Mandelbrot Set Explorer: Zoom, Julia Sets and Fractals

A free Mandelbrot set explorer: drag and zoom through the famous fractal in your browser, see the Julia set of any point live, trace the orbit under the pointer, try the Burning Ship, Tricorn and Multibrot sets, cycle the colours and test any number with the point calculator.

Julia set of this point
Point under the pointer–move over the picture
Width of the view–
Centre–

Drag to move · scroll, pinch or double-click to zoom · click a point to inspect it. In the Mandelbrot picture each point is a number c; the colour tells how fast the orbit of z → z² + c runs away. Black points never do.

Colours
Colour speed:
Colour shift:
Detail (iterations):
Quality

The Mandelbrot set

Take a complex number c and start with z = 0. Square it, add c, and repeat: z → z² + c. For some c the numbers stay small forever; for the others they shoot off to infinity. The Mandelbrot set is the set of all c that stay small. It is drawn black here. Everything else is coloured by how quickly the orbit escapes, and the colours trace out the unimaginably intricate edge of the set.

iteration \(\displaystyle z_{n+1} = z_n^2 + c,\qquad z_0 = 0\)
in the set when \(\displaystyle |z_n| \le 2\ \text{ for every } n\)
Julia set \(\displaystyle z_{n+1} = z_n^2 + c,\qquad z_0 = \text{the point},\ \ c\ \text{fixed}\)

What you are looking at

  • The cardioid and the bulbs. The big heart-shaped region is the set of c whose orbit settles to one fixed point. The round bulb to its left is where the orbit settles into a 2-cycle, the bulbs above and below into 3-cycles, and so on: every bulb has its own period, and between them sit infinitely many smaller copies of the whole set (look for the little “Mandelbrots”).
  • Smooth colours. A plain escape count gives bands. This explorer uses a fractional count (the “normalised iteration count”), which gives smooth gradients you can cycle through with Colour speed and Colour shift.
  • Julia sets. Every point c of the plane has its own Julia set, a different fractal. If c is inside the Mandelbrot set that Julia set is one connected piece; if c is outside it falls apart into dust. The little window shows the Julia set of the point under your pointer, and “Open this point's Julia set” explores it full size.
  • Other fractals. The same recipe with a twist gives the Burning Ship (take absolute values first), the Tricorn (use the complex conjugate) and the Multibrot sets (cube or fourth power instead of the square).
  • The orbit. Tick “Show the orbit under the pointer” to see the path 0, c, c² + c, … of the point you are on: it spirals in for points inside the set and flings out for the rest.

Facts and limits

The set sits inside a disc of radius 2 (if |z| ever exceeds 2, the orbit escapes). Its area is about 1.5066, but its boundary is so crinkly that it has dimension 2, like a filled region, even though it is a curve (Shishikura, 1998). It is connected, but all attempts to prove that it is also locally connected (the MLC conjecture) have so far failed.

This page computes with ordinary 64-bit numbers, so you can zoom in to a view about 10-13 wide (a magnification of over 1013: if the whole set were the size of the Earth you would be looking at a patch smaller than a grain of sand) before the numbers run out of digits and the picture turns into blocks. Zooming further needs special arbitrary-precision methods that this page does not use. Interior points that never escape are drawn black after the chosen number of iterations; near the edge a few points may be coloured as outside that are really inside, and “Detail (iterations)” is the dial for that.

Things people ask

  • Why is it a fractal? Zoom in anywhere on the edge and you find new structure, never a smooth curve, and the structure keeps repeating in new variations at every scale.
  • Is it infinite? The edge is infinitely detailed. The set itself fits in a box 2.5 wide and 2.5 tall.
  • Why do some zooms go black or blocky? Either the iteration count is too low for the depth (raise Detail) or you reached the limit of 64-bit numbers.
  • Can I save a picture? Yes: the camera button saves exactly what you see, and “Copy link” saves the place so you can come back to it.

Everything runs on your device; nothing is uploaded. Black Hole Simulator · 2D graphing calculator · CAS calculator · More MES tools

Comments
Advertisement