Gallery 2D




New for 2024:

Mini-Mandelbrot sets frequently pop up when we deal with various dynamical systems, not necessarily quadratic, and many of them come with intricate backgrounds as shown in the example below. Many fractal artists hunt for such mini-Mandelbrot sets as the backgrounds come in an endless varieties of patterns and dazzle the viewers.



A Mini-Mandelbrot Set

See Figure 3.3 of Fractal Home for Detail.



As shown in § 5 of the main website, we can find Julia sets with interesting shapes from atoms of the Mandelbrot set ℳ and we can do the same with a mini-Mandelbrot set ℳ ' using the canonical one-to-one correspondence between the atoms of ℳ and the atoms of ℳ '. For example, the following image shows the Julia set of a parameter, which is just outside of ℳ ' but near an atom around one of its "ears" in the preceding image. Here, we look at ℳ ' as a "Warty Snowman."







That's the idea used in the images shown below. They are given by a different mini-Mandelbrot set and reflect its dazzling backgrounds.















































New for 2023:


The fractals shown below feature the Julia sets of parameters p chosen from near the cusp (0.25, 0) of the Mandelbrot set. We call them "Lions" from the way the "global" figures appear, although each of them has "local" parts that look like "Elephants." Whether they are lions or elephants, our object is to show the beauty and intricacies of some of the Julia sets generated by quadratic dynamical systems.

Each of the Julia sets shown below is plotted by goldish colors that are intended to look like "gold dust," if it is a Cantor set comprising totally disconnected infinitely many points, or "gold filaments," if it is connected in just one piece. According to the Fatou-Julia Theorem for any quadratic dynamical system (with a unique critical point), these are the only two possibilities for the Julia sets. Even though the two situations are completely opposite of each other, we often find it hard to tell whether or not a Julia set is connected by just looking at its computer-generated images.







"Cloisonné Lion I"

Technical Description: The Julia Set of p = (0.27, 0.004)
On a z-Canvas Centered at z0 = (0, 0)
Generated by zn+1 = zn2 + p








"Cloisonné Lion II"

Technical Description: The Julia Set of p = (0.2691079, 0.004)
On a z-Canvas Centered at z0 = (0, 0)
Generated by zn+1 = zn2 + p








"Cloisonné Lion III"

Technical Description: The Julia Set of p = (0.2691, 0.004)
On a z-Canvas Centered at z0 = (0, 0)
Generated by zn+1 = zn2 + p








"Cloisonné Lion IV"

Technical Description: The Julia Set of p = (0.251594, 0.0001)
On a z-Canvas Centered at z0 = (0, 0)
Generated by zn+1 = zn2 + p



Let's look at the Julia set shown above in "Cloisonné Lion IV." First off, it is a totally disconnected Cantor set of points. Is it easy to tell from the picture? Not really.

The non-goldish colors are used to paint the complement of the filled-in Julia set, hence, the purplish color at the center of the image indicates that the orbit of z0 = (0, 0) diverges to ∞. Since the orbit coincides with the critical orbit of p, it follows that p does not belong to the Mandelbrot set, or equivalently, the Julia set of p is a Cantor set by the alternative definition of the Mandelbrot set. In short, we can tell if the Julia set is connected or not by looking at the color of the image at z0 = (0, 0) on some occasions.

So, math helps sometimes. Another fun fact in fractal plotting is that changing the value of a parameter p very slightly may result in a drastically different image. Compare the parameters of the images shown in "Cloisonné Lion IV" and "Cloisonné Lion V." An ugly fractal may turn to beauty by a tiny modification, so we should always be patient!





"Cloisonné Lion V"

Technical Description: The Julia Set of p = (0.251594, 0.00011)
On a z-Canvas Centered at z0 = (0, 0)
Generated by zn+1 = zn2 + p








"Cloisonné Lion VI"

Technical Description: The Julia Set of p = (0.25000316315, 0.00000000895)
On a z-Canvas Centered at z0 = (0, 0)
Generated by zn+1 = zn2 + p



The Julia sets shown in "Cloisonné Lion VI" and "Cloisonné Lion VII"are discussed in § 5 of Stories about Fractal Plotting. Despite their similar appearance, the former is a totally disconnected Cantor set while the latter is connected as one piece.





"Cloisonné Lion VII"

Technical Description: The Julia Set of p = (0.25000316374967, 0.00000000895972)
On a z-Canvas Centered at z0 = (0, 0)
Generated by zn+1 = zn2 + p








"Gold Lion I"

Technical Description: The Julia Set of p = (0.281220509396875, 0.011380997984375)
On a z-Canvas Centered at z0 = (0, 0)
Generated by zn+1 = zn2 + p




The parameters p for "Gold Lion I" and "Gold Lion II" are chosen from Figure 3.1 of Stories about Fractal Plotting. The former is from the interior of one of the mini Mandelbrot sets and the latter from the complement of the Mandelbrot set. Because of the alternative definition of the Mandelbrot set, it again follows that the Julia set in "Gold Lion I" is connected as one piece while the one in "Gold Lion II" is a totally disconnected Cantor set. Observe that the patterns and textures of the two Julia sets look identical.





"Gold Lion II"

Technical Description: The Julia Set of p = (0.28122084115, 0.011380987775)
On a z-Canvas Centered at z0 = (0, 0)
Generated by zn+1 = zn2 + p







"Jady Lion I"

Technical Description: The Julia Set of p = (0.29605, 0.01885)
On a z-Canvas Centered at z0 = (0, 0)
Generated by zn+1 = zn2 + p









"Jady Lion II"

Technical Description: The Julia Set of p = (0.296555,0.020525)
On a z-Canvas Centered at z0 = (0, 0)
Generated by zn+1 = zn2 + p




The Julia sets named "Jady Lion I" and "Jady Lion II" shown above are both connected. In each of them, the jady colors illustrate the filled-in Julia set indicating that its interior is not empty. It implies that its boundary, namely the Julia set, is not a Cantor set and that the Julia set was born from a circular atom of the Mandelbrot set. What are the periods of the atoms? They are 28 = 14 × 2 and 42 = 14 × 3, respectively.








"Turquoise Lion I"

Technical Description: The Julia Set of p = (0.281215625, 0.0113825)
On a z-Canvas Centered at z0 = (0, 0)
Generated by the Mandelbrot Equation








"Turquoise Lion II"

Technical Description: The Julia Set of p = (0.2513363, 0.0000889)
On a z-Canvas Centered at z0 = (0, 0)
Generated by zn+1 = zn2 + p








"Ruby Lion"

Technical Description: The Julia Set of p = (0.282311250, 0.012143125)
On a z-Canvas Centered at z0 = (0, 0)
Generated by the Mandelbrot Equation









"Esmeralda Lion"

Technical Description: The Julia Set of p = (0.281150625,0.011546875)
On a z-Canvas Centered at z0 = (0, 0)
Generated by the Mandelbrot Equation












New for 2022:


The next seven images are Mandelbrot fractals given by the logistic equation zn+1 = fp(zn ) = p(1 - zn) zn and the initial value z0 = 0.1 or 0.2. They are called

"Pearly Elephants"

to feature the special effects on the boundary of the logistic set given by the initial value that is different from the critical point z0 = 0.5 of fp. The first image shown below comes from a small rectangular neighborhood of p = (3.001273, 0.075920) and all other fratals are found in a nearby vicinity.































































Other 2D Fractals:



Please visit Stories about Fractal Art where many of our fractals are shown as examples.