Drag, scroll and pinch to dive into the Mandelbrot set, rendered live on your graphics card. Switch to any Julia set, try eight color palettes, jump to famous spots like Seahorse Valley and download the view as a wallpaper up to 4K.
Last updated:
Drag to pan, scroll or pinch to zoom, double-click to zoom in.
Advertisement
Advertisement
How to use the Mandelbrot Set Explorer
Explore: drag the image to pan, scroll or pinch to zoom, or double-click a spot to zoom in 3× (Shift + double-click zooms out).
Jump somewhere with the Famous places menu, or type a center and zoom and press Go to coordinates.
Try a Julia set: open the Julia set tab, pick a preset, or press Pick c on the Mandelbrot set and click any point.
Style it: choose a palette, adjust color density and shift, and raise max iterations if edges look blobby when zoomed in.
Save a wallpaper: choose a size such as 4K or Phone and press Download PNG.
How the picture is computed
Each pixel stands for a complex number. For the Mandelbrot set the explorer takes that number as c, starts from z = 0 and repeats z → z² + c. If |z| grows past 16 (|z|² > 256) the point has escaped and gets a color from the palette based on the number of steps; if it never escapes within the iteration limit it is treated as inside the set and drawn black. The large main cardioid and the circle to its left are detected with exact formulas and skipped, which makes the whole view much faster.
Written out with real numbers, one step of z² + c with z = x + yi and c = a + bi is: x' = x² − y² + a and y' = 2xy + b.
Famous Mandelbrot and Julia locations
Name
Center or constant
What you see
Main cardioid
around −0.25 + 0i
The heart-shaped body of the set, points whose orbit settles to one fixed value
Period-2 bulb
circle of radius ¼ around −1 + 0i
The large disc on the left, orbits that alternate between two values
Seahorse Valley
about −0.745 + 0.113i
Spiraling seahorse tails between the cardioid and the period-2 bulb
Elephant Valley
about 0.29 + 0.015i
Rows of trunk-like spirals on the right of the cardioid
Antenna mini Mandelbrot
about −1.7549 + 0i
A small complete copy of the whole set on the left-hand spike
Douady rabbit (Julia)
c = −0.123 + 0.745i
A Julia set made of "rabbits" with three ears around each point
Dendrite (Julia)
c = i
A branching tree with no interior
San Marco (Julia)
c = −0.75
A shape like the domes of St Mark's Basilica reflected in water
Tips for great fractal wallpapers
The most detailed areas are always along the boundary of the set. Zoom toward thin filaments and spirals rather than into the black interior or the empty outside.
If colors look noisy deep in a zoom, lower the color density; if large areas are one flat color, raise it.
For phone wallpapers, choose a tall size: the export keeps the center and the zoom of the shorter side, so a tall image shows more above and below.
Copy link saves the exact view in the address so you can come back to it or share it.
Frequently asked questions
What is the Mandelbrot set?
It is the set of complex numbers c for which the sequence z₀ = 0, zₙ₊₁ = zₙ² + c stays bounded. Points inside the set are drawn black. Points outside escape to infinity, and they are colored by how many steps it takes |z| to pass a cutoff, which creates the colorful bands around the edge.
How is a Julia set different from the Mandelbrot set?
Both use z → z² + c. For the Mandelbrot set, c changes across the picture and z starts at 0. For a Julia set, c is one fixed number and the starting z changes across the picture. Choosing c inside the Mandelbrot set gives a connected Julia set; choosing c outside gives a scattered dust of points.
Why do the edges look blobby when I zoom in?
The iteration limit is too low for that depth, so points that would eventually escape are wrongly drawn as inside. Raise Max iterations or keep the automatic option on. Higher values show more detail but take longer to draw.
How far can I zoom?
The fast WebGL mode uses 32-bit floating point numbers, which run out of precision around ×100,000. Beyond that the explorer switches automatically to 64-bit double precision on the CPU, which goes to about ×10¹³ before the image breaks into blocks. Deep views render progressively, so give them a moment.
What does smooth coloring do?
Instead of using the whole number of iterations, which produces visible steps between color bands, smooth coloring adds a fraction based on how far |z| went past the cutoff: ν = n + 1 − log₂(log |z|). The result is a continuous gradient.
Can I use the images as wallpapers?
Yes. The images are generated from math in your own browser, so they are yours to use. Choose a size such as 1920 × 1080, 3840 × 2160 (4K) or a phone size; the export keeps your current center and zoom.
Where is the Seahorse Valley?
Seahorse Valley is the deep cleft between the main cardioid and the large circular bulb on its left, around −0.745 + 0.113i. Zooming there reveals spirals shaped like seahorse tails. Elephant Valley is the cleft on the right side of the cardioid near 0.29 + 0.015i.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.