Visualizing 4D Fractal Slices

This application visualizes 2D slices of a 4-dimensional fractal object. The fractal is based on the iteration of the complex-number equation Z_n+1 = Z_n² + C, which is famous for generating the Mandelbrot and Julia sets.

Here, we explore a 4D space where both the complex parameter C and the initial value Z₀ can vary. The four dimensions are the real and imaginary parts of both numbers: (Cr, Ci, Zr, Zi). These correspond to our visualization's (X, Y, Z, W) axes.

How it Works:

• 4D Space: Every point in our 4D space represents a unique fractal definition. The point (x, y, z, w) maps to (Cr, Ci, Zr, Zi), defining the constant `C = x + iy` and the starting point `Z₀ = z + iw`.
• Slicing: Each canvas shows a 2D slice of this 4D space. The intersection of all slices is shown by a crosshair. For example:
  • The CrCi (XY) slice shows a Mandelbrot-like set. Here, `Cr` and `Ci` vary across the canvas. If the slice offsets `Zr` and `Zi` are set to 0, you get the classic Mandelbrot set.
  • The ZrZi (ZW) slice shows a Julia set. Here, `Zr` and `Zi` vary across the canvas. The `Cr` and `Ci` values are fixed by the slice offsets, and each combination reveals a different Julia set.
  • Other canvases show hybrid slices through this fascinating 4D object.
• Interaction: You can explore the fractal by panning the view center, changing the slice offsets, zooming, and rotating the entire 4D space. The color of each pixel is determined by how quickly the iteration `Z = Z² + C` escapes to infinity.

Controls:

• Direct Canvas Interaction:
  • Zoom: Use the mouse wheel over any canvas to zoom in or out.
  • Set Slice Offsets: Click and drag on any canvas to move its crosshair, which directly changes the slice offsets. This allows you to explore different "layers" of the 4D fractal.
• Slice View Layout:
  • The slice views can be reordered by clicking and dragging their titles.
  • Each slice view has a rotation slider below it to rotate the 4D space in that specific plane.
• Control Panel:
  • Fractal Parameters: Adjust `Max Iterations` for detail and `Bailout Radius`.
  • View: Adjust the `Zoom` and pan the 4D view with the `Center X/Y/Z/W` sliders. `Reset Center` [C] moves the view back to the origin. `Recenter Crosshair` [F] moves the view to the current crosshair position.
  • Slice Offsets: Fine-tune the crosshair position with sliders. `Reset Offsets` [O] moves the crosshair back to the center of the view.
  • Rotation Actions: `Apply Rotations` [A] "bakes" the current slider rotations into the base orientation. `Reset Rotations` [R] reverts all rotations to the default.
  • State Management: Save and load your exploration. `Copy/Import View State` handles the core view (center, zoom, rotation). `Export/Import Full State` saves and loads a complete JSON file with all settings, including the layout of the slice views.