The Share button (bottom right, with Save at the bottom left, in the single-window views) makes an image of what you are looking at (at most twice the display size, so it is quick to make and easy to send) and opens your device's share sheet so you can send it to someone. The message includes a link. Anyone who opens the link sees the same fractal in this program, with the same type, power, position, zoom, rotation, and colors. On a computer without a share sheet, Share copies the link and offers to save the image instead. Links only work when this page is opened from a web address, not from a file on your computer.
The selector at the top left chooses the fractal. Mandelbrot is the classic set: repeat z → z² + c starting from z = 0. Nova is a Newton fractal: each step applies Newton's method for z³ − 1 and then adds c, z → z − (z³ − 1) / (3z²) + c, starting from z = 1 (one of the three roots of z³ − 1).
In Nova, a point is colored by how quickly the steps settle down to a fixed value, and it is black if they never settle within the iteration limit. With Nova selected, Julia shows the Nova Julia set for the c you choose (c = 0 gives the classic Newton fractal), and Orbits shows the path that starts at z = 1 and is pulled toward the roots.
Magnet 1 is the Magnet type I fractal, named for its origin in the physics of magnetism and drawn in Ultra Fractal and elsewhere: z → ((z² + c − 1)/(2z + c − 2))², starting from z = 0. Every starting point is drawn toward the same attracting fixed point, z = 1, so, as with Nova, a point is colored by how quickly the steps settle there (or by how quickly they run off to infinity), and it is black if neither happens within the iteration limit. Magnet 2 is its sibling, Magnet type II: z → ((z³ + 3(c − 1)z + (c − 1)(c − 2)) / (3z² + 3(c − 2)z + (c − 1)(c − 2) + 1))², which settles onto z = 1 even faster. Julia shows the Julia set for the c you choose, and Orbits shows the path of z = 0. The Magnet Julia sets are large, so they open zoomed out. The Magnet fractals have no exponent, so the Power button is hidden while one of them is selected.
Celtic is a variation on the Mandelbrot set: z → |Re(z²)| + i·Im(z²) + c, starting from z = 0. Each step squares z as usual and then folds the real part of the result to its absolute value, which gives the set its sharp, knotted look. It is colored the same way as the Mandelbrot set, and Julia and Orbits work as for the others. As an experiment, the 2 in the formula can be changed with the Power button, which gives |Re(zd)| + i·Im(zd) + c; the usual Celtic fractal is d = 2. The view does not move when the exponent changes, so a larger or smaller set may need a pan or zoom to find.
Pointed Celtic is a cousin of Celtic: z → |Re(z)·|Re(z)| + Im(z)²| + 2i·Re(z)·Im(z) + c, starting from z = 0. Where Celtic squares z and then folds the real part, this rule keeps the sign of the real part when it squares it (x·|x|, so the left half of the plane is treated differently from the right) and adds Im(z)² to it instead of subtracting, then folds the result. The set ends in a sharp point on the right and is knotted on the left, and its Julia sets are tilted, pointed shapes. It is colored like the Mandelbrot set, and Julia and Orbits work as for the others.
As an experiment, the Power button changes the 2 in the Pointed Celtic rule. Written for any exponent d, the rule is: on the left half of the plane (Re z < 0) it is Celtic, |Re(zd)| + i·Im(zd) + c, and on the right half (Re z ≥ 0) the real part becomes the length of zd, |zd| + i·Im(zd) + c. For d = 2 that is exactly the formula above; the other exponents are one way of extending it, and the set may need a pan or zoom to find.
The Power button opens a slider for the exponent: the d in z → zd + c for Mandelbrot, the p in Newton's method for zp − 1 for Nova, the d in |Re(zd)| + i·Im(zd) + c for Celtic, and the d in the Pointed Celtic rule for Pointed Celtic. Drag it anywhere from 1 to 9 (at 1 the Mandelbrot rule z → z + c always escapes, so almost nothing is left of the set). Whole numbers give the classic higher-power sets (4, for example, has three-fold symmetry), and the values between them morph smoothly from one set to the next, with a faint seam along the negative axis where a fractional power is not single-valued. The slider eases onto each whole number so it is easy to land on one, but only within a few pixels, so values just beside a whole number stay within reach. The arrow keys move it in steps of 0.01 (0.1 with Shift) when it has focus. The 10x button next to the value stretches the slider so that its whole length covers just one step between whole numbers (2 to 3, for example, if the power is 2.something), which gives you fine control over fractional exponents, down to three decimal places. The button then reads −10x, and pressing it returns to the normal 1 to 9 scale. On the 10x scale the arrow keys move in steps of 0.001 (0.01 with Shift). Fractional powers take more computing, which is quick on your graphics chip but slower once you zoom deep enough for the precision renderer to take over.
Zoom, pan, rotate, Save, the keyboard shortcuts, and all the color controls work the same for both. Switching type resets the view, and each type remembers its own c, Power, Iterations and color settings.
The same palette colors the orbit, stepping through it from the first point to the last, and one eighth of the Density setting sets how many times it repeats along the way. Julia opens a second window showing the Julia set for the point c you choose in the Mandelbrot set. Orbits shows the path that point takes when you repeat the Mandelbrot rule, with the same path traced faintly over the Mandelbrot set. Julia and Orbits can be on at the same time: the second window then shows the Julia set, and the orbit is drawn over both the Mandelbrot set and the Julia set. Press a button again to turn that mode off; when both are off, you are back to the Mandelbrot set alone. The enlarge icon (four corner marks) at the top right of the second window enlarges it to fill the screen, and pressing it again returns to two windows. Tapping or clicking a window never changes the view. The ? button is available in the Mandelbrot-only view and whenever Julia is on, but is hidden while Orbits is on by itself; turn Julia on or Orbits off to bring it back.
Fifty-nine of the 60 palettes are a selection from the hand-made palettes of Matt Fisher's Making Fractal Wallpapers project (techmatt.github.io/fractals), shared under the CC0 1.0 public domain dedication; Electric is this explorer's own original.
In the two-window view, the lock icon at the bottom left of the Mandelbrot window freezes c: the Julia set and orbit stay as they are while you zoom and pan the Mandelbrot set freely (one finger now pans instead of choosing c). Press the lock icon again to unlock, and c follows your pointer or finger again. Returning to the Mandelbrot set alone also unlocks c.
Whenever Orbits is on, a slider at the bottom of the second window sets how many steps of the orbit to draw, and Animate grows the orbit one step at a time, in the orbit window and also over the Mandelbrot and Julia windows when Julia is on as well. In the Mandelbrot-only and enlarged views, Save downloads the image (or opens your device's share sheet), and the copy button next to it copies a link to the current view. The grid, coordinates, and zoom readout are never part of a saved image.
Each point c on the plane is tested with one rule: start at z = 0 and repeat z → z² + c. If z stays bounded forever, c is in the Mandelbrot set and is drawn black. Otherwise z escapes, and the color shows how many steps that took.
Every c also has its own Julia set. It uses the same rule, but c is held fixed and the starting z varies across the plane. Choose c inside the Mandelbrot set and its Julia set is one connected piece; choose c outside and it shatters into dust. The most intricate Julia sets come from c near the edge.
An orbit is the sequence z₀ = 0, z₁, z₂, … for one c. Inside the set it settles toward a point or loops through a cycle; outside, it flies away.
For Nova, c plays the same role as in the Mandelbrot set: the Nova "Mandelbrot" view maps every c, using the starting point z = 1, and each c has its own Nova Julia set where the starting point varies instead. Mini copies of the Mandelbrot set appear along the boundaries of the Nova view.
Speed and precision: shallow zooms are drawn by your graphics chip. Once you zoom past a few thousand times, the explorer switches to a slower but more precise renderer (turn on Show Axes, Formula, Zoom to see which one is drawing), which keeps working to about 10¹² times, the limit of standard double-precision numbers. Saved images are rendered at full quality either way.
The Fractal Foundation is a 501c3 nonprofit based in Albuquerque, New Mexico. We use the beauty of fractals to inspire interest in Science, Math and Art. We develop engaging “Fractivities” that we use in our educational outreach, and which you can download at our website.
Our award winning planetarium show “First Friday Fractals” is a hit in NM and can be seen at several other planetariums around the world. Contact us for licensing information.
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We use technology to awe and inspire. We hope you enjoy exploring fractals with this web app. Please make beautiful fractals and save them for yourself, and especially share them with others, so that they too can explore the beauty of fractals!
Help us share the beauty of fractals, and help us improve this program. Feedback welcome: feedback@fractalfoundation.org
This explorer counts how it is used, so the Fractal Foundation can see how many people use it and which features matter. The counts are anonymous. For each visit the explorer records: that it was opened, and whether that was from a shared link; each time an image is shared or saved, which type of fractal and which window it was, and how it went; which features were used during the visit (such as Julia, Orbits, and palettes) and for how long; a random ID kept on your device, which is not linked to you and only lets a returning device be recognized; whether you are on a phone, tablet or computer; and your country, which the server works out from your device’s time zone (the time zone itself is thrown away).
It never records your name or email address, the position or settings of the fractal you are looking at, or your images. It does not read or store your IP address, although the web server’s ordinary logs, like any website’s, contain it. Counts are used only to make totals, and are deleted after two years.