Initial Construction and Preliminary Properties of Chunpinative Geometry

Overview

This content takes the form of a systematic constructive record of a geometric theory, formally proposing the core axiom of "Chunpinative Geometry" and establishing a set of foundational concepts—including "Chunpinative Space" and "Chunpinatively Measurable Space"—on the basis of this axiom. Additionally, it deduces preliminary properties of the space and presents a key lemma proposed by a collaborator. From the perspective of mainstream geometric theory (encompassing both Euclidean and non-Euclidean geometries), the core axiom of Chunpinative Geometry—asserting that "an angle is equivalent to its supplement"—represents a deliberate departure from the fundamental logical relationships that govern angle arithmetic in conventional geometric systems. Specifically, it abandons the well-established rule that "the supplement of an angle is equal to $180^\circ$ minus the measure of the original angle," replacing it with a direct equivalence relation. It is important to acknowledge that certain preliminary property descriptions in this initial construction lack the full rigor of formal mathematical derivation, as they serve as tentative deductions to frame the theory’s structure. As indicated by contextual cues embedded in the naming and framing, this work is a playful, commemorative simulation of a mathematical theory: it adopts the rigorous framework of axiom-based geometric system construction to pay lighthearted tribute to the author’s junior high school mathematics teacher, Chunping Ding (with the term "Chunpinative" derived from the teacher’s given name "Chunping"), and to classmate CZL, who contributed the key lemma to the theory.

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1. Core Axiom and Basic Definitions of Chunpinative Geometry

1.1 Foundational Axiom: The Dual-Complementary Principle (D-C-P Axiom)

The core axiom of Chunpinative Geometry, formally designated the Dual-Complementary Principle (abbreviated as the D-C-P Axiom), is explicitly and rigorously stated as follows: An angle is axiomatically equivalent to its supplement. Within the theoretical context of this axiom, the term "angle" is defined as the geometric figure formed by two non-coincident, intersecting line segments or rays in a given geometric space—consistent with the standard definition of an angle in elementary geometric theory, but constrained to the specific axiomatic framework of Chunpinative Geometry. The "supplement" (or "supplementary angle") retains its conventional geometric meaning: an angle whose measure, when added to the measure of the original angle, equals $180^\circ$ (or $\pi$ radians, when using the radian system of angle measurement). This axiom constitutes a deliberate break from the traditional geometric constraint that "the supplement of an angle is arithmetically equal to $180^\circ$ minus the measure of the original angle"; instead, it establishes a direct, unmediated equivalence between an angle and its supplement. As the foundational axiom of the entire theory, it serves as the logical starting point for constructing all subsequent definitions, properties, and lemmas within Chunpinative Geometry.

1.2 Definition of the Core Geometric Space: Chunpinative Space

1.2.1 Formal Definition: A geometric space is formally defined as a Chunpinative Space if and only if it satisfies the D-C-P Axiom—i.e., in every region and for every possible configuration of intersecting line segments or rays within the space, the angle formed is equivalent to its supplement. The broader theoretical system that investigates the structural characteristics, algebraic properties, and measurement of general sets within such spaces is collectively termed Chunpinative Geometry. The naming convention for both the space and the geometric theory is dual-purpose: it emphasizes the centrality of the D-C-P Axiom to the entire framework, while also carrying contextual commemorative significance tied to the individuals honored by the theory.

1.2.2 Scope of Application and Future Extensions: The initial definition of Chunpinative Space presented herein is formulated within a general, unrestricted geometric framework, designed to apply to the most common cases of two-dimensional and three-dimensional spaces. However, special cases of geometric spaces—including zero-dimensional spaces (isolated points), one-dimensional spaces (lines or curves), and discrete spaces (countable sets of non-continuous points)—introduce unique constraints that are not fully addressed in this preliminary construction. These special cases will require additional axiomatic supplements, restrictive conditions, and tailored definitions in subsequent extensions of the theory, which will be the focus of detailed follow-up research aimed at generalizing Chunpinative Geometry to a more comprehensive set of geometric contexts.

1.3 Preliminary Properties of Chunpinative Spaces

By applying the D-C-P Axiom to the formal definition of Chunpinative Space, the following three preliminary properties can be deduced. These properties are not exhaustive but serve as foundational building blocks for further investigations into the structural and algebraic characteristics of Chunpinative Spaces:

• Property i (Space Carrier and Coordinate Representation): A general Chunpinative Space is defined over the Cartesian product $X \times Y$, where $X$ and $Y$ are non-empty sets. In line with conventional geometric practice, $X$ and $Y$ are typically number sets (such as the set of real numbers $\mathbb{R}$, the set of integers $\mathbb{Z}$, or other ordered number fields) or point sets in traditional Euclidean space. This Cartesian product structure implies that every element (referred to as a "point") within the Chunpinative Space can be uniquely represented by an ordered pair $(x, y)$ where $x \in X$ and $y \in Y$. This coordinate representation provides a critical foundation for analytical studies of the space, enabling the use of algebraic tools to investigate geometric relationships within Chunpinative Geometry. The specific forms of $X$ and $Y$ (e.g., whether they are continuous or discrete, finite or infinite) are not fixed universally but are determined by the specific research objectives and contextual constraints of the Chunpinative Space under investigation.

• Property ii (Algebraic Structure: Commutativity and Non-Closure Under Addition): All Chunpinative Spaces exhibit the structure of an Abelian (commutative) space with respect to the standard binary operation of vector or point addition. Formally, this means that for any two elements $a$ and $b$ within a Chunpinative Space $\mathcal{C}$, the commutative law holds: $a + b = b + a$. This property aligns with the algebraic structure of many conventional geometric spaces (such as Euclidean space) and ensures consistency with basic algebraic operations. However, a distinguishing feature of Chunpinative Spaces is their non-closure under addition: unlike Euclidean space, where the sum of two vectors (or points) remains within the space, the result of adding two elements in a Chunpinative Space may not belong to the space itself. A rigorous proof of this non-closure property is reserved as an exercise for further exploration; the core strategy for such a proof involves constructing explicit counterexamples that leverage the D-C-P Axiom to demonstrate cases where the sum of two elements violates the axiomatic constraints of Chunpinative Space.

• Property iii (Rigidity and Structural Invariance): Every Chunpinative Space is inherently rigid, a property primarily manifested in its resistance to deformation. In traditional geometric terminology, "rigidity" refers to the property of a geometric figure or space wherein the distances between points (and thus the shape and size of the figure) remain unchanged when subjected to external forces or transformations that do not involve stretching or compression. In the context of Chunpinative Spaces, this rigidity is a direct consequence of the uniqueness of angle relationships imposed by the D-C-P Axiom: since an angle is equivalent to its supplement, the measure of any angle in the space is uniquely determined (for real-valued angle measures, this unique value is $90^\circ$, as $x = 180^\circ - x$ implies $x = 90^\circ$). This unique angle constraint fixes the relative positional relationships between line segments, rays, and planes within the space, preventing arbitrary deformation or reconfiguration without violating the D-C-P Axiom. Verification of this rigidity property can be concretely achieved by constructing a specific instance of a Chunpinative Space (such as the specialized space referenced in Lemma 1.3) and analyzing its structural invariance under standard geometric transformations (e.g., translations, rotations, or shears).

1.4 Key Lemma: The CZL Lemma

Lemma Statement (1.3 Lemma): There exists at least one Chunpinative Space in which a right triangle with a vertex angle of $100^\circ$ can be embedded. This lemma is proposed by classmate CZL and is formally designated the CZL Lemma in recognition of this contribution. The CZL Lemma represents a critical breakthrough in the study of the specific structural characteristics of Chunpinative Spaces, as it confirms the existence of specialized Chunpinative Spaces that diverge significantly from both Euclidean and non-Euclidean spaces—wherein a right triangle (defined as a triangle containing a $90^\circ$ angle) cannot have a vertex angle of $100^\circ$ due to the constraints of the triangle angle sum theorem.

Relationship to General Chunpinative Spaces: The specialized Chunpinative Space described in the CZL Lemma is explicitly a special case of the general Chunpinative Space defined in Property i (1.2.1 i). While the general Chunpinative Space is defined over the unrestricted Cartesian product $X \times Y$, the space in the CZL Lemma corresponds to a specific realization of $X \times Y$ with constrained forms of $X$ and $Y$ (e.g., non-standard number sets or modified coordinate systems). This specialized space thus embodies the general structure of Chunpinative Spaces under particular contextual conditions, demonstrating the flexibility and diversity of the theory’s framework.

1.5 Construction Method of Specialized Chunpinative Spaces

The construction of the specialized Chunpinative Space that satisfies the CZL Lemma—i.e., a space in which a right triangle with a $100^\circ$ vertex angle can exist—requires the application of the mathematical technique of global analytic continuation. Analytic continuation is a well-established method in mathematical analysis, typically used to extend the domain of a function beyond its original definition while preserving key analytical properties (such as holomorphy or continuity). In the context of Chunpinative Space construction, this technique is adapted to transcend the constraints of the traditional Cartesian product $X \times Y$, enabling the expansion and modification of the space’s structure to accommodate the unique geometric configuration specified by the CZL Lemma.

Formally, the process of global analytic continuation transforms the original general Chunpinative Space (defined over $X \times Y$) into a mapping from $X \times Y$ to $\xi(X \times Y)$, where $\xi(X \times Y)$ (abbreviated as $\xi$ for brevity when no ambiguity arises) denotes the extended, specialized Chunpinative Space post-continuation. The core of this mapping lies in the redefinition of angle measurement rules within the space: while the D-C-P Axiom is retained as the foundational constraint, the method of assigning numerical measures to angles is modified to allow for the coexistence of a right angle (consistent with the D-C-P Axiom) and a $100^\circ$ vertex angle in the same triangle. This modification breaks the traditional triangle angle sum theorem (which requires the sum of angles in a triangle to be $180^\circ$ in Euclidean space), thereby enabling the existence of the specialized right triangle specified by the CZL Lemma.

1.6 Definition of Chunpinatively Measurable Spaces

Formal Definition (1.4 Definition): Let $\xi$ be a non-empty Chunpinative Space obtained via the method of global analytic continuation (i.e., $\xi \neq \emptyset$), and let $R$ be a $\sigma$-algebra consisting of subsets of $\xi$. The ordered pair $(\xi, R)$ is formally defined as a Chunpinatively Measurable Space. The $\sigma$-algebra $R$ must satisfy the three standard axioms of $\sigma$-algebras in measure theory: 1. The entire specialized Chunpinative Space $\xi$ is an element of $R$ (i.e., $\xi \in R$); 2. The $\sigma$-algebra is closed under complementation: if a subset $A$ of $\xi$ is an element of $R$ (i.e., $A \in R$), then the complement of $A$ with respect to $\xi$ (denoted $\xi \setminus A$) is also an element of $R$; 3. The $\sigma$-algebra is closed under countable unions: for any countable sequence of subsets $A_1, A_2, \dots, A_n, \dots$ where each $A_k \in R$ (for $k = 1, 2, 3, \dots$), their union $\bigcup_{k=1}^\infty A_k$ is also an element of $R$. This definition of the $\sigma$-algebra provides a rigorous mathematical foundation for the subsequent definition of measures on Chunpinative Spaces, serving as an essential prerequisite for the study of the measurement of general sets (such as line segments, regions, or surfaces) within the framework of Chunpinative Geometry.

2. Supplementary Notes on Chunpinative Geometry

It is important to explicitly note that the content presented herein constitutes an initial, preliminary attempt to construct the theoretical framework of Chunpinative Geometry. As such, it may contain limitations and areas for improvement, including (but not limited to) incomplete logical derivations for certain preliminary properties, potential imprecisions in the formalization of edge-case concepts, and the absence of comprehensive proofs for all assertions. The core purpose of this initial construction is not to present a fully mature, peer-reviewed mathematical theory, but rather to convey creative ideas through a playful, commemorative adaptation of the rigorous framework of axiom-based geometric system construction. A revised, expanded, and fully self-consistent version of Chunpinative Geometry has since been refined and systematically organized in January,2025. This improved version enhances the theoretical rigor, logical consistency, and comprehensiveness of the entire system, and is accessible via the following link: https://www.runnelzhang.com/achieved/paper-cpg.