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The $Z$-Curve as an $n$-Dimensional Hypersphere: Properties and Analysis

Diego Vazquez Gonzalez, Hsing-Kuo Pao

Abstract

In this research, we introduce an algorithm that produces what appears to be a new mathematical object as a consequence of projecting the \( n \)-dimensional \( Z \)-curve onto an \( n \)-dimensional sphere. The first part presents the algorithm that enables this transformation, and the second part focuses on studying its properties.

The $Z$-Curve as an $n$-Dimensional Hypersphere: Properties and Analysis

Abstract

In this research, we introduce an algorithm that produces what appears to be a new mathematical object as a consequence of projecting the -dimensional -curve onto an -dimensional sphere. The first part presents the algorithm that enables this transformation, and the second part focuses on studying its properties.
Paper Structure (20 sections, 1 theorem, 23 equations, 18 figures, 24 tables, 1 algorithm)

This paper contains 20 sections, 1 theorem, 23 equations, 18 figures, 24 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Methodology
  3. BitDistance
  4. Projecting onto an $N$-Dimensional Sphere
  5. Analysis and Properties
  6. Numerical Properties of the Elements in $\mathbb{T}$
  7. The Number of Elements in $\mathbb{T}$
  8. Summation of the Elements in $\mathbb{T}$
  9. Equivalent Position and the Minimum Number of Knowable Subsets
  10. Summation of the Elements in $\mathbf{X}^D_K$
  11. Spatial Properties of $\mathbb{T}$
  12. Operations in $\Pi^{AB}_n$
  13. Internal Structure of $\mathbb{T}$
  14. Dictionary Isomorphism
  15. Acknowledgments
  16. ...and 5 more sections

Key Result

Theorem 3.5

For any dimension $D$ and scaling factor $K$, the integer $\Upsilon$ satisfying Equation eq:sum_equation is given by:

Figures (18)

  • Figure 1: The image shows how the values in $\textbf{X}_3^2$, when following the Z-order pattern, result from the intersection of the binary representations of the numbers associated with each row and column, creating the Morton coordinate system. To enhance readability and interpretation, different typographical styles are used for the bits in each dimensional component based on their position.
  • Figure 2: This image plots the set of numbers in $\mathbf{X}^2_3$, showing where each number is placed on the Cartesian plane after calculating the $BD$ in each dimension. We can observe that after being plotted, they are placed in the space following the $Z$-pattern. The numbers that start with the same $D$ bits share the same color, corresponding to whether they begin with 00, 01, 10, or 11.
  • Figure 5: On the surface of the green circle are all the numbers contained in $\mathbf{X}^2_3$, and the green radius connects the origin to the position of the number 47. The surface of the blue circle contains numbers that share the first two bits with 47, where the BitDistance is calculated ignoring the first two bits. The blue radius connects the position of 47 in the green circle to its position in the blue circle. The same idea applies to the final smaller circle.
  • Figure 6: In the left image are the values in $\mathbf{X}^2_1$ after calculating their BitDistance. In the right image, the values in $\mathbf{X}^2_1$ had been projected into a circle. The color of the number depends on whether the number starts with 00, 01, 10, or 11.
  • Figure 7: In the left image are the values in $\mathbf{X}^2_2$ after calculating their BitDistance. In the right image, the values in $\mathbf{X}^2_2$ had been projected into a circle. The color of the number depends on whether the number starts with 00, 01, 10, or 11.
  • ...and 13 more figures

Theorems & Definitions (8)

  • Definition 2.1: BitDistance
  • Definition 3.1: Subset
  • Conjecture 3.2
  • Definition 3.3: Equivalent Position
  • Conjecture 3.4
  • Theorem 3.5
  • Definition 3.8: Subset Generator
  • Conjecture 3.9