01 / EDIT THE RULE
DRAG TO CHANGEA tiny rule. A world to explore.
Fractal Explorer.
Start with a line. Shape its middle. Watch your small rule create a whole new pattern.
YOUR FIRST FRACTAL
One small change. A whole new pattern.
Select the orange middle, then drag it. The right side repeats your shape on every smaller line.
START FROM
02 / LIVE RESULT
Your fractal
Iteration 3
Visible detail: iteration 3 of 3
Tiny bends need a closer look. Use Reveal this iteration to follow the curve into smaller bends. The visible-detail readout tells you which levels you can see; the total iteration count can be higher.
+ follows a bend · Scroll or pinch to zoom where you point · Drag to move.Keyboard: + / − to zoom, arrows to move, Home to fit.
Adjust the selected middle section
Line pieces64
Total length1×
Add iterations to repeat your rule on smaller pieces.
The middle is flat, so repeating this rule still makes a straight line. Lift it to create a pattern.
Select a handle, then use arrow keys for small changes. Shift + arrow makes a bigger change. The endpoints stay fixed. Use Fit to see the whole result; this is a finite construction.
CHOOSE AN INVESTIGATION
Where will the rule take you?
Six starting points. Make a prediction, then test it.
01 / EXPLORE THE MAPS
One setting connects two worlds.
Every point chooses a setting c.
Calculating map…
Keep c fixed. Try a starting point z₀.
Calculating map…
THE LINK BETWEEN THE MAPSc = 0 + 0i
Tap a map to investigate a point. Drag to pan · scroll or use ＋/− to zoom. Colours show escape speed; dark pixels have not escaped within the limit.
Fine controls Colour, rendering and exact view coordinates
Jump to a known pattern, then change its settings.
02 / FOLLOW ONE POINT
Watch the rule do its work.
MANDELBROT JOURNEYStart at zero. Test the chosen c.
The point's journey
Real → · imaginary ↑Lines connect the steps in order. They are not a physical path.
Before the first turn
WHAT HAPPENS
Choose a point to see its journey.
WHY IT HAPPENS
The same square-and-add rule is repeated.
Current value · z0
Distance from zero · |z|0
THE EVIDENCE
Does the distance keep growing?
Pin a journey, change one setting, and compare what happens.
Exact steps See the numbers behind the diagrams
03 / PREDICT, TEST, EXPLAIN
One rule, a whole map
WHAT TO WATCH
Your note is kept when you save this setup. Change one thing, then compare the result.
THE RULE
Two coordinates. One number.
The horizontal coordinate is real; the vertical coordinate is imaginary. Squaring combines both: (a + bi)² = (a² − b²) + (2ab)i. It does not simply square each coordinate.
THE TWO MAPS
Setting or starting point?
Mandelbrot asks: “What happens from zero with this c?” Julia asks: “With c fixed, what happens from this starting point?” Click between the maps to test the difference.
THE COLOURS
Beautiful, but not a proof.
Colours show how quickly a point escapes. Dark means no escape was detected within the limit. More steps can reveal a late escape; changing the palette changes only the colours.
Does a new palette move the boundary?