# The Tri-Quarter Framework: Unifying Complex Coordinates with Topological and Reflective Duality across Circles of Any Radius

Nathan O. Schmidt

► To cite this version:

Nathan O. Schmidt. The Tri-Quarter Framework: Unifying Complex Coordinates with Topological and Reflective Duality across Circles of Any Radius. 2025. hal-05010951v3

HAL Id: hal-05010951

https://hal.science/hal-05010951v3

Preprint submitted on 14 Jun 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# The Tri-Quarter Framework: Unifying Complex Coordinates with Topological and Reflective Duality across Circles of Any Radius

Nathan O. Schmidt

Cold Hammer Research & Development LLC, Eagle, Idaho, USA

nate.o.schmidt@coldhammer.net

June 13, 2025

## Abstract

In this paper, we introduce the *Tri-Quarter Topological Duality Theorem*, the foundation of a novel mathematical framework that unifies complex, Cartesian, and polar coordinate systems on the complex plane $C$ while equipping the circle $T_r$ of radius $r > 0$ with a new topological property. Our framework integrates a generalized coordinate system—where real and imaginary components are assigned unique phase pairs—with a structured orientation that elevates $T_r$ to an active separator with intrinsic directional properties. We prove that $T_r$, as the boundary zone, exhibits topological duality with the inner zone $X_{-,r}$ ($||\vec{x}|| < r$) and outer zone $X_{+,r}$ ($||\vec{x}|| > r$), ensuring consistent separation between inner and outer radial directions across $T_r$ with a phase pair map encoding additional information. We also introduce the *Escher Tri-Quarter Reflective Duality Theorem*, proving reflective duality across $T_r$ via a circle inversion map that preserves phase pairs while swapping $X_{-,r}$ and $X_{+,r}$. Moreover, these phase pair assignments provide a combinatorial classification of directional structures in $C$, enhancing the topological analysis. This approach offers insights into topological separation, orientation, and reflection, facilitating analysis of systems with circular symmetry, with potential applications in fields such as black hole physics, signal processing, and other areas reliant on complex domain partitioning. A case study on quadrant-based transformations demonstrates streamlined directional mappings, geometric elegance, unified classification, and computational efficiency in $C$. For instance, in the case study, the Tri-Quarter approach consistently reduces the number of comparison conditional checks from a maximum of 7 in the standard method to a maximum of 4, significantly enhancing computational efficiency. A software tool visualizes some of these concepts, with future work aimed at exploring practical implementations.

## 1 Introduction

The complex plane $C$, viewed as a complex 1D Riemann surface, serves as a foundational structure in mathematics and its applications, bridging algebraic, geometric, and analytic perspectives [1, 2]. In some contexts, traditional coordinate systems, such as Cartesian and polar, while powerful, treat the circle of radius $r > 0$ as a mere boundary without fully exploiting its topological role. In this work, we address this gap by introducing the Tri-Quarter Topological Duality Theorem, which establishes a generalized complex-Cartesian-polar coordinate system while applying a trichotomy—a three zone partition of $C$ based on the circle of radius $r > 0$—to the four quadrants of $C$ to achieve a new topological property.

This paper includes two main results. Our first main result demonstrates that the circle $T_r$ of radius $r > 0$ is topologically dual to both the inner zone $X_{-,r}$ ($||\vec{x}|| < r$) and outer zone $X_{+,r}$ ($||\vec{x}|| > r$), with a structured orientation that ensures consistent directional separation across $C$—more specifically, between the inner radial direction (approaching $T_r$ from $X_{-,r}$) and the outer radial direction (approaching $T_r$ from $X_{+,r}$), which is combined with a novel phase pair assignment based on well-defined quadrant and axis boundary rules. This framework unifies three coordinate representations and elevates the circle's role from a passive boundary to an active separator with intrinsic properties. By assigning discrete phase pairs to the real and imaginary components, our
framework establishes a combinatorial encoding of geometric directions, unifying algebraic and geometric perspectives. Our second main result leverages a circle inversion map [3] to demonstrate reflective duality across $T_r$ to preserve the assigned phase pairs while swapping $X_{-,r}$ and $X_{+,r}$—the Escher Tri-Quarter Reflective Duality Theorem—inspired by the artistic symmetry of M.C. Escher [4].

The significance lies in its potential to refine analyses, computations, modeling, and design in fields that rely on complex numbers and complex domain spatial partitioning, such as offering a nuanced approach to handling precise boundary conditions, directional properties, and reflective properties across $\mathbb{C}$. For instance, potential applications may include deciphering the quasi-normal modes of black holes and enhancing the computational efficiency of signal processors. To demonstrate its practical utility, a case study in Section 7 applies the framework to quadrant-based transformations, revealing its capacity to unify classification and leverage geometric structure for efficient and consistent directional analysis across $\mathbb{C}$. Specifically, the framework reduces the computational complexity by requiring only up to 4 comparison conditional checks compared to the standard approach's up to 7, making it particularly advantageous for applications in computer science and signal processing where efficiency is critical.

In Section 2, we define the coordinate system, followed by the structured orientation in Section 3, the topological zones in Section 4, the topological duality proof in Section 5, the reflective duality proof in Section 6, and the aforementioned case study in Section 7. In Section 8, we conclude with a discussion of its strengths, software animation tool, and potential future applications.

## 2 Generalized Complex-Cartesian-Polar Coordinate System

Let $\mathbb{C}$ be the complex plane and complex 1D Riemann surface [1, 2] with the standard topology. For a complex number $x = x_R + x_I i \in \mathbb{C}$, we define the complex vector $\vec{x} = \vec{x}_R + \vec{x}_I \in \mathbb{C}$ as a position-vector on $\mathbb{C}$ with Cartesian coordinate form

$$ (x_R, x_I)_{\text{Cartesian}} = (x_R, x_I)_C, \qquad (1) $$

where $C$ denotes Cartesian form. Vectors are denoted with arrows (e.g., $\vec{x}$) throughout this paper for consistency. Here $\vec{x}_R$ and $\vec{x}_I$ are axis-aligned orthogonal components that we treat interchangeably as both vectors and points in specific subsets of $\mathbb{C}$. Specifically, we define

$$ \begin{aligned} \vec{x}_R &\in \mathbb{R} \times \{0\} = \{(x_R, 0)_C \mid x_R \in \mathbb{R}\} \\ \vec{x}_I &\in \{0\} \times \mathbb{I} = \{(0, x_I)_C \mid x_I \in \mathbb{R}\}, \end{aligned} \qquad (2) $$

where $\mathbb{I} = i\mathbb{R}$ denotes the imaginary axis. Therefore, $\vec{x}_R = (x_R, 0)_C$ is a “real Euclidean 2-vector” aligned with the real axis, and $\vec{x}_I = (0, x_I)_C$ is an “imaginary Euclidean 2-vector” aligned with the imaginary axis. Throughout this framework, this dual interpretation allows $\vec{x}_R$ and $\vec{x}_I$ to serve as both individual vectors and representative points within their respective sets, enabling their use interchangeably with the complex number $x$ and its vector representation $\vec{x}$, which they form as a linear combination (emphasizing that $\vec{x} = \vec{x}_R + \vec{x}_I$ is the direct sum of these vectors). As we will demonstrate in this paper, by subtly defining the components $\vec{x}_R$ and $\vec{x}_I$ to be orthogonal axis-aligned vectors—instead of just scalars—we achieve some useful properties such as orthogonality and direction.

Throughout this paper, we consider $X = \mathbb{C} \setminus \{(0, 0)_C\}$ to exclude the origin where phase assignments are undefined. This focuses our framework on zones that are critical to the main results.

Next, $\vec{x} \in X$ is a Euclidean 2-vector with norm $||\vec{x}|| \in [0, \infty)$ and phase $\arg(\vec{x}) = \langle \vec{x} \rangle \in [0, 2\pi)$, which are respectively analogous to magnitude and direction in conventional notation with polar coordinate form

$$ (||\vec{x}||, \langle \vec{x} \rangle)_{\text{Polar}} = (||\vec{x}||, \langle \vec{x} \rangle)_P, \qquad (3) $$
where $P$ denotes polar form, to give

$$ 
\begin{aligned}
x_{\mathbb{R}} &= ||\vec{x}|| \cos\langle\vec{x}\rangle \\
x_{\mathbb{I}} &= ||\vec{x}|| \sin\langle\vec{x}\rangle
\end{aligned} 
\qquad (4) 
$$

with Pythagorean form

$$ ||\vec{x}||^2 = x_{\mathbb{R}}^2 + x_{\mathbb{I}}^2. \qquad (5) $$

(Note: From this point forward, for conciseness, for the phase angle notation of a given $\vec{x} \in X$, we will just say $\langle\vec{x}\rangle$ instead of $\arg\vec{x}$ also.) Thus, similarly to $\vec{x}$, for the axis-aligned orthogonal vectors $\vec{x}_{\mathbb{R}}$ and $\vec{x}_{\mathbb{I}}$, we also obtain the polar coordinate form

$$ 
\begin{aligned}
(\|\vec{x}_{\mathbb{R}}\|, \langle\vec{x}_{\mathbb{R}}\rangle)_P \\
(\|\vec{x}_{\mathbb{I}}\|, \langle\vec{x}_{\mathbb{I}}\rangle)_P,
\end{aligned} 
\qquad (6) 
$$

such that $\|\vec{x}_{\mathbb{R}}\|, \|\vec{x}_{\mathbb{I}}\| \in [0, \infty)$, $\langle\vec{x}_{\mathbb{R}}\rangle \in \{0, \pi\}$, and $\langle\vec{x}_{\mathbb{I}}\rangle \in \{\frac{\pi}{2}, \frac{3\pi}{2}\}$. Henceforth, the Pythagorean form of Equation (5) becomes

$$ ||\vec{x}||^2 = ||\vec{x}_{\mathbb{R}}||^2 + ||\vec{x}_{\mathbb{I}}||^2 = x_{\mathbb{R}}^2 + x_{\mathbb{I}}^2. \qquad (7) $$

Thus, for all $\vec{x} \in X$ we define the 2D generalized complex-Cartesian-polar coordinate system as

$$ \vec{x} = \vec{x}_{\mathbb{R}} + \vec{x}_{\mathbb{I}} = (x_{\mathbb{R}}, x_{\mathbb{I}})_C = (\|\vec{x}\| \cos\langle\vec{x}\rangle, \|\vec{x}\| \sin\langle\vec{x}\rangle)_C = (\|\vec{x}\|, \langle\vec{x}\rangle)_P \qquad (8) $$

unifying complex, Cartesian, and polar representations. This unification is further enriched by the phase pair assignments and structured orientation defined in Section 3, which provide directional properties that are critical to our topological analysis.

## 3 Structured Orientation

Building on the foundation of the unified coordinate representation in Equation (8), for any $\vec{x} \in X$, we now assign phase pairs to establish a structured orientation across $X$. We define the *quadrant phase pair assignments* as

$$ 
\begin{aligned}
\text{I:} \quad & \langle\vec{x}\rangle \in (0, \frac{\pi}{2}) \iff (x_{\mathbb{R}} > 0) \land (x_{\mathbb{I}} > 0) \iff (\langle\vec{x}_{\mathbb{R}}\rangle = 0) \land (\langle\vec{x}_{\mathbb{I}}\rangle = \frac{\pi}{2}) \iff (0, \frac{\pi}{2})_{\phi} \\
\text{II:} \quad & \langle\vec{x}\rangle \in (\frac{\pi}{2}, \pi) \iff (x_{\mathbb{R}} < 0) \land (x_{\mathbb{I}} > 0) \iff (\langle\vec{x}_{\mathbb{R}}\rangle = \pi) \land (\langle\vec{x}_{\mathbb{I}}\rangle = \frac{\pi}{2}) \iff (\pi, \frac{\pi}{2})_{\phi} \\
\text{III:} \quad & \langle\vec{x}\rangle \in (\frac{3\pi}{2}, \pi) \iff (x_{\mathbb{R}} < 0) \land (x_{\mathbb{I}} < 0) \iff (\langle\vec{x}_{\mathbb{R}}\rangle = \pi) \land (\langle\vec{x}_{\mathbb{I}}\rangle = \frac{3\pi}{2}) \iff (\pi, \frac{3\pi}{2})_{\phi} \\
\text{IV:} \quad & \langle\vec{x}\rangle \in (\frac{3\pi}{2}, 2\pi) \iff (x_{\mathbb{R}} > 0) \land (x_{\mathbb{I}} < 0) \iff (\langle\vec{x}_{\mathbb{R}}\rangle = 0) \land (\langle\vec{x}_{\mathbb{I}}\rangle = \frac{3\pi}{2}) \iff (0, \frac{3\pi}{2})_{\phi},
\end{aligned} 
\qquad (9) 
$$

where $\phi(\vec{x}) = (\langle\vec{x}_{\mathbb{R}}\rangle, \langle\vec{x}_{\mathbb{I}}\rangle)_{\phi}$ denotes the phase pair assigned by the coordinate system. Then to maintain continuity and ensure that no boundary remains undefined, we define the *axis boundary phase pair assignments* as

$$ 
\begin{aligned}
\text{“East”:} \quad & \langle\vec{x}\rangle = 0 \iff (x_{\mathbb{R}} > 0) \land (x_{\mathbb{I}} = 0) \iff (\langle\vec{x}_{\mathbb{R}}\rangle = 0) \land (\langle\vec{x}_{\mathbb{I}}\rangle = 0) \iff (0, 0)_{\phi} \\
\text{“North”:} \quad & \langle\vec{x}\rangle = \frac{\pi}{2} \iff (x_{\mathbb{R}} = 0) \land (x_{\mathbb{I}} > 0) \iff (\langle\vec{x}_{\mathbb{R}}\rangle = \frac{\pi}{2}) \land (\langle\vec{x}_{\mathbb{I}}\rangle = \frac{\pi}{2}) \iff (\frac{\pi}{2}, \frac{\pi}{2})_{\phi} \\
\text{“West”:} \quad & \langle\vec{x}\rangle = \pi \iff (x_{\mathbb{R}} < 0) \land (x_{\mathbb{I}} = 0) \iff (\langle\vec{x}_{\mathbb{R}}\rangle = \pi) \land (\langle\vec{x}_{\mathbb{I}}\rangle = 0) \iff (\pi, 0)_{\phi} \\
\text{“South”:} \quad & \langle\vec{x}\rangle = \frac{3\pi}{2} \iff (x_{\mathbb{R}} = 0) \land (x_{\mathbb{I}} < 0) \iff (\langle\vec{x}_{\mathbb{R}}\rangle = \frac{\pi}{2}) \land (\langle\vec{x}_{\mathbb{I}}\rangle = \frac{3\pi}{2}) \iff (\frac{\pi}{2}, \frac{3\pi}{2})_{\phi},
\end{aligned} 
\qquad (10) 
$$

where each of these assigned phase pairs uniquely corresponds to each axis direction, aligning with $\langle\vec{x}\rangle$ and maintaining a structured orientation. To establish a consistent rule for handling the edge cases when $x_{\mathbb{R}} = 0$ or $x_{\mathbb{I}} = 0$, we can simply set $\langle\vec{x}_{\mathbb{R}}\rangle = \frac{\pi}{2}$ or $\langle\vec{x}_{\mathbb{I}}\rangle = 0$, respectively, to ensure that, for a given component, each of the three possible assignments is unique. Thus, from Equations (9–10) we derive the following *phase assignment rules*:
• For the real component $\vec{x}_{\mathbb{R}}$:

◦ If $x_{\mathbb{R}} > 0$, then set $\langle \vec{x}_{\mathbb{R}} \rangle = 0$.

◦ If $x_{\mathbb{R}} = 0$, then set $\langle \vec{x}_{\mathbb{R}} \rangle = \frac{\pi}{2}$.

◦ If $x_{\mathbb{R}} < 0$, then set $\langle \vec{x}_{\mathbb{R}} \rangle = \pi$.

• For the imaginary component $\vec{x}_{\mathbb{I}}$:

◦ If $x_{\mathbb{I}} > 0$, then set $\langle \vec{x}_{\mathbb{I}} \rangle = \frac{\pi}{2}$.

◦ If $x_{\mathbb{I}} = 0$, then set $\langle \vec{x}_{\mathbb{I}} \rangle = 0$.

◦ If $x_{\mathbb{I}} < 0$, then set $\langle \vec{x}_{\mathbb{I}} \rangle = \frac{3\pi}{2}$.

The phase assignment rules serve as “building blocks” for the phase pair assignments. They maintain consistency across all vectors—including the axis-bound vectors—and indicate the absence of direction in a zero-magnitude component.

In standard complex analysis, the phase (argument) of the zero vector is undefined because it has no direction. However, in this framework, we adopt the convention such that when a component is zero (e.g., $x_{\mathbb{R}} = 0$ or $x_{\mathbb{I}} = 0$), then its corresponding phase is set (e.g., $\langle \vec{x}_{\mathbb{R}} \rangle = \frac{\pi}{2}$ or $\langle \vec{x}_{\mathbb{I}} \rangle = 0$) to ultimately ensure that, for each quadrant or axis, the assigned phase pair is unique. This choice simplifies the phase pair assignments, especially on the real and imaginary axes, and ensures that for all $\vec{x} \in X$ there exists a well-defined phase pair $\phi(\vec{x}) = (\langle \vec{x}_{\mathbb{R}} \rangle, \langle \vec{x}_{\mathbb{I}} \rangle)_{\phi}$. This convention is consistently applied throughout the paper and does not affect the main results because the origin is excluded from our analysis.

These phase pair assignments establish a structured orientation that not only unifies the complex-Cartesian-polar coordinate representations but also upgrades $X$ with consistent directional properties. Furthermore, in the context of algebraic combinatorics, these phase pair assignments offer a novel combinatorial classification of directional structures in $X$. By mapping each vector $\vec{x} \in X$ to a pair $(\langle \vec{x}_{\mathbb{R}} \rangle, \langle \vec{x}_{\mathbb{I}} \rangle)_{\phi}$ based on the signs of its $x_{\mathbb{R}}$ and $x_{\mathbb{I}}$ components, we effectively partition $X$ into a finite set of directional categories. This discrete encoding not only unifies geometric and algebraic perspectives but also invites further exploration of symmetry and transformation properties through the lens of combinatorial algebra. This system forms the foundation for the topological partitioning into three distinct zones in Section 4, where the circle’s role as a separator is empowered by these orientations.

# 4 Topological Zones

For any $r > 0$ and $\vec{x} \in X$, we use trichotomy of the norm $||\vec{x}||$ to partition $X$ into three distinct topological zones:

(1) the inner zone $X_{-,r}$,

(2) the boundary zone $T_r$, and

(3) the outer zone $X_{+,r}$.

Thus, for $\vec{x} \in X$ we know that precisely one of the following trichotomy conditions must be satisfied

$$ 
\begin{aligned}
& ||\vec{x}|| < r \iff \vec{x} \in X_{-,r} \\
& ||\vec{x}|| = r \iff \vec{x} \in T_r \\
& ||\vec{x}|| > r \iff \vec{x} \in X_{+,r},
\end{aligned} 
\quad (11) $$

where clearly $X_{-,r} \cap T_r = T_r \cap X_{+,r} = X_{-,r} \cap X_{+,r} = \emptyset$ and $X_{-,r} \cup T_r \cup X_{+,r} = X$, such that

$$ 
\begin{aligned}
& X_{-,r} = \{\vec{x} \in X : ||\vec{x}|| < r\} \\
& T_r = \{\vec{x} \in X : ||\vec{x}|| = r\} \\
& X_{+,r} = \{\vec{x} \in X : ||\vec{x}|| > r\}
\end{aligned} 
\quad (12) $$
because the boundary zone $T_r$ is a circle of radius $r$ and 1-manifold, which serves as the separator between $X_{-,r}$ and $X_{+,r}$. So the four quadrants of $X$ are partitioned into three zones via norm $||\vec{x}||$ trichotomy—a “tri-quarter” if you will.

**Definition 4.1 (Tri-Quarter Topological Duality).** Let $X = \mathbb{C} \setminus \{(0,0)_C\}$ be equipped with the standard topology. For any $r > 0$, define $X_{-,r} = \{\vec{x} \in X : ||\vec{x}|| < r\}$, $T_r = \{\vec{x} \in X : ||\vec{x}|| = r\}$, and $X_{+,r} = \{\vec{x} \in X : ||\vec{x}|| > r\}$, where $X_{-,r} \cup T_r \cup X_{+,r} = X$ and $X_{-,r} \cap X_{+,r} = \emptyset$. The circular boundary zone $T_r$ with radius $r$ is topologically dual to the inner zone $X_{-,r}$ and outer zone $X_{+,r}$ if:

(1) $T_r = \partial X_{-,r}$ and $T_r = \partial \bar{X}_{+,r}$, where $\bar{X}_{+,r}$ denotes the closure of $X_{+,r}$ in $X$,

(2) $T_r$, as a 1-manifold, has a structured orientation defined by a map $\phi: T_r \to \{0, \frac{\pi}{2}, \pi\} \times \{0, \frac{\pi}{2}, \frac{3\pi}{2}\}$, assigning phase pairs $(\langle \vec{x}_{\mathbb{R}} \rangle, \langle \vec{x}_{\mathbb{I}} \rangle)_{\phi}$ based on quadrant and axis boundary phase assignments per Equations (9-10), and

(3) the structured orientation $\phi$ is constant along each ray that emanates from the origin $(0,0)_C$ in $\mathbb{C}$ across $X$, enabling the norm $||\vec{x}||$ to consistently separate the inner radial direction (approaching $T_r$ from $X_{-,r}$ as $||\vec{x}|| \to r^-$) from the outer radial direction (approaching $T_r$ from $X_{+,r}$ as $||\vec{x}|| \to r^+$).

For the upcoming theorem and proof in Section 5, this duality captures $T_r$'s role as a separator with an intrinsic directional structure to unify the topological and geometric properties of the partition.

# 5 Tri-Quarter Topological Duality

In this section, we prove the first of the two main results of this paper. The topological duality of $T_r$ with $X_{-,r}$ and $X_{+,r}$ not only formalizes the $T_r$'s role as a separator but also integrates the structured orientation to provide a unified framework for analyzing radial directions in the complex plane. This duality lays the groundwork for exploring additional symmetries, such as the reflective duality addressed in the subsequent section.

**Theorem 5.1 (Tri-Quarter Topological Duality).** Let $X = \mathbb{C} \setminus \{(0,0)_C\}$ be equipped with the standard topology and the generalized complex-Cartesian-polar coordinate system of Equation (8), where $\vec{x} = \vec{x}_{\mathbb{R}} + \vec{x}_{\mathbb{I}}$ for $\vec{x} \in X$, with $\vec{x}_{\mathbb{R}} = (x_{\mathbb{R}}, 0)_C \in \mathbb{R} \times \{0\}$, $\vec{x}_{\mathbb{I}} = (0, x_{\mathbb{I}})_C \in \{0\} \times \mathbb{I}$, and phase pairs $(\langle \vec{x}_{\mathbb{R}} \rangle, \langle \vec{x}_{\mathbb{I}} \rangle)_{\phi}$ assigned as:

* if $x_{\mathbb{R}} > 0$ then $\langle \vec{x}_{\mathbb{R}} \rangle = 0$, if $x_{\mathbb{R}} = 0$ then $\langle \vec{x}_{\mathbb{R}} \rangle = \frac{\pi}{2}$, if $x_{\mathbb{R}} < 0$ then $\langle \vec{x}_{\mathbb{R}} \rangle = \pi$,

* if $x_{\mathbb{I}} > 0$ then $\langle \vec{x}_{\mathbb{I}} \rangle = \frac{\pi}{2}$, if $x_{\mathbb{I}} = 0$ then $\langle \vec{x}_{\mathbb{I}} \rangle = 0$, if $x_{\mathbb{I}} < 0$ then $\langle \vec{x}_{\mathbb{I}} \rangle = \frac{3\pi}{2}$,

per the phase assignment rules following Equations (9-10). For any $r > 0$, define $X_{-,r} = \{\vec{x} \in X : ||\vec{x}|| < r\}$, $T_r = \{\vec{x} \in X : ||\vec{x}|| = r\}$, $X_{+,r} = \{\vec{x} \in X : ||\vec{x}|| > r\}$. Then $T_r$ is topologically dual to $X_{-,r}$ and $X_{+,r}$, where the structured orientation ensures consistent separation between the inner radial direction (approaching $T_r$ from $X_{-,r}$) and the outer radial direction (approaching $T_r$ from $X_{+,r}$) across $X$ with respect to $T_r$.

**Proof.** Let $X = \mathbb{C} \setminus \{(0,0)_C\}$ be equipped with the standard topology, and for any $r > 0$, define $X_{-,r}$, $T_r$, $X_{+,r}$ as above. We proceed in steps to verify each condition of Definition 4.1:
• Step 1: Verify Partition and Boundaries on X

From the Jordan Curve Theorem [5, 6] we obtain the boundary zone $T_r = \{\vec{x} \in X : \|\vec{x}\| = r\}$
(a simple closed curve) that separates X into two disjoint open sets: the inner zone $X_{-,r}$
and the outer zone $X_{+,r}$, where $X \setminus T_r = X_{-,r} \cup X_{+,r}$ and $X_{-,r} \cap X_{+,r} = \emptyset$, such that:

◦ $X_{-,r}$ is open, so $\partial X_{-,r} = \{\vec{x} \in X : \|\vec{x}\| = r\} = T_r$ (every neighborhood of $\vec{x} \in T_r$ intersects $X_{-,r}$ and $X_{+,r}$), and

◦ $X_{+,r}$ is open, with $\bar{X}_{+,r} = \{\vec{x} \in X : \|\vec{x}\| \ge r\}$, so $\partial \bar{X}_{+,r} = T_r$ (similar reasoning).

This satisfies condition (1) of topological duality. ☑

• Step 2: Verify Structured Orientation on $T_r$

We define $\phi : T_r \to \{0, \frac{\pi}{2}, \pi\} \times \{0, \frac{\pi}{2}, \frac{3\pi}{2}\}$ by $\phi(\vec{x}) = (\langle \vec{x}_R \rangle, \langle \vec{x}_I \rangle)_\phi$, where for $\vec{x} = (x_R, x_I)_C \in T_r$ with $\|\vec{x}\| = r$:

◦ if $x_R > 0$ then $\langle \vec{x}_R \rangle = 0$, if $x_R = 0$ then $\langle \vec{x}_R \rangle = \frac{\pi}{2}$, if $x_R < 0$ then $\langle \vec{x}_R \rangle = \pi$,

◦ if $x_I > 0$ then $\langle \vec{x}_I \rangle = \frac{\pi}{2}$, if $x_I = 0$ then $\langle \vec{x}_I \rangle = 0$, if $x_I < 0$ then $\langle \vec{x}_I \rangle = \frac{3\pi}{2}$.

Since $T_r$ is a 1-manifold, we parametrize it as $\vec{x} = (\|\vec{x}\| \cos\langle\vec{x}\rangle, \|\vec{x}\| \sin\langle\vec{x}\rangle)_C = (r \cos\langle\vec{x}\rangle, r \sin\langle\vec{x}\rangle)_C$, with $\langle\vec{x}\rangle \in [0, 2\pi)$. Then we check the structured orientation as $\langle\vec{x}\rangle \to 2\pi^{-}$:

◦ “East” positive real axis boundary at $\langle \vec{x} \rangle = 0$: $\vec{x} = (r, 0)_C$, $x_R > 0$, $x_I = 0$, so $\phi(\vec{x}) = (0, 0)_\phi$.

◦ Quadrant I ($\langle \vec{x} \rangle \in (0, \frac{\pi}{2})$): $x_R > 0, x_I > 0$, so $\phi(\vec{x}) = (0, \frac{\pi}{2})_\phi$.

◦ “North” positive imaginary axis boundary at $\langle \vec{x} \rangle = \frac{\pi}{2}$: $\vec{x} = (0, r)_C$, $x_R = 0$, $x_I > 0$, so $\phi(\vec{x}) = (\frac{\pi}{2}, \frac{\pi}{2})_\phi$.

◦ Quadrant II ($\langle \vec{x} \rangle \in (\frac{\pi}{2}, \pi)$): $x_R < 0, x_I > 0$, so $\phi(\vec{x}) = (\pi, \frac{\pi}{2})_\phi$.

◦ “West” negative real axis boundary at $\langle \vec{x} \rangle = \pi$: $\vec{x} = (-r, 0)_C$, $x_R < 0$, $x_I = 0$, so $\phi(\vec{x}) = (\pi, 0)_\phi$.

◦ Quadrant III ($\langle \vec{x} \rangle \in (\pi, \frac{3\pi}{2})$): $x_R < 0, x_I < 0$, so $\phi(\vec{x}) = (\pi, \frac{3\pi}{2})_\phi$.

◦ “South” negative imaginary axis boundary at $\langle \vec{x} \rangle = \frac{3\pi}{2}$: $\vec{x} = (0, -r)_C$, $x_R = 0$, $x_I < 0$, so $\phi(\vec{x}) = (\frac{\pi}{2}, \frac{3\pi}{2})_\phi$.

◦ Quadrant IV ($\langle \vec{x} \rangle \in (\frac{3\pi}{2}, 2\pi)$): $x_R > 0, x_I < 0$, so $\phi(\vec{x}) = (0, \frac{3\pi}{2})_\phi$.

Thus, $\phi$ is constant on each open arc of $T_r$ and adjusts at axis boundaries (e.g., $\langle \vec{x} \rangle = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}$). So it is piecewise continuous, which suffices for a structured orientation on a 1-manifold. This satisfies condition (2) of topological duality. ☑

• Step 3: Verify Inner and Outer Radial Directional Separation across $T_r$

For $\vec{x} \in T_r$, fix $\langle \vec{x} \rangle \in [0, 2\pi)$. We define two paths approaching $T_r$:

◦ Inner radial path from $X_{-,r}$: $\vec{x} = (\|\vec{x}\| \cos\langle\vec{x}\rangle, \|\vec{x}\| \sin\langle\vec{x}\rangle)_C, \|\vec{x}\| \to r^-$, so $\|\vec{x}\| < r$.

◦ Outer radial path from $X_{+,r}$: $\vec{x} = (\|\vec{x}\| \cos\langle\vec{x}\rangle, \|\vec{x}\| \sin\langle\vec{x}\rangle)_C, \|\vec{x}\| \to r^+$, so $\|\vec{x}\| > r$.

Then we compute $\phi$ along both paths:

◦ For $\langle \vec{x} \rangle \in (0, \frac{\pi}{2})$, we obtain $\vec{x} = (\|\vec{x}\| \cos\langle \vec{x} \rangle, \|\vec{x}\| \sin\langle \vec{x} \rangle)_C$, $x_R > 0$, $x_I > 0$, $\phi = (0, \frac{\pi}{2})_\phi$; as $\|\vec{x}\| \to r^-$, $\vec{x} \to (r \cos\langle \vec{x} \rangle, r \sin\langle \vec{x} \rangle)_C \in T_r$, where $\phi = (0, \frac{\pi}{2})_\phi$ remains constant.
◦ Similarly, as $||\vec{x}|| \to r^+$, $\phi = (0, \frac{\pi}{2})_\phi$.

The phase pair $(\langle \vec{x}_R \rangle, \langle \vec{x}_I \rangle)_\phi$ is constant, where its direction is distinguished by $||\vec{x}||$:

◦ a “contractive” ($< r$) inner radial direction from $X_{-,r}$, and

◦ an “expansive” ($> r$) outer radial direction from $X_{+,r}$.

This holds for all $\langle \vec{x} \rangle \in [0, 2\pi)$ as it adjusts at axis boundaries (e.g., if $\langle \vec{x} \rangle = 0$ then $\phi = (0, 0)_\phi$). This structured orientation ensures consistent separation between the inner radial direction (approaching $T_r$ from $X_{-,r}$ as $||\vec{x}|| \to r^-$) and the outer radial direction (approaching $T_r$ from $X_{+,r}$ as $||\vec{x}|| \to r^+$) across $X$ with respect to $T_r$. This satisfies condition (3) of topological duality. ☐

Thus, $T_r$ is topologically dual to $X_{-,r}$ and $X_{+,r}$.

<figure><img src="image_2.png" /></figure>

Now that we've established the topological duality of the circular boundary zone $T_r$ with respect to the inner and outer zones, we now turn to a complementary symmetry that leverages this structured orientation to achieve a reflective duality across $T_r$ for swapping between the inner and outer zones.

# 6 Escher Tri-Quarter Reflective Duality

In this section, we prove the second of the two main results of this paper. Building upon the structured orientation and topological duality established with the Tri-Quarter Topological Duality Theorem in the previous section, we now establish a reflective duality across the circle $T_r$ of radius $r > 0$ in the complex plane $X = \mathbb{C} \setminus \{(0, 0)_C\}$, inspired by M.C. Escher's artistic symmetry [4]. This duality is achieved through a circle inversion map [3], which preserves the structured orientation defined by the phase pair map $\phi$ while swapping the inner zone $X_{-,r}$ and outer zone $X_{+,r}$.

**Definition 6.1 (Circle Inversion Map).** For any $r > 0$, the circle inversion map $\iota_r : X \to X$ is defined as

$$ \iota_r(\vec{x}) = \frac{r^2 \vec{x}}{||\vec{x}||^2}, $$

where $\vec{x} \in X$ and $||\vec{x}||$ is the Euclidean norm [3].

**Definition 6.2 (Escher Tri-Quarter Reflective Duality).** Let $X = \mathbb{C} \setminus \{(0, 0)_C\}$ be equipped with the standard topology. For any $r > 0$, define $X_{-,r} = \{\vec{x} \in X : ||\vec{x}|| < r\}$, $T_r = \{\vec{x} \in X : ||\vec{x}|| = r\}$, and $X_{+,r} = \{\vec{x} \in X : ||\vec{x}|| > r\}$, where $X_{-,r} \cup T_r \cup X_{+,r} = X$ and $X_{-,r} \cap X_{+,r} = \emptyset$. Let the circular boundary zone $T_r$ with radius $r$ be topologically dual to both the inner zone $X_{-,r}$ and outer zone $X_{+,r}$ as in Theorem 5.1. Then $T_r$ exhibits reflective duality between $X_{-,r}$ and $X_{+,r}$ if there exists a map $\iota_r$ such that:

(1) $\iota_r(\vec{x}) = \vec{x}$ for all $\vec{x} \in T_r$,

(2) $\iota_r(X_{-,r}) = X_{+,r}$ and $\iota_r(X_{+,r}) = X_{-,r}$,

(3) $\phi(\iota_r(\vec{x})) = \phi(\vec{x})$ for all $\vec{x} \in X$, and

(4) $\iota_r$ is an involution, i.e., $\iota_r^{-1} = \iota_r$.
**Theorem 6.3 (Escher Tri-Quarter Reflective Duality).** Let $X = \mathbb{C} \setminus \{(0,0)_C\}$ be equipped with the standard topology. For any $r > 0$, define $X_{-,r} = \{\vec{x} \in X : ||\vec{x}|| < r\}$, $T_r = \{\vec{x} \in X : ||\vec{x}|| = r\}$, and $X_{+,r} = \{\vec{x} \in X : ||\vec{x}|| > r\}$, where $X_{-,r} \cup T_r \cup X_{+,r} = X$ and $X_{-,r} \cap X_{+,r} = \emptyset$. Let $T_r$ be topologically dual to $X_{-,r}$ and $X_{+,r}$ as in Theorem 5.1. Then the circle inversion map $\iota_r$ establishes reflective duality across $T_r$ between $X_{-,r}$ and $X_{+,r}$ that satisfies the conditions:

$$ (1) \quad \iota_r(\vec{x}) = \vec{x} \text{ for all } \vec{x} \in T_r, $$

$$ (2) \quad \iota_r(X_{-,r}) = X_{+,r} \quad \text{and} \quad \iota_r(X_{+,r}) = X_{-,r}, $$

$$ (3) \quad \phi(\iota_r(\vec{x})) = \phi(\vec{x}) \text{ for all } \vec{x} \in X, \text{ and} $$

$$ (4) \quad \iota_r^{-1} = \iota_r. $$

**Proof.** Let $X = \mathbb{C} \setminus \{(0,0)_C\}$ be equipped with the standard topology, and for any $r > 0$, define $X_{-,r}$, $T_r$, $X_{+,r}$ to satisfy Theorem 5.1 as above. Let $\iota_r : X \to X$ be given by Definition 6.1. We proceed in steps to verify each condition of Definition 6.2:

**(1) Verify Fixed Points on $T_r$**

For $\vec{x} \in T_r$, we have $||\vec{x}|| = r$. Thus,

$$ \iota_r(\vec{x}) = \frac{r^2 \vec{x}}{||\vec{x}||^2} = \frac{r^2 \vec{x}}{r^2} = \vec{x}. \qquad (13) $$

This satisfies condition (1) of reflective duality. ☑

**(2) Verify Zone Swapping between $X_{-,r}$ and $X_{+,r}$**

* For $\vec{x} \in X_{-,r}$, $||\vec{x}|| < r$. Then

$$ ||\iota_r(\vec{x})|| = \left\| \frac{r^2 \vec{x}}{||\vec{x}||^2} \right\| = \frac{r^2}{||\vec{x}||} > r \implies \iota_r(\vec{x}) \in X_{+,r}. \qquad (14) $$

* For $\vec{x} \in X_{+,r}$, $||\vec{x}|| > r$. Then

$$ ||\iota_r(\vec{x})|| = \left\| \frac{r^2 \vec{x}}{||\vec{x}||^2} \right\| = \frac{r^2}{||\vec{x}||} < r \implies \iota_r(\vec{x}) \in X_{-,r}. \qquad (15) $$

This satisfies condition (2) of reflective duality. ☑

**(3) Verify Phase Pair Preservation**

We define $\phi : T_r \to \{0, \frac{\pi}{2}, \pi\} \times \{0, \frac{\pi}{2}, \frac{3\pi}{2}\}$ by $\phi(\vec{x}) = (\langle\vec{x}_R\rangle, \langle\vec{x}_I\rangle)_\phi$, which depends on the signs of the $x_R$ and $x_I$ components. Then

$$ \left( \iota_r(\vec{x}) = \frac{r^2}{||\vec{x}||^2} \vec{x} \right) \wedge (r > 0) \wedge (||\vec{x}||^2 > 0) \implies \frac{r^2}{||\vec{x}||^2} > 0. \qquad (16) $$

Since $\frac{r^2}{||\vec{x}||^2} > 0$, the transformation $\iota_r(\vec{x}) = \frac{r^2}{||\vec{x}||^2} \vec{x}$ scales $\vec{x}$ without changing the signs of $x_R$ and $x_I$, ensuring that $\phi(\iota_r(\vec{x})) = \phi(\vec{x})$. This satisfies condition (3) of reflective duality. ☑
(4) Verify Involution Property

Compute $\iota_r(\iota_r(\vec{x}))$:

$$ \iota_r(\iota_r(\vec{x})) = \iota_r \left( \frac{r^2 \vec{x}}{\|\vec{x}\|^2} \right) = \frac{r^2 \cdot \frac{r^2 \vec{x}}{\|\vec{x}\|^2}}{\left\| \frac{r^2 \vec{x}}{\|\vec{x}\|^2} \right\|^2} = \frac{\frac{r^4 \vec{x}}{\|\vec{x}\|^2}}{\frac{r^4}{\|\vec{x}\|^2}} = \vec{x}. $$

Thus, $\iota_r^2 = \text{id}$, so $\iota_r^{-1} = \iota_r$. This satisfies condition (4) of reflective duality. ☑

Therefore, $\iota_r$ establishes reflective duality across $T_r$. ☐

This reflective duality not only preserves the phase pair orientation but also swaps the inner and outer zones, offering a new perspective on symmetry in the complex plane. The circle inversion map $\iota_r$ acts as an involution that preserves the structured orientation of $\phi$ across the circular boundary zone $T_r$ and provides a mathematical realization of the reflective symmetry found in Escher's artwork, where the complex plane is “reflected”.

# 7 Case Study: Quadrant-Based Transformations in the Complex Plane

In this case study, we explore the application of quadrant-based transformations to vectors in the complex plane $X = \mathbb{C} \setminus \{(0,0)_C\}$, emphasizing the theoretical and computational advantages of the Tri-Quarter framework over traditional methods. This investigation leverages the generalized complex-Cartesian-polar coordinate system and structured orientation developed in Sections 2 and 3, showcasing their utility in handling directional transformations with geometric elegance and consistency. This case study was developed with assistance from the AI tool Grok.

## 7.1 Problem Statement

Consider a vector $\vec{x} = \vec{x}_R + \vec{x}_I \in X$, where $\vec{x}_R = (x_R, 0)_C \in \mathbb{R} \times \{0\}$ and $\vec{x}_I = (0, x_I)_C \in \{0\} \times \mathbb{I}$, with $x_R, x_I \in \mathbb{R}$. We define a transformation $\tau: X \to X$ based on the quadrant or axis position of $\vec{x}$:

* **Quadrant I** ($x_R > 0, x_I > 0$): $\tau(\vec{x}) = 2\vec{x}$, doubling the vector's magnitude while preserving its direction.

* **Quadrant III** ($x_R < 0, x_I < 0$): $\tau(\vec{x}) = \vec{x}$, leaving the vector unchanged.

* **Quadrants II or IV** ($x_R < 0, x_I > 0$ or $x_R > 0, x_I < 0$): $\tau(\vec{x}) = i\vec{x}$, rotating the vector 90° counterclockwise.

* **Axes**: Vectors on the positive real or imaginary axes ($x_R > 0, x_I = 0$ or $x_R = 0, x_I > 0$) follow Quadrant I rules, yielding $\tau(\vec{x}) = 2\vec{x}$; those on the negative axes ($x_R < 0, x_I = 0$ or $x_R = 0, x_I < 0$) follow Quadrant III rules, yielding $\tau(\vec{x}) = \vec{x}$.

This transformation exemplifies piecewise-defined mappings common in complex analysis and geometric transformations, where quadrant-specific rules reflect directional properties. For instance, $\vec{x} = (2,3)_C$ in Quadrant I becomes $\tau(\vec{x}) = 2\vec{x} = (4,6)_C$, while $\vec{x} = (-1,-2)_C$ in Quadrant III remains $\tau(\vec{x}) = \vec{x} = (-1,-2)_C$. Boundary cases, such as $\vec{x} = (0,4)_C$ on the positive imaginary axis, yield $\tau(\vec{x}) = 2\vec{x} = (0,8)_C$. The challenge lies in applying these rules efficiently and consistently, particularly at axis boundaries.
## 7.2 Standard Approach

The conventional method determines $\tau(\vec{x})$ by explicitly evaluating the signs of $x_R$ and $x_I$ to classify $\vec{x}$'s quadrant, supplemented by separate checks for axis alignment. A pseudocode representation illustrates this:

```pseudocode
// Standard Approach Pseudo-Code

def get_standard Approach_transformation(x):
    // decompose vector into real and imaginary components
    x_R = real_part(x)
    x_I = imag_part(x)

    // execute nested logic, apply transformation, and grab transformed result
    if (x_R > 0) and (x_I > 0):
        // quadrant I
        return 2*x
    else if (x_R < 0) and (x_I < 0):
        // quadrant III
        return x
    else if (x_R == 0) or (x_I == 0):
        // axes
        if (x_R > 0) or (x_I > 0):
            // positive axes
            return 2*x
        else:  // (x_R < 0) or (x_I < 0) -- negative axes
            return x
    else:  // quadrant II or quadrant IV
        return i*x
```

This standard approach, while straightforward, is computationally intensive, requiring up to 7 comparison conditional checks per vector. For example, $\vec{x} = (3,0)_C$ triggers the axis condition, necessitating additional logic to confirm $x_R > 0$, yielding $\tau(\vec{x}) = 2\vec{x} = (6,0)_C$. The method's reliance on disjoint rules for quadrants and axes introduces complexity and potential inconsistency, especially in theoretical extensions or higher-dimensional analogs, where the number of conditions would scale unfavorably. In computer science applications, such as real-time systems or large-scale data processing, this inefficiency can lead to increased computational costs and slower performance, making the standard approach less scalable and more resource-intensive.

## 7.3 Tri-Quarter Framework Approach

The Tri-Quarter framework, rooted in the structured orientation of Section 3, offers a unified and efficient alternative:

(1) **Orthogonal Decomposition:** Express $\vec{x} = \vec{x}_R + \vec{x}_I$, where $\vec{x}_R = (x_R, 0)_C$ and $\vec{x}_I = (0, x_I)_C$, aligning with the real and imaginary axes, respectively.

(2) **Phase Pair Assignment:** Assign $(\langle\vec{x}_R\rangle, \langle\vec{x}_I\rangle)_\phi$ per the phase assignment rules:

* If $x_R > 0$, then $\langle\vec{x}_R\rangle = 0$; if $x_R = 0$, then $\langle\vec{x}_R\rangle = \frac{\pi}{2}$; if $x_R < 0$, then $\langle\vec{x}_R\rangle = \pi$.

* If $x_I > 0$, then $\langle\vec{x}_I\rangle = \frac{\pi}{2}$; if $x_I = 0$, then $\langle\vec{x}_I\rangle = 0$; if $x_I < 0$, then $\langle\vec{x}_I\rangle = \frac{3\pi}{2}$.

(3) **Transformation Application:** Define $\tau$ based on the phase pair and whether the vector lies on an axis:

* If $(\langle\vec{x}_R\rangle, \langle\vec{x}_I\rangle)_\phi = (0,0)_\phi$, then $\tau(\vec{x}) = 2\vec{x}$ (Positive real axis).

* If $(\langle\vec{x}_R\rangle, \langle\vec{x}_I\rangle)_\phi = (0, \frac{\pi}{2})_\phi$, then $\tau(\vec{x}) = 2\vec{x}$ (Quadrant I).
* If $\langle\langle \vec{x}_R \rangle, \langle \vec{x}_I \rangle\rangle_\phi = (\frac{\pi}{2}, \frac{\pi}{2})_\phi$, then $\tau(\vec{x}) = 2\vec{x}$ (Positive imaginary axis).

* If $\langle\langle \vec{x}_R \rangle, \langle \vec{x}_I \rangle\rangle_\phi = (\pi, \frac{\pi}{2})_\phi$, then $\tau(\vec{x}) = i\vec{x}$ (Quadrant II).

* If $\langle\langle \vec{x}_R \rangle, \langle \vec{x}_I \rangle\rangle_\phi = (\pi, 0)_\phi$, then $\tau(\vec{x}) = \vec{x}$ (Negative real axis).

* If $\langle\langle \vec{x}_R \rangle, \langle \vec{x}_I \rangle\rangle_\phi = (\pi, \frac{3\pi}{2})_\phi$, then $\tau(\vec{x}) = \vec{x}$ (Quadrant III).

* If $\langle\langle \vec{x}_R \rangle, \langle \vec{x}_I \rangle\rangle_\phi = (\frac{\pi}{2}, \frac{3\pi}{2})_\phi$, then $\tau(\vec{x}) = \vec{x}$ (Negative imaginary axis).

* If $\langle\langle \vec{x}_R \rangle, \langle \vec{x}_I \rangle\rangle_\phi = (0, \frac{3\pi}{2})_\phi$, then $\tau(\vec{x}) = i\vec{x}$ (Quadrant IV).

The Tri-Quarter framework's efficiency is best illustrated through pseudo-code, which demonstrates how the phase pair assignments streamline the transformation process. By decomposing the vector into its real and imaginary components and assigning phase pairs based on their signs, the framework enables a direct mapping to the appropriate transformation, eliminating the need for extensive conditional branching.

```c
// Tri-Quarter Approach Pseudo-Code

def get_triquarter ApProach_transformation(x):
    // decompose vector into real and imaginary components
    x_R = real_part(x)
    x_I = imag_part(x)

    // assign real phase based on signs and zero conditions
    if (x_R > 0):
        phi_R = 0
    else if (x_R == 0):
        phi_R = pi/2
    else:
        phi_R = pi

    // assign imaginary phase based on signs and zero conditions
    if (x_I > 0):
        phi_I = pi/2
    else if (x_I == 0):
        phi_I = 0
    else:
        phi_I = 3*pi/2

    // assign phase pair
    phase_pair = (phi_R, phi_I)

    // initialize dictionary of transformations based on phase pairs
    transformations = {
        (0, 0): lambda x: 2*x,      // positive real axis
        (0, pi/2): lambda x: 2*x,    // quadrant I
        (pi/2, pi/2): lambda x: 2*x,  // positive imaginary axis
        (pi, pi/2): lambda x: i*x,   // quadrant II
        (pi, 0): lambda x: x,        // negative real axis
        (pi, 3*pi/2): lambda x: x,   // quadrant III
        (pi/2, 3*pi/2): lambda x: x, // negative imaginary axis
    }
```
(0, 3*pi/2): lambda x: i*x // quadrant IV
}

// apply transformation and grab transformed result
return transformations[phase_pair](x)

This Tri-Quarter approach only requires up to 4 comparison conditional checks per vector. Specifically, it involves up to 2 comparison conditional checks for the real component (1 for positivity and 1 for zero) and up to 2 comparison conditional checks for the imaginary component (similarly for positivity and zero), totaling up to 4 comparison conditional checks to assign the phase pair. Once the phase pair is determined, the transformation is accessed directly from a predefined mapping and executed in constant time without additional comparison conditional checks. This contrasts sharply with the standard approach, which requires up to 7 comparison conditional checks due to its nested conditional structure, making the Tri-Quarter method notably more efficient.

## 7.4 Examples

To illustrate the consistency and efficiency of the Tri-Quarter framework, consider the following examples, comparing both approaches by sampling each quadrant and axis. For each example, we detail the sequence and counts of comparison conditional checks executed by each method in a concise table format, highlighting the computational advantages of the Tri-Quarter approach.

* $\vec{x}_1 = (5, 0)_C$ (Positive real axis)

<table><thead><tr><th>Standard Approach</th><th>Tri-Quarter Approach</th></tr></thead><tbody><tr><td>Checks:<br>○ $x_R > 0$ (true)<br>○ $x_I > 0$ (false)<br>○ $x_R < 0$ (false)<br>○ $x_R == 0$ (false)<br>○ $x_I == 0$ (true)<br>○ $x_R > 0$ (true)<br>Result: $\tau(\vec{x}_1) = 2\vec{x}_1 = (10, 0)_C$<br>Total checks: 6</td><td>Checks:<br>○ $x_R > 0$ (true)<br>○ $x_I > 0$ (false)<br>○ $x_I == 0$ (true)<br>Phase pair: $(0, 0)_\phi$<br>Result: $\tau(\vec{x}_1) = 2\vec{x}_1 = (10, 0)_C$<br>Total checks: 3</td></tr></tbody></table>

* $\vec{x}_2 = (3, 4)_C$ (Quadrant I)

<table><thead><tr><th>Standard Approach</th><th>Tri-Quarter Approach</th></tr></thead><tbody><tr><td>Checks:<br>○ $x_R > 0$ (true)<br>○ $x_I > 0$ (true)<br>Result: $\tau(\vec{x}_2) = 2\vec{x}_2 = (6, 8)_C$<br>Total checks: 2</td><td>Checks:<br>○ $x_R > 0$ (true)<br>○ $x_I > 0$ (true)<br>Phase pair: $(0, \frac{\pi}{2})_\phi$<br>Result: $\tau(\vec{x}_2) = 2\vec{x}_2 = (6, 8)_C$<br>Total checks: 2</td></tr></tbody></table>

* $\vec{x}_3 = (0, 3)_C$ (Positive imaginary axis)
<table><thead><tr><th>Standard Approach</th><th>Tri-Quarter Approach</th></tr></thead><tbody><tr><td>Checks:<br>○ x<sub>R</sub> > 0 (false)<br>○ x<sub>R</sub> < 0 (false)<br>○ x<sub>R</sub> == 0 (true)<br>○ x<sub>I</sub> > 0 (true)<br>Result: τ(x̃<sub>3</sub>) = 2x̃<sub>3</sub> = (0, 6)<sub>C</sub><br>Total checks: 4</td><td>Checks:<br>○ x<sub>R</sub> > 0 (false)<br>○ x<sub>R</sub> == 0 (true)<br>○ x<sub>I</sub> > 0 (true)<br>Phase pair: (π/2, π/2)<sub>φ</sub><br>Result: τ(x̃<sub>3</sub>) = 2x̃<sub>3</sub> = (0, 6)<sub>C</sub><br>Total checks: 3</td></tr></tbody></table>

• $\vec{x}_4 = (-2, 3)_C$ (Quadrant II)

<table><thead><tr><th>Standard Approach</th><th>Tri-Quarter Approach</th></tr></thead><tbody><tr><td>Checks:<br>○ x<sub>R</sub> > 0 (false)<br>○ x<sub>R</sub> < 0 (true)<br>○ x<sub>I</sub> < 0 (false)<br>○ x<sub>R</sub> == 0 (false)<br>○ x<sub>I</sub> == 0 (false)<br>Result: τ(x̃<sub>4</sub>) = ĩx<sub>4</sub> = (-3, -2)<sub>C</sub><br>Total checks: 5</td><td>Checks:<br>○ x<sub>R</sub> > 0 (false)<br>○ x<sub>R</sub> == 0 (false)<br>○ x<sub>I</sub> > 0 (true)<br>Phase pair: (π, π/2)<sub>φ</sub><br>Result: τ(x̃<sub>4</sub>) = ĩx<sub>4</sub> = (-3, -2)<sub>C</sub><br>Total checks: 3</td></tr></tbody></table>

• $\vec{x}_5 = (-4, 0)_C$ (Negative real axis)

<table><thead><tr><th>Standard Approach</th><th>Tri-Quarter Approach</th></tr></thead><tbody><tr><td>Checks:<br>○ x<sub>R</sub> > 0 (false)<br>○ x<sub>R</sub> < 0 (true)<br>○ x<sub>I</sub> < 0 (false)<br>○ x<sub>R</sub> == 0 (false)<br>○ x<sub>I</sub> == 0 (true)<br>○ x<sub>R</sub> > 0 (false)<br>○ x<sub>I</sub> > 0 (false)<br>Result: τ(x̃<sub>5</sub>) = x̃<sub>5</sub> = (-4, 0)<sub>C</sub><br>Total checks: 7</td><td>Checks:<br>○ x<sub>R</sub> > 0 (false)<br>○ x<sub>R</sub> == 0 (false)<br>○ x<sub>I</sub> > 0 (false)<br>○ x<sub>I</sub> == 0 (true)<br>Phase pair: (π, 0)<sub>φ</sub><br>Result: τ(x̃<sub>5</sub>) = x̃<sub>5</sub> = (-4, 0)<sub>C</sub><br>Total checks: 4</td></tr></tbody></table>

• $\vec{x}_6 = (-1, -1)_C$ (Quadrant III near boundary)

<table><thead><tr><th>Standard Approach</th><th>Tri-Quarter Approach</th></tr></thead><tbody><tr><td>Checks:<br>○ x<sub>R</sub> > 0 (false)<br>○ x<sub>R</sub> < 0 (true)<br>○ x<sub>I</sub> < 0 (true)<br>Result: τ(x̃<sub>6</sub>) = x̃<sub>6</sub> = (-1, -1)<sub>C</sub><br>Total checks: 3</td><td>Checks:<br>○ x<sub>R</sub> > 0 (false)<br>○ x<sub>R</sub> == 0 (false)<br>○ x<sub>I</sub> > 0 (false)<br>○ x<sub>I</sub> == 0 (false)<br>Phase pair: (π, 3π/2)<sub>φ</sub><br>Result: τ(x̃<sub>6</sub>) = x̃<sub>6</sub> = (-1, -1)<sub>C</sub><br>Total checks: 4</td></tr></tbody></table>

• $\vec{x}_7 = (0, -2)_C$ (Negative imaginary axis)
<table><thead><tr><th>Standard Approach</th><th>Tri-Quarter Approach</th></tr></thead><tbody><tr><td>Checks:<br>○ x<sub>R</sub> > 0 (false)<br>○ x<sub>R</sub> < 0 (false)<br>○ x<sub>R</sub> == 0 (true)<br>○ x<sub>R</sub> > 0 (false)<br>○ x<sub>I</sub> > 0 (false)<br>Result: τ(x̃<sub>7</sub>) = x̃<sub>7</sub> = (0, -2)<sub>C</sub><br>Total checks: 5</td><td>Checks:<br>○ x<sub>R</sub> > 0 (false)<br>○ x<sub>R</sub> == 0 (true)<br>○ x<sub>I</sub> > 0 (false)<br>○ x<sub>I</sub> == 0 (false)<br>Phase pair: (π/2, 3π/2)<sub>φ</sub><br>Result: τ(x̃<sub>7</sub>) = x̃<sub>7</sub> = (0, -2)<sub>C</sub><br>Total checks: 4</td></tr></tbody></table>

• $\vec{x}_8 = (4, -5)_C$ (Quadrant IV)

<table><thead><tr><th>Standard Approach</th><th>Tri-Quarter Approach</th></tr></thead><tbody><tr><td>Checks:<br>○ x<sub>R</sub> > 0 (true)<br>○ x<sub>I</sub> > 0 (false)<br>○ x<sub>R</sub> < 0 (false)<br>○ x<sub>R</sub> == 0 (false)<br>○ x<sub>I</sub> == 0 (false)<br>Result: τ(x̃<sub>8</sub>) = ĩx<sub>8</sub> = (5, 4)<sub>C</sub><br>Total checks: 5</td><td>Checks:<br>○ x<sub>R</sub> > 0 (true)<br>○ x<sub>I</sub> > 0 (false)<br>○ x<sub>I</sub> == 0 (false)<br>Phase pair: (0, 3π/2)<sub>φ</sub><br>Result: τ(x̃<sub>8</sub>) = ĩx<sub>8</sub> = (5, 4)<sub>C</sub><br>Total checks: 3</td></tr></tbody></table>

The examples demonstrate that overall the Tri-Quarter framework consistently requires fewer comparison conditional checks than the standard approach, with reductions ranging from 2 to 4 checks per vector. This efficiency, combined with the framework's ability to handle boundary cases seamlessly through unique phase pair assignments, underscores its computational superiority. Moreover, the Tri-Quarter approach maintains accuracy while simplifying the transformation process, making it a robust alternative for both theoretical and applied contexts.

## 7.5 Advantages of the Framework

The Tri-Quarter framework offers several theoretical and practical advantages over the standard approach, rooted in its structured orientation and phase pair assignments:

* **Unified Classification:** The phase pair $((\langle \vec{x}_R \rangle, \langle \vec{x}_I \rangle))_\phi$ encodes all directional information based on the signs and zero conditions of $x_R$ and $x_I$. This eliminates the need for separate quadrant and axis logic, reducing the standard approach's maximum of 7 comparison conditional checks down to a maximum of 4 comparison conditional checks for phase pair assignment, making it more efficient and more consistent. For instance, $\vec{x} = (0, 3)_C$ is assigned $(\frac{\pi}{2}, \frac{\pi}{2})_\phi$, directly mapping to $\tau(\vec{x}) = 2\vec{x}$ for the positive imaginary axis.

* **Geometric Elegance:** By decomposing $\vec{x}$ into orthogonal components $\vec{x}_R$ and $\vec{x}_I$, the framework leverages the natural symmetry of $\mathbb{C}$. This decomposition aligns transformations with the axes' intrinsic directions, enhancing conceptual clarity. For example, the doubling of vectors on the positive axes and in Quadrant I reflects uniform scaling along rays from the origin, seamlessly integrated via phase pairs.

* **Consistency Across Boundaries:** The phase pair system naturally incorporates axis points with unique assignments. For example, the negative imaginary axis is assigned $(\frac{\pi}{2}, \frac{3\pi}{2})_\phi$, distinguishing it from Quadrant IV's $(0, \frac{3\pi}{2})_\phi$, ensuring precise transformation application without additional conditional logic.
* **Computational Efficiency:** The framework relies solely on sign evaluations and zero checks, avoiding computationally expensive operations like arctan or norm calculations. This efficiency is particularly beneficial for large datasets or real-time applications, as demonstrated in the examples where overall the Tri-Quarter approach requires fewer comparison conditional checks.

* **Scalability and Generalization:** The phase pair system may be extended to finer angular partitions or higher dimensions, offering a scalable alternative to traditional conditional logic. This adaptability makes it suitable for advanced mathematical modeling and computer science applications, such as multi-dimensional data classification.

* **Mathematical Elegance and Symmetry:** The discrete phase pairs introduce a symmetry that mirrors the rotational properties of the complex plane, potentially aligning with group-theoretic concepts. This elegance enhances the framework's theoretical appeal and facilitates deeper mathematical exploration.

* **Topological Insights:** The discrete partitioning induced by phase pairs provides a novel perspective on the complex plane's topology, enabling the study of transformation properties and discontinuities through a combinatorial lens.

* **Integration with Pure Mathematical Concepts:** The framework's directional classification without norm computations could simplify analyses in harmonic analysis, dynamical systems, and other areas, offering a computationally efficient alternative for theoretical investigations.

* **Compatibility with Reflective Duality:** The preservation of phase pairs under circle inversion ensures consistent transformation behavior across the boundary $T_r$, enhancing the framework's robustness in applications involving symmetry and reflection.

* **Facilitation of Combinatorial Modeling:** The discrete nature of phase pairs enables efficient combinatorial modeling, ideal for applications in computer science such as lattice path counting or symmetry analysis, reducing complexity and potential errors.

## 7.6 Theoretical Implications and Future Directions

The Tri-Quarter framework's phase pair assignments establish a discrete topological partitioning that parallels continuous phase mappings, offering a bridge between geometric and analytic perspectives in complex analysis. This structure suggests potential connections to:

* **Symmetry and Duality:** The preservation of phase pairs under the circle inversion map $\iota_r$ (Section 6) underscores the framework's compatibility with reflective symmetries. For example, applying $\iota_r$ to a vector in $X_{-,r}$ maps it to $X_{+,r}$ while maintaining the same phase pair, facilitating symmetric transformations across the boundary $T_r$.

* **Directional Analysis:** In fields like harmonic analysis or signal processing, where directional properties are crucial, the framework's efficient classification via phase pairs could optimize algorithms by eliminating norm computations. For instance, vectors in Quadrant IV, such as $\vec{x} = (4, -5)_C$ with phase pair $(0, \frac{3\pi}{2})_\phi$, can be directly transformed without angular calculations.

* **Topological Extensions:** Integrating the radial zones $(X_{-,r}, T_r, X_{+,r})$ with quadrant-based transformations could yield hybrid mappings that leverage both magnitude and direction. For example, a transformation $\tau'$ that scales vectors differently based on their zone
and quadrant could be naturally expressed within the Tri-Quarter framework, enhancing its applicability to complex dynamical systems.

Future research might explore these hybrid transformations, leveraging the full Tri-Quarter structure to uncover new properties in complex dynamics or geometric function theory. Additionally, extending the framework to higher-dimensional spaces or other manifolds could reveal broader applications in mathematics and computer science.

## 7.7 Case Study Summary

This case study demonstrates that the Tri-Quarter framework streamlines quadrant-based transformations by unifying classification through phase pair assignments, reducing computational complexity, and enhancing geometric intuition. The framework's advantages—unified classification, computational efficiency, and scalability—position it as a superior alternative to the standard approach, particularly for applications requiring precise directional analysis without trigonometric overhead. Its compatibility with topological and reflective properties further underscores its value, inviting exploration into broader symmetries and applications in both mathematics and computer science.

## 8 Conclusion

The strength, flexibility, and novelty of the Tri-Quarter framework, anchored by the Tri-Quarter Topological Duality Theorem and the Escher Tri-Quarter Reflective Duality Theorem, lie in its unification of complex, Cartesian, and polar coordinate systems on the complex plane $C$. This framework integrates a generalized coordinate system with a structured orientation via phase pair assignments, elevating the circle $T_r$ of any radius $r > 0$ from a passive boundary to an active separator with intrinsic directional properties. The topological duality ensures consistent separation between inner and outer radial directions across $T_r$, while the reflective duality, achieved through circle inversion, preserves phase pairs and swaps the inner and outer zones, offering a fresh perspective on symmetry in the complex plane.

The framework's structured orientation provides a combinatorial classification of directional structures through discrete phase pairs, uniquely assigned based on the signs and zero conditions of the real and imaginary components. This discrete encoding not only unifies geometric and algebraic perspectives but also enhances computational efficiency, as demonstrated in the case study on quadrant-based transformations. The Tri-Quarter approach consistently reduces the number of comparison conditional checks from a maximum of 7 in the standard approach to a maximum of 4, making it particularly valuable for applications in fields like computer science and signal processing where efficiency is paramount.

To animate the Tri-Quarter Topological Duality Theorem and give a visual depiction of the additional encoded data layer it serves, we created a software tool with open-source code, available online at [7]. It is a Python script that should execute on any machine equipped with Python 3.8+. See Figure 1 for a screenshot of the animation, which provides a tangible demonstration of the framework's concepts.

This framework's unification of coordinate systems, structured orientation, topological duality, and reflective duality offers a versatile tool for analyzing systems with circular symmetry and directional properties. Its ability to simplify complex directional mappings and enhance computational efficiency, as shown in the case study, underscores its potential to inspire novel methodologies across mathematics and computer science. Moreover, the framework's compatibility with topological and reflective properties suggests broader applications, such as refining models in complex dynamics, harmonic analysis, and geometric function theory, or improving the efficiency of algorithms in machine learning and cryptography.
Looking ahead, can these results of the Tri-Quarter framework be further developed and applied to help decipher complex phenomena like quark confinement [8] and black holes [9, 10, 11]? Can it enhance the efficiency, accuracy, and scalability of technologies such as quantum computing hardware, medical imaging devices, or wireless communication networks? For instance, in quantum computing, where phase relationships are critical, this framework might optimize qubit modeling across boundaries, potentially advancing gate design or error correction. While the current work focuses on the complex plane, extending the structured orientation and phase pair assignments to higher-dimensional spaces like $C^n$ or other manifolds with symmetry—such as tori or spheres—could yield new insights into multivariable complex analysis, algebraic topology, and beyond. These extensions remain open challenges, inviting future research to deepen our understanding of symmetry, separation, and duality in mathematics and computer science.

In summary, the Tri-Quarter framework provides a robust foundation for both theoretical exploration and practical implementation. Its novel perspective on the complex plane, combined with its computational advantages and interdisciplinary potential, positions it as a powerful tool for advancing knowledge and technology, encouraging further investigation into its full capabilities.

## Acknowledgements

I wish to thank my friend Dr. Christian Corda of Prato, Italy for his guidance and sharing of knowledge over the years, and for his helpful feedback on this paper.

I also wish to thank the AI tool Grok, created by xAI, for its assistance in refining ideas, testing concepts, and generating text for the case study. The original theorems, proofs, and core contributions remain the author's own.

## Author's Note

I earned a B.S. in computer science and a minor in mathematics at Eastern Oregon University, and then a dual M.S. in computer science and mathematics at Boise State University. From 2007 to 2017, I engaged in research work in various capacities to apply computer science and mathematics to various fields such as machine learning, robotics, bioinformatics, high energy physics, quantum gravity, sustainable energy, and cryptography. I see much interdisciplinary overlap across these great fields. I'm currently working full-time as a software development engineer and doing some unpaid after-hours research as a hobby. This is my first math or science paper after taking an eight-year pause from research.

I originally came up with ideas for the generalized complex-Cartesian-polar coordinate system and structured orientation back in 2012 when I was working on quark confinement [8]. Furthermore, during that time, Dr. Andrej Inopin of Kharkiv, Ukraine, shared with me the topological connections to M.C. Escher's work. In [8] the core ideas are basically there but we never had the opportunity and realization to fully test, refine, and finalize them into a consolidated theorems and proofs with legit terminology until now.

Then, just recently, upon revisiting these ideas from 2012 and focusing to further develop them, I began to experiment with Grok v3 as an assistive tool. I wrote latex code with text and equations to formulate my ideas into written thought experiments, and then I asked Grok to pretend to be a mathematician and scientist, and then I fed my latex code directly into Grok and asked it to help validate or refute my various thought experiments. I also asked it to help me look for holes or gaps in my work. It found things that I missed and I found things that it missed. Then I'd fix it. Rinse and repeat. Through this process, I was able to test, refine, and advance my ideas at a faster velocity.

Thus far, I've discovered that the key to using Grok (or any artificial intelligence) to effectively help as an assistant (with technical disciplines) resides in one's ability to creatively and rigorously question, prompt, and apply the methods of science and mathematics. I think of Grok as a friendly,
powerful piece of heavy machinery, where I'm the operator and it is a force multiplier that helps
get the job done.

<figure><img src="image_3.png" /><figcaption>FIGURE 1. A screenshot from the software animation tool [7], visualizing the three topological zones ($X_{-,r}$, $T_r$, $X_{+,r}$) and the structured orientation on a circle of radius $r$ in the complex plane.</figcaption></figure>
References

[1] Bernhard Riemann. *Grundlagen für eine allgemeine Theorie der Funktionen einer veränderlichen complexen Größe*. PhD thesis, Universität Göttingen, Göttingen, Germany, 1851. Doctoral dissertation.

[2] Lars V. Ahlfors. *Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable*. McGraw-Hill, New York, 1st edition, 1953. Later editions exist, e.g., 3rd ed., 1979.

[3] Tristan Needham. *Visual Complex Analysis*. Oxford University Press, Oxford, 1997. ISBN 978-0-19-853446-4. See Chapter 3 for a detailed discussion of circle inversion and Möbius transformations.

[4] Bruno Ernst. *The Magic Mirror of M.C. Escher*. Random House, New York, 1976. ISBN 0-394-73217-0. Includes an image of “Hand with Reflecting Sphere” (Self-Portrait in Spherical Mirror), lithograph by M.C. Escher, January 1935.

[5] Camille Jordan. *Cours d'Analyse de l'École Polytechnique, Tome Premier*. Gauthier-Villars, Paris, 1st edition, 1887. First appearance of the Jordan Curve Theorem; later editions (e.g., 1893) refine the presentation.

[6] James R. Munkres. *Topology*. Prentice Hall, Upper Saddle River, NJ, 2nd edition, 2000. ISBN 978-0-13-181629-9.

[7] Nathan O. Schmidt. Tri-Quarter Toolbox Github Repository. Accessed: 2025-03-07. https://github.com/nathanoschmidt/tri-quarter-toolbox/.

[8] A. E. Inopin and N. O. Schmidt. Proof of Quark Confinement and Baryon-Antibaryon Duality: I: Gauge Symmetry Breaking in Dual 4D Fractional Quantum Hall Superfluidic Space-Time. *Hadronic Journal*, 35(5):469, 2012. URL https://vixra.org/abs/1208.0219. Preprint available at viXra:1208.0219.

[9] C. Corda, S. H. Hendi, R. Katebi, and N. O. Schmidt. Effective State, Hawking Radiation and Quasi-Normal Modes for Kerr Black Holes. *JHEP*, 06:008, 2013. doi: 10.1007/JHEP06(2013)008.

[10] C. Corda, S. H. Hendi, R. Katebi, and N. O. Schmidt. Hawking Radiation – Quasi-Normal Modes Correspondence and Effective States for Nonextremal Reissner-Nordstrom Black Holes. *Advances in High Energy Physics*, 2014(1):527874, 2014. doi: https://doi.org/10.1155/2014/527874. URL https://onlinelibrary.wiley.com/doi/abs/10.1155/2014/527874.

[11] C. Corda, S. H. Hendi, R. Katebi, and N. O. Schmidt. Initiating the Effective Unification of Black Hole Horizon Area and Entropy Quantization with Quasi-Normal Modes. *Adv. High Energy Phys.*, 2014:530547, 2014. doi: 10.1155/2014/530547.