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

Abstract

We introduce a simple iterative geometric construction on the integer lattice $\mathbb{Z}^2$ consisting of paired unit squares that share a single corner. At each step, a new square is constructed outwardly on the hypotenuse of the isosceles right triangle formed by the outer edges adjacent to the shared corner. This process generates two interlocking affine spiral orbits on $\mathbb{Z}^2$: a positive orbit {P_n} starting at (0,0) and a negative orbit $\{N_n\}$ starting at (-1,2). Both sequences satisfy linear recurrences driven by multiplication by the Gaussian integer $1+i$. We show that the paired points remain symmetric with respect to the fixed midpoint $M=(-1/2,1)$ for all $n$, satisfying the invariant $P_n + N_n = (-1,2)$. Extending the iteration backward under the associated inverse maps produces an iterated function system whose attractor is similar to the Twindragon fractal, providing a concrete lattice-based viewpoint on its geometry. In addition, the paired points yield the normalized integer sequence $a(n) = (|P_n|^2 + |N_n|^2)/5 = 2^(n+1) + 1 - 2*Re((1+i)^n)$, which is always an integer and appears as A396151 in the OEIS.