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Is Parallel Postulate Necessary?

Chengpu Wang, Alice Wang

TL;DR

This paper questions the necessity of Euclid's parallel postulate and advocates a Modernized Euclidean Axiom Set that omits the postulate while relying on the 4th axiom, $A_4$, that all right angles are equal. It introduces a metric-based foundational framework with axioms Measurable, Continuous, and Boundless, and defines new constructs such as round objects and angle-distance concepts. A central contribution is the Right Ratio, a local curvature proxy that links to Gaussian curvature and potentially to space-time curvature, offering a simpler analytic alternative to traditional curvature tensors. The work discusses theoretical refinements, potential astrophysical applications (e.g., curvature measurements near points or via interferometry), and educational implications, arguing for a more intuitive and locally informative approach to geometry.

Abstract

As a much later addition to the original Euclidean geometry, the parallel postulate distinguishes non-Euclidean geometries from Euclidean geometry. This paper will show that the parallel postulate is unnecessary because the 4th Euclidean axiom can already achieve the same goal. Furthermore, using the 4th Euclidean axiom can measure space curvature locally on manifold, while using the parallel postulate cannot.

Is Parallel Postulate Necessary?

TL;DR

This paper questions the necessity of Euclid's parallel postulate and advocates a Modernized Euclidean Axiom Set that omits the postulate while relying on the 4th axiom, , that all right angles are equal. It introduces a metric-based foundational framework with axioms Measurable, Continuous, and Boundless, and defines new constructs such as round objects and angle-distance concepts. A central contribution is the Right Ratio, a local curvature proxy that links to Gaussian curvature and potentially to space-time curvature, offering a simpler analytic alternative to traditional curvature tensors. The work discusses theoretical refinements, potential astrophysical applications (e.g., curvature measurements near points or via interferometry), and educational implications, arguing for a more intuitive and locally informative approach to geometry.

Abstract

As a much later addition to the original Euclidean geometry, the parallel postulate distinguishes non-Euclidean geometries from Euclidean geometry. This paper will show that the parallel postulate is unnecessary because the 4th Euclidean axiom can already achieve the same goal. Furthermore, using the 4th Euclidean axiom can measure space curvature locally on manifold, while using the parallel postulate cannot.
Paper Structure (22 sections, 15 theorems, 5 figures)

This paper contains 22 sections, 15 theorems, 5 figures.

Table of Contents

  1. Introduction
  2. The Modernized Euclidean Axiom Set
  3. Notation
  4. Measurable
  5. Continuous
  6. Boundless
  7. Topologically Simplest
  8. Strictly Monotonic
  9. Angle Measurement
  10. Flat
  11. Proof for Theorem (perpendicular)
  12. Proof for Theorem (flat)
  13. Beyond Euclidean Geometry
  14. Right Ratio
  15. Isometric and Isotopic
  16. ...and 7 more sections

Key Result

Theorem 2.9

$\{ C, D \div \underline{*AB*} \}_2: C, E \div \underline{*AB*} \iff \underline{*DE*} \cap \underline{*AB*} = \oslash$, which is defined as $D$ and $E$ being on the same side of $\underline{*AB*}$.

Figures (5)

  • Figure 1: Demonstration of Axiom (strictly monotonic).
  • Figure 2: Left: Form a square in a Euclidean surface. Right: Stack identical squares side by side to form rectangles
  • Figure 3: Proof of Theorem (perpendicular).
  • Figure 4: Proof of Theorem (flat).
  • Figure 5: The right ratios at the centers of the inside surface and the outside surface of a torus Non-Euclidean Geometry. The x-axis is the angle measuring distance normalized as the angle from the center crossing the torus. The torus has an inner radius $2$ and an outer radius $3$.

Theorems & Definitions (43)

  • Definition 2.1: distance
  • Definition 2.2: round
  • Definition 2.3: straight segment
  • Definition 2.4: ray
  • Definition 2.5: straight line
  • Definition 2.6: opposite side
  • Definition 2.7: 2-dimensional space
  • Definition 2.8: 2-sided 2-dimensional space: surface
  • Theorem 2.9: same side
  • Theorem 2.10: perpendicular
  • ...and 33 more