac{\pi}{2} = \prod_{n=1}^{\infty} \frac{(2n)^2}{(2n-1)(2n+1)}

4.2 Transformation Rules

Core idea: Each term of the series corresponds to a spatial "infinitesimal layer". A three-dimensional periodic structure is formed by term-by-term superposition.

Formal definition:

\mathcal{\Pi}^{(II)}(f(x)) = \lim_{N \to \infty} \sum_{n=0}^{N} c_n \cdot \Phi_n(x,y,z)

where:

- f(x) denotes a two-dimensional periodic curve (or periodic function);

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- c_n stands for the series coefficient;

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- \Phi_n represents the n-th three-dimensional infinitesimal element (helical turn, ripple layer).

Simplified form for helix generation:

Given a two-dimensional periodic curve y = f(x) with period T = 2\pi R:

\mathcal{\Pi}^{(II)}(f) =

\begin{cases}

x = R\cos\theta \\

y = R\sin\theta \\

z = f(\theta)

\end{cases}, \quad \theta \in [0, 2\pi)

This is the parametric equation of a cylindrical helix, where f(\theta) is a periodic function.

4.3 Typical Transformation Examples

- y = \sin x (period 2\pi) → Sinusoidal helix: one turn per 2\pi interval

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- Constant function (arbitrary period) → Cylindrical surface: the series degenerates into a constant term

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- Sawtooth wave (period T) → Sawtooth helix: formed by progressive superposition

4.4 Special Status of Ramanujan Series

The 1/\pi series proposed by Ramanujan features extremely fast convergence. Within Channel II, its properties correspond to the following physical interpretations:

- The convergence rate of each term reflects the packing density of helical structures.

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- Modular forms embedded in coefficients correspond to higher-dimensional symmetries.

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- It can be regarded as an advanced optimized version of Channel II.

Detailed discussions are presented in Paper 3-3 and Paper 3-4 of this series.

4.5 Applicable Scenarios

Applicable cases: Periodic curves (sine curve, cosine curve, triangular wave); helical structures (spring, screw thread); ripple surfaces and periodically concave-convex structures.

Inapplicable cases: Aperiodic objects (Channel I is recommended); closed figures of finite length (Channel I provides a more direct solution).

5. Channel III: Integral & Complex-variable π → Global Field Transformation

5.1 Basis of Expressions

Channel III adopts integral and complex-variable expressions of π:

Gaussian integral:

\int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi}

Complex-variable form (Euler's formula):

e^{i\pi} + 1 = 0

Gamma function form:

\Gamma\left(\frac{1}{2}\right) = \sqrt{\pi}

5.2 Transformation Rules

Core idea: Map a two-dimensional field (scalar field, probability field, wave field) into a three-dimensional field, where π acts as a normalization constant or phase factor.

Formal definition for scalar field elevation:

\mathcal{\Pi}^{(III)}\big(\phi_2(x,y)\big) = \phi_3(x,y,z) = \phi_2(x,y) \cdot K(z; \pi)

The kernel function K(z; \pi) satisfies:

\int_{-\infty}^{\infty} K(z; \pi) dz = \sqrt{\pi} \quad \text{or} \quad \int_{-\infty}^{\infty} K^2(z; \pi) dz = \sqrt{\pi}

Typical kernel functions:

- Gaussian kernel: K(z) = e^{-\pi z^2}

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- Exponential kernel: K(z) = e^{-|z|/\sqrt{\pi}}

5.3 Typical Transformation Examples

- Gaussian distribution \mathcal{N}(0,\sigma^2) combined with Gaussian kernel → 3D Gaussian sphere (application in probability fields)

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- Plane wave e^{i(k_x x + k_y y)} extended via complex exponential → Spatial wave e^{i(k_x x + k_y y + k_z z)} (application in wave optics)

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- Circularly symmetric potential field V(r) processed via rotational integral → Spherically symmetric potential field V(R) (application in quantum mechanics)

5.4 Special Status of Euler's Formula

Euler's formula e^{i\pi} + 1 = 0 serves as the minimal instance of Channel III:

- It can be interpreted as dimensional reduction mapping from zero dimension (point) to one dimension (unit circle in complex plane), and further to two dimension (complex plane).

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- π acts as a phase measure, representing a rotation of 180 degrees.

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- The term "+1" denotes the closure condition for returning to the real axis.

Relevant contents are elaborated in Paper E1-4 and Paper E4-4 of this series.

5.5 Applicable Scenarios

Applicable cases: Probability fields, Gaussian processes; wave functions, quantum fields; potential fields, diffusion fields; field mapping requiring continuous symmetry.

Inapplicable cases: Discrete geometric objects (Channel I is recommended); non-smooth fields (preprocessing is required).

6. Collaboration and Conversion between Channels

6.1 Composite Transformation

Different channels can be applied in series:

Channel I → Channel II: Generate a rotating solid first, then attach periodic textures

\mathcal{\Pi}^{(II)} \circ \mathcal{\Pi}^{(I)}(G_2) = \text{Textured rotating solid}

Channel III → Channel I: Construct field distribution first, then generate geometric solids

\mathcal{\Pi}^{(I)} \circ \mathcal{\Pi}^{(III)}(\phi_2) = \text{Rotating solid of field equipotential surface}

6.2 Decision Tree for Channel Selection

1. Classify the input object. If it is a discrete geometric figure, proceed to Step 2; if it is a field or function distribution, jump to Step 4.

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2. For discrete geometric figures: Check whether a rotational symmetry axis exists. If yes, select Channel I; if no, proceed to Step 3.

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3. Check whether the object presents periodic or helical features. If yes, select Channel II; if no, the object is outside the domain of the \mathcal{\Pi} operator.

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4. For field or function distributions: Check whether integral or complex-variable formulations are available. If yes, select Channel III; if no, perform preprocessing and rejudge.

6.3 Handling Overlapping Cases

Certain objects satisfy the conditions of multiple channels simultaneously. It is recommended to choose the most direct channel to avoid unnecessary complexity:

- Circle-to-sphere transformation: Channel I is optimal, Channel III is marginally applicable → Select Channel I

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- Sinusoidal helix: Channel II is applicable, Channel I is not → Select Channel II

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- Gaussian field: Channel III is optimal, Channel II is marginally applicable → Select Channel III

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- Spherical harmonics: Channel I and Channel II are marginally applicable, Channel III is optimal → Select Channel III

General principle: Prioritize the most direct channel.

7. Unified Criteria for Channel Selection

7.1 Three Core Criteria

Geometric Criterion

- Channel I: Equipped with a rotation axis
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- Channel II: Possesses periodic structures
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- Channel III: Described as field distribution

Expression Criterion

- Channel I: π appears as a geometric ratio
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- Channel II: π appears as the sum of infinite series
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- Channel III: π appears as an integral value

Output Criterion

- Channel I: Closed rotating solids
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- Channel II: Open periodic surfaces
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- Channel III: Continuous spatial fields

7.2 Rapid Decision Logic

1. Is the input a bounded closed figure?
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- Yes → Select Channel I
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- No → Proceed to the next question
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2. Does the input repeat periodically along a certain direction?
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- Yes → Select Channel II
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- No → Proceed to the next question
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3. Can the input be described by probability or field functions?
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- Yes → Select Channel III
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- No → The object is outside the domain of the \mathcal{\Pi} operator

 

8. Conclusion and Positioning of Subsequent Papers

8.1 Contributions of This Paper

1. Taxonomic contribution: This paper systematically classifies diverse expressions of π into three functional channels for the first time.
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2. Rule formulation contribution: Rigorous transformation rules and mathematical formulations are established for each channel.
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3. Decision-making contribution: Complete criteria and logical flow for channel selection are provided.
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4. Collaboration contribution: Composite transformation rules between different channels are discussed.

8.2 Position within the 19-Paper Framework

- Paper 1-1 (Definition): Establishes the symbolic foundation of the operator
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- Paper 1-2 (Axiom): Presents the curl-preservation constraint
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- This paper (1-3): Constructs the operational framework of three channels
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- Paper 2-1 ~ 2-4 (Geometric Verification): Verify the validity of Channel I respectively
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- Paper 3-3 & 3-4 (Ramanujan Series): Further develop the theories of Channel II
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- Paper 4-1 ~ 4-3 (Higher Dimension & Field Theory): Extend the application of Channel III

8.3 Preview of Subsequent Papers

- Paper 2-1 (Cylinder): Dual-channel implementation of Channel I (geometric rotation + periodic expansion)
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- Paper 2-2 (Torus): Transformation of double-radius rotating solids via Channel I
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- Paper 2-3 (Helix): Full implementation of Channel II
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- Paper 2-4 (Ellipsoid): Generalization of Channel I considering eccentricity

 

References

(Omitted)

 

Appendix: Quick Reference for the Three Channels

- Channel I (\mathcal{\Pi}^{(I)}): Geometric π; Representative formula \pi = C/2r; Transformation: rotation by 2\pi around an axis; Typical outputs: sphere, cylinder, cone, ellipsoid; Curl preservation is explicitly satisfied.
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- Channel II (\mathcal{\Pi}^{(II)}): Series-type π; Representative formula \displaystyle \frac{\pi}{4} = \sum_{n=0}^{\infty}\frac{(-1)^n}{2n+1}; Transformation: superposition of periodic infinitesimals; Typical outputs: helix, ripple, screw thread; Curl preservation manifests as periodicity preservation.
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- Channel III (\mathcal{\Pi}^{(III)}): Integral & complex-variable π; Representative formula \displaystyle \int_{-\infty}^{\infty} e^{-x^2}dx = \sqrt{\pi}; Transformation: field distribution multiplied by kernel function; Typical outputs: Gaussian field, wave field; Curl preservation manifests as radial preservation.

 

Next Paper: Paper E1-4 Euler's Identity as the Minimal Closed Instance of the \mathcal{\Pi} Operator