The Advanced Mandelbrot Period-Bulb Calculator graphically calculates the period of bulbs along the boundary while inspecting the Mandelbrot set. This interactive explorer lets you click any point to identify the period of the surrounding hyperbolic component (the period tells you how many steps it takes for a point’s orbit to repeat when iterating the Mandelbrot formula). When detection succeeds, the tool reports the estimated period and, optionally, an approximate rotation number. It can also display the corresponding Julia set for each point selected.
Special note: The technical details that follow are based on information provided by the AI assistant used to develop this application.
How to use this app
- Click any point in the Mandelbrot set (black region) or the colored areas around it.
- The app analyzes its period, displays its value and shows you the orbit path.
- The orbit visualization shows how the point’s values cycle.
- Zoom in by clicking “Zoom to Point” button after selecting a point to explore that region in detail.
- Use “Reset view” to return to the full Mandelbrot set.
- Pan by clicking and dragging.
- Switch between different color modes and palettes to highlight different aspects of the fractal.
- Export image to save your current view as a PNG file.
- Use the detail level slider to control the computational precision, from fast rendering (quick exploration and low detail), to extreme precision (for deep zooms and ultra detail).
Tip #1: Start with lower detail levels for smooth navigation, then increase detail when you want to analyze specific points more accurately. Higher detail levels take longer to compute but reveal finer structures.
Tip #2: The large circular bulb on the left has period 2. Try clicking it first to see how period detection works!
Fun Fact: The boundary of the Mandelbrot set is infinitely complex and has a fractal dimension of approximately 2. This means it’s more than a one-dimensional line but less than a two-dimensional area. It exists somewhere in between!
Advanced features
Next to the Analysis tab, there are three more with additional features:
- Julia Set preview: Every point in the complex plane has a corresponding Julia set. When you select a point, the app shows what its Julia set looks like. Points inside the Mandelbrot set have connected Julia sets, while points outside have disconnected “dust” Julia sets.
- Bookmarks: Save interesting locations to revisit later. Each bookmark stores the exact coordinates, zoom level, and view bounds so you can return to your favorite spots.
- Jump to famous locations: Jump to well-known regions of the Mandelbrot set, such as Seahorse Valley, Elephant Valley, and various spiral formations. These presets help you discover the most visually striking areas.
Getting the best results
- Click near the center of a bulb for clearest period detection.
- For very small bulbs, zoom in first, then increase detail level.
- Try different color modes to see different features more clearly.
- Watch the Julia set preview. Connected sets indicate you’re inside the Mandelbrot set.
- Export images before zooming further to preserve interesting views.
- Still need help? Try the tutorial: Click the “Show Tutorial” button for an interactive walkthrough of all features.
Technical details
The app iterates the Mandelbrot formula and tracks the orbit of complex numbers. It detects when the orbit enters a repeating cycle by checking if any value repeats within a small tolerance. The period is determined by counting how many steps occur before the cycle repeats.
The detail slider controls two parameters: Render iterations determines how many steps to check before deciding a point is in the set (affects visualization), while Period detection determines how many steps to analyze when looking for repeating cycles (affects accuracy). Both scale exponentially, ranging from around 100 iterations at low detail to over 25,000 at maximum detail.
Limitations
- Very deep zooms may encounter JavaScript’s floating-point precision limits.
- Extremely high periods (100+) can be slow to detect and may be approximate.
- Points very close to the boundary may give inconsistent results.
- Performance varies by device. Mobile devices may struggle with ultra-high detail.
Troubleshooting
The app is running slowly. Lower the detail slider. High detail levels can be very computationally intensive, especially on mobile devices or when deeply zoomed in.
I’m getting incorrect or inconsistent periods. Try increasing the detail level. Points near bulb boundaries or very small structures need higher iteration counts for accurate detection. Also, extremely deep zooms may hit JavaScript’s floating-point precision limits.
The visualization looks blocky or pixelated. Increase the detail level for smoother rendering. Low detail settings use fewer iterations, which can create a stepped appearance in the color gradients.
Clicking doesn’t show a period. You may have clicked outside the Mandelbrot set. Points that escape to infinity don’t have a period. Try clicking on the black regions or the circular bulbs attached to the main shape.
The zoom is too deep and everything looks wrong. Click “Reset View” to return to the full Mandelbrot set. Very deep zooms may encounter precision limitations.