Historical exploratory interpretation, qualified October 20, 2026: these modular grids and their coordinate transformations provide mathematical examples of symmetry and periodicity. A reflection, rotation or reversible coordinate relabeling alone does not supply a quantum cellular automaton or a local unitary quantum-evolution rule. This article provides no specified quantum state space, local quantum update construction or validation of a physical model. The earlier descriptions of seeds as core unitaries, and of these grids as established QCA, quantum-field, topological-material or transport models, should therefore be read as unvalidated proposals. The same limit applies to suggested quantum computing, cryptographic, AI and materials uses: no application benefit or implemented capability is established here. See the current family definitions (https://nsquares.org/family.html) and research results and limits (https://nsquares.org/research.html) for the mathematical scope. The QCA, physics and application research lanes remain parked.
History & naming, arithmetic formulas, QCA context, and colorized grids for each orientation. All indices are 1-based.
Naglich Squares are RL-style modular grids parameterized by a size N and a kernel k. Starting from a simple “row–times–column” rule modulo N, one can normalize, flip, and rotate the grid to reveal a family of visually distinct, algebraically related patterns. This page documents the Forward/Backward family and shows how all eight views derive from just two seeds.
We use a quadrant-style orientation name plus a family tag:
TLB, TRB, BLB, BRB.TLF, TRF, BLF, BRF.“Forward” means each row begins with k (via a constant cyclic column shift −1).
// Conventions (1-indexed)
wrapN(x, N) = ((x - 1) % N) + 1 // 1..N wraparound
H(c, N) = N - c + 1 // horizontal mirror
V(r, N) = N - r + 1 // vertical mirror
// Canonical seeds
A_TRB(r,c,k,N) = ((r - 1) * c + (k - 1)) % N + 1 // RL base (Backward seed)
A_TLF(r,c,k,N) = ((r - 1) * wrapN(c - 1, N) + (k - 1)) % N + 1 // k-anchored (Forward seed)
// Forward family (from TLF)
A_TRF(r,c,k,N) = A_TLF(r, H(c,N), k, N) // horizontal mirror
A_BLF(r,c,k,N) = A_TLF(V(r,N), c, k, N) // vertical mirror
A_BRF(r,c,k,N) = A_TLF(V(r,N), H(c,N), k, N) // 180° rotation
// Backward family (from TRB)
A_TLB(r,c,k,N) = A_TRB(r, H(c,N), k, N) // horizontal mirror
A_BLB(r,c,k,N) = A_TRB(V(r,N), H(c,N), k, N) // 180° rotation
A_BRB(r,c,k,N) = A_TRB(V(r,N), c, k, N) // vertical mirrorAll eight orientations are dihedral images of one of two cores (TRB or TLF). The Forward normalization is a constant cyclic column shift (−1) that moves the row’s k from column N to column 1.
Grid size N×N.
Row anchor. Max auto-updates to N.
Exports high-res PNGs using the same palette & labels.
Historical exploratory interpretation, qualified October 20, 2026: these modular grids and their coordinate transformations provide mathematical examples of symmetry and periodicity. A reflection, rotation or reversible coordinate relabeling alone does not supply a quantum cellular automaton or a local unitary quantum-evolution rule. This article provides no specified quantum state space, local quantum update construction or validation of a physical model. The earlier descriptions of seeds as core unitaries, and of these grids as established QCA, quantum-field, topological-material or transport models, should therefore be read as unvalidated proposals. The same limit applies to suggested quantum computing, cryptographic, AI and materials uses: no application benefit or implemented capability is established here. See the current family definitions (https://nsquares.org/family.html) and research results and limits (https://nsquares.org/research.html) for the mathematical scope. The QCA, physics and application research lanes remain parked.
The phase parameter k is not thereby established as physical time or energy.
The modular formulas and coordinate reflections provide mathematical examples of periodicity and symmetry. Suggested hashing, signal-analysis, AI and engineering uses are unvalidated possibilities; no measured application benefit is established here.
Prime or composite N does not alone establish engineering utility.
The eight orientations of the Naglich Square framework form a complete dihedral family derived from two canonical seeds (TRB and TLF). Each formula is algebraically simple—expressible as modular multiplications with index reflections, rotations, or cyclic shifts—yet collectively they generate structurally rich, visually distinct, and reversible grids. This makes them well-suited for studying modular periodicity, symmetry, and chirality across discrete lattice systems.
Historical exploratory interpretation, qualified October 20, 2026: these modular grids and their coordinate transformations provide mathematical examples of symmetry and periodicity. A reflection, rotation or reversible coordinate relabeling alone does not supply a quantum cellular automaton or a local unitary quantum-evolution rule. This article provides no specified quantum state space, local quantum update construction or validation of a physical model. The earlier descriptions of seeds as core unitaries, and of these grids as established QCA, quantum-field, topological-material or transport models, should therefore be read as unvalidated proposals. The same limit applies to suggested quantum computing, cryptographic, AI and materials uses: no application benefit or implemented capability is established here. See the current family definitions (https://nsquares.org/family.html) and research results and limits (https://nsquares.org/research.html) for the mathematical scope. The QCA, physics and application research lanes remain parked.
Historical hypothetical directions, qualified October 21, 2026. This list is not an active roadmap or a statement of implemented capabilities; the related research lanes remain parked.
Summary: this page demonstrates modular formulas and coordinate transformations. QCA, physical-model, cryptographic, AI and materials benefits are not established; the related research lanes remain parked.