Calculator inputs
Use the form below to generate a polygon-based chaos game dataset and plot it immediately after submission.
Example data table
| Vertices | Jump ratio | Rule | Warm-up | Typical outcome |
|---|---|---|---|---|
| 3 | 0.50 | Any vertex | 20 | Classic Sierpinski-style triangular pattern. |
| 4 | 0.50 | Cannot repeat or choose adjacent | 20 | Square-based structure with strong symmetry. |
| 5 | 0.50 | Cannot repeat the previous vertex | 50 | Pentagonal web with star-like void regions. |
| 6 | 0.33 | Cannot match either of the last two | 50 | Layered hexagonal texture with denser interiors. |
Formula used
Vertex construction
vi = (cx + R cos θi, cy + R sin θi)
θi = rotation − 90° + i × 360° / n
Chaos game update
xk+1 = (1 − r) xk + r vi,x
yk+1 = (1 − r) yk + r vi,y
Distance to center
dk = √((xk − cx)² + (yk − cy)²)
Here, n is the number of vertices, R is radius, r is jump ratio, and the selected vertex changes according to your movement rule.
How to use this calculator
- Choose the number of polygon vertices and set the polygon radius.
- Enter the center, starting point, rotation, jump ratio, and total iterations.
- Pick a movement rule to control how the next vertex is selected.
- Set a random seed if you want reproducible results and a warm-up count to ignore early transient points.
- Press Generate Chaos Game to see the result block, graph, summary metrics, and export buttons above the form.
Frequently asked questions
1) What does the chaos game generator calculate?
It builds vertices of a regular polygon, repeatedly jumps toward selected vertices, stores the resulting points, and summarizes the generated fractal pattern with tables, metrics, and plotting.
2) Why does the jump ratio matter so much?
The jump ratio controls how far each point moves toward the chosen vertex. Small changes can transform the attractor from sparse dust into dense, highly organized fractal structure.
3) What is the purpose of the warm-up value?
Warm-up removes early transient points that still depend heavily on the starting location. Skipping them often produces a cleaner final pattern and more stable summary statistics.
4) Why use a random seed?
A random seed lets you reproduce the exact same vertex choices and generated points later. This is useful when comparing rules, ratios, or different polygon sizes.
5) Can different movement rules create different fractals?
Yes. Restrictions such as avoiding repeats or adjacent vertices change the symbolic dynamics. That changes which regions receive points and can reveal very different geometric patterns.
6) Does the starting point change the final attractor?
Usually the long-run attractor remains the same for a fixed setup, but early points differ. That is why warm-up skipping is helpful when you want the stable structure only.
7) Why is the graph sometimes sampled?
Very large point sets can slow browsers. The plot uses a sampled subset for speed, while CSV export still includes the full generated dataset for analysis.
8) What can I do with the exported files?
CSV is ideal for spreadsheet analysis, filtering, and external plotting. PDF creates a compact report containing settings, summary values, and a preview table of generated points.