About

A binary square is an N × N grid where each cell holds exactly one of two states: 0 (off) or 1 (on). This simple constraint produces 2^N² possible configurations. For a modest 8 × 8 grid, that is 2⁶⁴ ≈ 1.84 × 10¹⁹ unique patterns. Getting symmetry right by hand is error-prone. A single misplaced cell breaks visual balance. This tool enforces symmetry constraints algorithmically so every toggle propagates to mirrored positions automatically.

The grid state maps directly to a binary string of length N², read in row-major order. You can import or export this string to reproduce patterns exactly. Export supports lossless PNG at any cell resolution and scalable SVG. Note: very large grids (N > 48) may produce large files. The tool approximates visual weight by counting active cells as a fraction of total area.

Formulas

Each cell at row r and column c maps to a linear index in a binary string:

i = r ⋅ N + c

The full pattern is a binary string B of length N² where B[i] ∈ {0, 1}. Fill density (visual weight) is computed as:

D = N²∑i=1 B[i]N² × 100%

Symmetry mirror indices for horizontal reflection:

Mirror of (r, c) → (r, N − 1 − c)

For vertical reflection:

Mirror of (r, c) → (N − 1 − r, c)

Quad symmetry combines both, adding the diagonal mirror (N − 1 − r, N − 1 − c).

Where N = grid dimension (rows = columns), r = row index (zero-based), c = column index (zero-based), B = binary string representation, D = fill density percentage, i = linear cell index.

Reference Data

Grid Size	Total Cells	Unique Patterns	Binary String Length	Common Use
4 × 4	16	65,536	16 bits	Icons, favicons
5 × 5	25	33,554,432	25 bits	QR alignment patterns
8 × 8	64	1.84 × 10¹⁹	64 bits	Chess boards, retro sprites
10 × 10	100	1.27 × 10³⁰	100 bits	Pixel art thumbnails
12 × 12	144	2.23 × 10⁴³	144 bits	Tile patterns
16 × 16	256	1.16 × 10⁷⁷	256 bits	Game sprites, emoji
20 × 20	400	2.58 × 10¹²⁰	400 bits	Detailed pixel art
24 × 24	576	9.87 × 10¹⁷³	576 bits	Toolbar icons
32 × 32	1,024	1.80 × 10³⁰⁸	1,024 bits	Standard icon size
48 × 48	2,304	> 10⁶⁹³	2,304 bits	High-res icons
64 × 64	4,096	> 10¹²³³	4,096 bits	Detailed sprites, textures

Frequently Asked Questions

When you toggle cell (r, c) with horizontal symmetry active, the tool simultaneously toggles (r, N − 1 − c). Vertical mirrors across the horizontal midline. Quad symmetry mirrors across both axes plus the diagonal opposite, toggling up to 4 cells per click. This guarantees mathematical symmetry without manual counting.

The binary string is a row-major sequence of 0s and 1s with length exactly N². For an 8 × 8 grid, it must be exactly 64 characters. Paste any valid binary string into the import field. If the length does not match a perfect square, the tool rejects it with an error. Leading zeros are significant.

Each cell is rendered at the pixel size set by the cell-size slider (default 16 pixels). An 8 × 8 grid at 16px per cell produces a 128 × 128px PNG. For print quality at 300DPI, set cell size to 32 or higher. The SVG export is resolution-independent and always crisp.

Yes. Random generation uses crypto.getRandomValues and then enforces the selected symmetry mode. With quad symmetry on a 16 × 16 grid, only the top-left 8 × 8 quadrant is randomized. The remaining three quadrants are mirrored copies. Fill density can be adjusted via the density slider before randomizing.

The patterns are mathematically generated from your input. No copyrighted assets are embedded. The output is a pure binary matrix visualization. You own the arrangement you create. However, if a pattern coincidentally resembles a trademarked symbol, standard trademark law applies independently of this tool.

The density slider sets a probability threshold per cell during random generation. With symmetry active, mirrored cells are forced to match their source cell, so the effective density converges toward the slider value but may deviate by several percentage points. For a 4 × 4 grid with quad symmetry, only 4 independent cells exist, making fine density control impossible. Larger grids produce closer approximations.