Fractal World
Where Mathematics Becomes Art
Fractals are the most obvious shape illustrating where mathematics becomes art. Though purely math in character, fractals instantly draw us in with their other worldly beauty. Eye-catching and wonderful. Our Fractal Generator tool allows you to explore five of the most common of fractal sets.
A fractal is not drawn. It is calculated. A mathematician takes a simple equation and maps it on 2D or 3D. What emerges from it the shape of infinity, self-similar at every scale, laced with spirals and loops and lacy that never repeats exactly and never ends.
The mathematician Benoit Mandelbrot, who works extensively with fractals coined the word fractal in 1975, liked fractals to the “the geometry of nature.” or the character of “rough.” The branching of a rivers, the meanders of coastline, a lightning bolt; all of them are fractals. Self-similarity down or up by scale. Why does it exist? It is the natural endpoint of a system that tends to the most efficient for survival.
So, we see that in fractals, the same efficiently written down no matter the size. What you zoom down, you do not meet a point where repeating of structure ends, you do not reach a resolution limit. The detail continues. Spirals appear inside spirals, and repeat within spirals. A year of zooming and you still would not meet the end.
drag to pan
right-click zooms out
tap to zoom in · double-tap out
Triangle to Select
How to Use Fractal World
Navigating the Fractal Plane
Our Fractal World tool lets you explore five fractal families. All are based on a mathematical equation in recursive form. For this tool we have Mandelbrot, Julia, Burning Ship, Tricorn and Cubic Z3
Use your mouse wheel to zoom toward or away from the cursor position, the tool zooms into exactly where you are pointing. Click and drag to pan across the plane. Right-click anywhere to zoom out by half. On a touch device, pinch to zoom and drag with one finger to pan.
The heads-up display in the upper left shows your current center coordinates in the complex plane and your zoom level relative to the default view. As you zoom deeper, the coordinate display updates in real time so you can record interesting locations.
Palettes
The six palettes were defined by graphics researcher Inigo Quilez. Each is a smooth iterated gradient so there are no hard edges.
Iterations
The iteration slider controls how many times the formula is applied before going on to the next iteration. Higher values show more detail but render slower. The default of 300 is a good balance.
Flow Mode
Flow animates cycles color across the fractal. Flow works on any fractal type. Press F or click the Flow button to toggle it. Press it again to stop.
Saving Your Work
Click Save PNG or press S to download the current view at full canvas resolution. The file is named with a timestamp so future saves do not overwrite each other. For high-resolution saves, expand your browser to full screen before saving.
Keyboard Shortcuts
Click the canvas first to give it focus, then: R resets the view to the default position and zoom. F toggles Flow animation. S saves the current frame as a PNG.
Frequently Asked Questions
What is a fractal?
A fractal is a figure that has a similar shape up and down the scale. The is computed mathematically and usually involves iterative computations.
What is a complex number and why are they needed in fractals?
We need complex numbers to define a fractal because fractals exist in two dimensions, and its numbers are expressed as having spin (rotation) and scale (size) at the same time. A real number can express one of these dimensions, but not two, so we have to use an imaginary number to express it in one figure.
Why do fractals iterate in forever?
It is never possible to reach the end of a fractal shape because zooming into any edge reveals the same complexity repeating, there is always more structure to find, . Unlike natural fractals, which stop the physical limit, a mathematical fractal has no smallest scale; it divides and recurses forever.
Can I use images saved from Fractal World in my own work?
Images made with our tool are yours. Use your renders freely for personal projects, art, writing, social media, or education. Attribution to GoRhyme is appreciated but not required.