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Ready. Draw a cut line inside the highlighted wedge sector!
Advanced Diagnostics & Keyboard Controls

Keyboard Shortcuts:
Enter or Space: Unfold / Fold Toggle
Z: Undo last stroke
C: Clear all cut paths
M: Toggle Mirror Sub-Fold mode

Active Transformations:
Group: Dihedral D6
Symmetry Elements: 12-fold reflection/rotation
Total Active Cut Strokes: 0

Overview: Spatial Geometry of N-Fold Radial and Dihedral Symmetry

The Symmetry Cut Paper Art Simulator is an advanced scientific playground that maps the spatial and mathematical coordinate transforms governing folded paper art, crystallographic point groups, and traditional Kirigami. When physical paper is folded, a continuous planar sheet of 360 degrees (2-pi radians) is structurally divided into repeating wedge sectors. Any physical cut executed on this single wedge is mirrored and rotated across the folded sheets, yielding beautifully complex mandalas, snowflakes, and rosettes upon unfolding. By digitizing this spatial reasoning process, users can interact with multi-dimensional geometric transforms, experimenting with rotation, reflection, and vector mapping in a risk-free, optimized terminal interface.

From a rigorous geometric standpoint, this visualizer simulates two primary families of 2D point groups: cyclic symmetry groups (denoted Cn) and dihedral symmetry groups (denoted Dn). The number of folds, N, represents the rotational order of the point group. In a cyclic group Cn, the design wedge spans an angle of exactly (2-pi / N) radians. When mirror sub-folding is introduced, the system transitions to the dihedral group Dn. This effectively divides the active wedge in half across its central angular bisector (at angle pi / N), adding a mirror plane. The result is a doubling of the total symmetry elements to 2N, allowing bilateral reflection across each crease line. This mathematical configuration is the foundation of structural crystal lattices, snowflake formations (D6 symmetry), and biological flower arrangements, illustrating the profound intersection of abstract mathematics and biological form.

This simulator serves as an interactive laboratory for exploring these transformations. The user draws inside a highlighted boundary slice, representing the open edge of the folded wedge paper. Real-time matrix projection maps the coordinates onto all other rotated and reflected sectors. The tool is optimized for educational environments, structural design analysis, and creative play, rendering complex generative geometry with hardware-accelerated precision.

How to Use: Workflow and Interactive Controls

The terminal control panel is engineered for clear, immediate configuration adjustment, allowing fluid transitions between design states. Below is the step-by-step workflow for configuring and generating symmetrical paper structures:

1. Draw Cuts: Using a mouse, pointer, or direct touch contact on mobile devices, click and drag inside the brightly highlighted wedge on the canvas. The vector points drawn within this sector represent scissors cutting through paper. If Concentric Guide Rings are active, they will display fine polar guidelines at intervals of twenty percent radial distance to assist with drawing exact curves.

2. Select Fold Order (N): Use the range slider in the control panel to set the fold order from N=2 up to N=12. Increasing N creates a narrower drawing wedge but denser, more complex radial rosettes. Changing this slider dynamically adjusts the canvas and updates the current active angle.

3. Configure Symmetry Settings: Enable or disable Mirror Sub-Fold to toggle between dihedral reflection (Dn) and cyclic rotation (Cn). Check Edge Snapping to automatically snap coordinate points falling within close proximity to the wedge boundaries or center, allowing users to cut clean holes or carve completely off the outer rim. Toggle Live Full Preview to display a ghosted, semi-transparent rendering of the fully unfolded design in real-time as you draw.

4. Unfold and Transition: Click the primary green action button, Unfold and Reveal Pattern, or hit Space or Enter on your keyboard to witness a smooth, fanned unfolding sequence. The segments physically rotate and separate to transition from a single slice into the completed circular art piece.

5. Diagnostics and Themes: Access the Advanced Diagnostics collapsible panel to observe active transformation matrices, geometric group designations, and raw cut vector lengths. Choose from stylized themes like Kirigami Red, Snowflake White, Cyberpunk Neon, or Golden Foil to adjust the high-contrast presentation.

Technical Details: Vector Mapping, Touch Normalization, and Audio sonification

The simulation visualizer operates on an HTML5 canvas scaled to match high-resolution screens. At startup, and upon any browser window resize event, the system queries the device pixel ratio (DPR). If this value is valid and finite, the internal dimensions of the canvas are multiplied by the DPR, and the canvas context is scaled by the same factor. This maintains high line fidelity on Retina displays and eliminates pixelated rendering artifacts. All coordinate calculations are performed inside a normalized polar coordinate space relative to the center of the paper, where radius R ranges from zero at the center to one at the outer circumference.

When a user interacts with the canvas via pointer events, the client coordinates are translated into the normalized space using the bounding client rectangle of the canvas element. To ensure safety and prevent divide-by-zero or infinite value conditions, checking limits are enforced:

let rx = x_client - centerX;
let ry = y_client - centerY;
if (isNaN(rx) || !isFinite(rx)) rx = 0;
if (isNaN(ry) || !isFinite(ry)) ry = 0;

Once computed, the radius (r) and the angle (theta) of the contact point are evaluated in polar space. If the coordinate falls within the active wedge bounds, it is logged into the stroke vector. Reflection inside the dihedral Dn sector uses an affine transformation matrix. To mirror a path across the wedge bisector (at angle alpha = pi / N), the rendering pipeline saves the current drawing context, rotates by alpha, scales the vertical axis by negative one, rotates back by negative alpha, and draws the vector path again. Fanning animations utilize a parametric variable (unfoldProgress) that interpolates from zero to one using a non-linear cubic easing function, dynamically altering the rotational steps of the drawn wedges during a requestAnimationFrame loop.

Audio feedback is synthesized natively via the Web Audio API using a sawtooth oscillator. Because browser security guidelines require user interaction to activate audio nodes, the AudioContext remains completely suspended until the user explicitly toggles the Audio button or starts drawing on the canvas. To assist the design process, the output frequency of the oscillator is dynamically mapped to the radial coordinates of the pen contact point: frequency equals 250 plus 850 times (one minus the normalized radius). This provides a progressive, rising frequency pitch as the user cuts closer to the delicate center fold, creating an immersive multisensory laboratory environment.

Future Directions: Procedural Export, 3D Mesh Shading, and Vector Exporting

The future development timeline for the Symmetry Cut simulator focuses on enhancing export fidelity and structural representation. First, we plan to implement a vector SVG export utility that converts the drawn canvas segments into layered paths. This will allow digital artists to export their designs directly to physical cutter plotters, CNC routers, or laser engraving systems. Second, we aim to introduce a WebGL-based three-dimensional rendering viewport. By applying vertex displacement shaders to a flat, tessellated triangular mesh, we can visually model the physical folding and unfolding process in 3D, complete with paper thickness, realistic drop shadows, and folding crease stresses. Lastly, a procedural design generator is planned to allow users to generate abstract mathematically procedurally generated snowflakes and star patterns based on customizable algorithmic variables.

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