Dual Affine Spiral Orbits on $\mathbb{Z}^2$ Generated by Paired Unit Squares

Technical Summary: Dual Affine Spiral Orbits on $\mathbb{Z}^2$ Generated by Paired Unit Squares

Problem Statement
The paper investigates a geometric construction on the integer lattice $\mathbb{Z}^2$ involving paired unit squares that share a single corner. The central problem is to characterize the iterative process where new squares are constructed outwardly on the hypotenuse of the isosceles right triangle formed by the outer edges adjacent to the shared corner. The author seeks to determine the algebraic structure of the resulting sequences of lattice points, their symmetries, and their relationship to known fractal geometries when the iteration is extended in both forward and backward directions.

Methodology
The study employs a combination of elementary Euclidean geometry, linear algebra, and complex analysis (specifically Gaussian integers).

1. Geometric Construction: The process begins with two unit squares sharing a corner at $P_0 = (0,0)$. At each step $n$, a new square is built on the hypotenuse of the triangle formed by the outer edges of the previous pair. This generates two distinct sequences of points: a "positive" orbit $\{P_n\}$ and a "negative" orbit $\{N_n\}$.
2. Algebraic Formulation: The geometric transformations are encoded using affine recurrences. The author identifies the integer lattice $\mathbb{Z}^2$ with the complex plane $\mathbb{C}$, mapping the transformation matrix to multiplication by the Gaussian integer $1+i$.
3. Closed-Form Derivation: By solving the linear recurrences, the author derives closed-form expressions for $P_n$ and $N_n$ in terms of powers of $(1+i)$.
4. Iterated Function System (IFS) Analysis: The iteration is extended backward ($n < 0$) by inverting the affine maps. The resulting inverse maps are analyzed as contractions to identify their attractor and compare it to the Twindragon fractal.

Key Contributions and Results

• Dual Spiral Orbits: The construction generates two interlocking affine spiral orbits on $\mathbb{Z}^2$.

  • The positive orbit starts at $P_0 = (0,0)$ and satisfies the affine recurrence $P_{n+1} = (1+i)P_n + (2+i)$.
  • The negative orbit starts at $N_0 = (-1, 2)$ and satisfies the linear recurrence $N_{n+1} = (1+i)N_n$.
  • Both sequences remain strictly on the integer lattice indefinitely.

• Symmetry and Invariants:

  • Fixed Midpoint: The orbits are symmetric with respect to a fixed non-lattice point $M = (-1/2, 1)$. For all $n$, the invariant $P_n + N_n = (-1, 2)$ holds, meaning $N_n$ is the point reflection of $P_n$ through $M$.
  • Geometric Interpretation: The points $P_n$ and $N_n$ sample logarithmic spirals centered at $N_0$ and $P_0$ respectively, with a scaling factor of $\sqrt{2}$ and a rotation of $\pi/4$ per step.

• New Integer Sequence: The paper defines a normalized integer sequence $a(n)$ derived from the squared Euclidean norms of the orbit points:
  $$a(n) = \frac{|P_n|^2 + |N_n|^2}{5} = 2^{n+1} + 1 - 2\text{Re}((1+i)^n)$$
  The author proves that $a(n)$ is always an integer. This sequence corresponds to OEIS A396151 and satisfies a specific linear recurrence of order 4. Geometrically, $a(n)$ represents an affine rescaling of the squared distance from the orbit points to the symmetry center $M$.

• Connection to the Twindragon Fractal:

  • When the iteration is extended backward ($n \to -\infty$), the inverse maps $T^{-1}$ and $S^{-1}$ form an Iterated Function System (IFS).
  • These maps are similarities with a contraction ratio of $1/\sqrt{2}$ and a rotation of $-\pi/4$.
  • The attractor of this IFS is shown to be similar to the Twindragon fractal, a self-affine tile associated with the base $1+i$ in the Gaussian integers.
  • Unlike the forward orbits which spiral outward, the backward orbits spiral inward toward the fixed points of the inverse maps ($P_0$ and $N_0$), accumulating on the Twindragon attractor. The midpoint $M$ serves as the center of symmetry for this attractor.

Significance and Claims
The paper claims that this elementary geometric construction on graph paper provides a concrete, lattice-based viewpoint on the dynamics of the Twindragon fractal.

• Novelty: The specific pairing construction, the resulting dual spiral orbits, and the integer sequence $a(n)$ (OEIS A396151) are presented as new contributions.
• Synthesis: The work bridges elementary geometry with algebraic number theory and fractal geometry. It demonstrates how a simple iterative rule on $\mathbb{Z}^2$ naturally leads to the self-affine tile structure of the Twindragon.
• Modesty: The author does not claim to solve open problems in fractal geometry but rather offers a "visual lattice viewpoint" that re-derives the classical Twindragon attractor from an elementary starting point. The paper suggests that such simple geometric origins may imply a richer underlying structure involving algebraic number theory than initially apparent.