Balanced folds
Scale 2.0 with minimum radius 0.4 and fixed radius 1.2. Start here to see the overall symmetry and the boundary between interior and escaping points.
Start with a folding experiment
These presets change the formula settings and reset the view. Compare one setting at a time to see how escape bands move around the folds.
Wider sphere fold
Keep scale at 2.0 and increase fixed radius to 1.5. Notice which colour bands shift when a larger region is affected by sphere inversion.
Stronger scaling
Increase scale to 2.6 with the original radii. Compare how quickly the bands and interior regions change as points are pushed outward each iteration.
What this Benesi explorer renders
This site’s implementation is a 2D escape-time view using box folds, sphere folds and scaling. Each pixel starts at a point in a plane; repeated folds determine how soon it crosses the bailout radius. The palette colours that escape count. This view does not provide a rotatable 3D Benesi surface.
Understand the controls
- Scale
- Multiplies the point after each fold. Change it in small steps to compare the resulting pattern.
- Minimum and fixed radius
- Set the regions used by the sphere fold. Keep minimum radius below fixed radius for a useful starting point; increasing either changes which points are rescaled.
- Iterations and bailout
- Iterations limit the number of folding steps. Bailout sets the escape threshold. More iterations can reveal changes near a boundary, while also changing the colour mapping and increasing GPU work.
- Rotation and offset
- Rotate the 2D view or move its centre. These change the region you inspect, rather than adding a third dimension.
Navigate on a phone or desktop
Use Zoom + and Zoom −, or open Controls to adjust zoom and horizontal or vertical offsets. Enable Interact to drag the picture and use a mouse wheel or pinch gesture to zoom. Disable Interact to scroll the page normally. Reset view restores the centre, angle and zoom while preserving your fractal settings.
Keep exploration responsive
If the image updates slowly, lower Resolution scale in Advanced before increasing iterations. Auto Animate changes folding settings continuously; switch it off to inspect a still pattern. Reset all parameters restores the formula and stops animation. A WebGL-capable browser is required; the guide remains available if rendering cannot start.
Explore the separate Benesi Pine Tree formula in 3D, or continue with the Benesi folding guide , compare Mandelbox folds in 3D , or explore Mandelbrot escape-time boundaries .