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Navigating the Mandelbrot Set

Many of the pictures shown on this website —such as the one above— are 2D images obtained from various areas of the renowned Mandelbrot Set. This emblematic figure, recognized as one of the most complex mathematical objects, is the epitome of the fractal art style. For many years, people have explored its intricacies in detail naming some of its features after recognizable shapes —an effective way to learn by association.

A few years back —probably up until the end of a certain era of the Internet— Paul Derbyshire, a master of M-set cartography, had a web page called: Quick guide to the Mandelbrot Set. It identified regions within the standard figure using colloquial names, with thumbnails to illustrate their whereabouts. I used to link that tutorial on my old website (actually, I still have the link) because I thought it was a good way for beginners to jump to cool-looking spots that almost guarantee nice shots.

Unfortunately, the page disappeared some time ago —as far as I know, its author no longer maintains a website. I’m not as well-versed as Derbyshire when it comes to this subject, but here’s my attempt at compiling a similar list. I won’t go into the math behind these regions; I’m more focused on their visual characteristics and location relative to the main figure. These features repeat indefinitely, gaining additional details the deeper you zoom. They also replicate within each minibrot, where the process starts all over again.

Mandelbrot set tour guide

In the following descriptions, the numbers in square brackets identify each spot on the Mandelbrot set. Keep in mind that this figure is symmetric about the x-axis, so labeled areas on the upper part have corresponding regions on the lower section. In Figure 1 (below), some features are marked on both sides to make this more apparent.

Figure 1. Mandelbrot set features
Figure 1. Mandelbrot set features
  • Main Cardioid, a.k.a the Continent [1] – This is the large central body, also referred to as a period-1 component. The name “cardioid” derives from its heart-like shape.
  • Bulb, Bud or Lobe – Round shapes attached to pretty much everything and everywhere around the edges. They are classified depending on their periodic cycle as:
    • Period-2 [2] – Primary bulb, to the left of the main cardioid; also known as The Big Disk or The Big Circle.
    • Period-3 [3] – To me, this is where symmetry unravels, as this periodic bulb appears at both upper and lower sides of the figure, mirroring one another.
    • Period-4 [4]. This one appears three times: twice to the right of both period-3 bulbs following the edge toward Elephant Valley (see below), and once to the left of the period-2 bulb.
    • And the list of periodic bulbs goes on and on, forming a bifurcation pattern.
  • Tendrils or filaments [5] – All those lightning-like threads visible everywhere —these tendrils connect other features (see Mini Mandelbrot).
  • Elephant Valley [6] – The rightmost cusp marking the heart’s cleft. Spiral structures in this area resemble elephant trunks, inspiring the name.
  • Seahorse Valley [7] – The cusp formed between the main cardioid and the period-2 bulb. Zooming into this area reveals spiral structures that look like seahorses—that’s how it got its name.
  • Double Spiral Valley [8] – An example of this feature appears on the western side of Seahorse Valley. As its name suggests, it’s full of spirals with two arms. Similar structures can be found throughout the figure, and as additional arms emerge, they evolve in number into Triple Spiral Valley, Quad Spiral Valley, and so on.
  • Scepter Valley [9] – Another cusp, this time between the period-2 and period-4 bulbs, to the left of the main cardioid. Sometimes called Seahorse Valley West, this region features spiky structures resembling scepters, hence the name.
  • Double Scepter Valley [10] – A zone similar to Scepter Valley, atop the period-3 bulb above the main cardioid, where a pair of scepters mirror each other.
  • Antenna or Needle [11] – A long, pointy filament extending opposite the main cardioid; it is full of ever-shrinking copies of the Mandelbrot set arranged like pearls on a string.
  • Mini Mandelbrot, a.k.a. Minibrot, Midget, Baby Mandelbrot, Baby Brot, Mandelbrotie [12] – Smaller satellite version of the main set; one of the most obvious examples of self-similarity on the Mandelbrot fractal. Like bulbs, they too are related to a particular period. Each minibrot, also called island or mu-molecule, contains all the features already mentioned, repeating the structure on a smaller scale. Because of their abundance, I should have placed this feature higher on the list (maybe in third place?), but I wanted to end this trip (pun intended) with one of the best “items” in the whole figure (in my opinion of course). They range from perfect copies, to distorted, stretched or skewed ones. All of them are surrounded by intricate tendrils and spirals, making each one quite unique.

Features 1-4 and 12 include some mathematical reasoning in their description (specifically regarding the periodicity), but that is necessary, as they represent key elements within the Mandelbrot set —see Devaney (n. d.), Fowler & McGuinness (2019), and McClure (2023) for a scientific overview. The rest are informal names given to them due to common usage. Additional labels exist, often inspired by how certain structures appear, repeat or behave rather than pinpointing specific locations. For example, there are octopus arms for spirals appearing around minibrots, snake pits for areas full of twisting tendrils, galaxies or whirlpools for seas of spirals, paperclips for spiral loops that bend back onto themselves; eyes, backbones, rings, clouds, and the list goes on.

Illustrated guide

Let’s now zoom in on these features. I’ll follow the same order in which I introduced them, prioritizing structure over style. I will skip the main cardioid, periodic bulbs and tendrils, since they’re clearly visible in the graph above.

Elephant Valley
Figure 1. Elephant Valley.

Elephant Valley cusp; this is the cleft forming the heart-shaped contour of the main cardioid (right side of the figure).

Elephant Valley
Figure 3. Elephant head and trunk.

The “elephant trunk” that gives its name to Elephant Valley is clearly visible in this image. Notice how the bulbs march along the edge.

Seahorse Valley
Figure 4. Seahorse Valley.

Seahorse Valley cusp: The black right side is the main cardioid or Continent, and the left side is the period-2 bulb, also known as The Big Disk or The Big Circle. Inside the cleft, the right side corresponds to Seahorse Valley proper, while the left shore delineates Double Spiral Valley.

Seahorse Valley and Double Spiral Valley
Figure 5. Seahorse Valley (right) and Double Spiral Valley (left).

A closer look at a section of Seahorse Valley: Now both the seahorse shape (on the right) and the double-arm spiral (left) are clearly defined.

Seahorse
Figure 6. Seahorse.

A closer inspection reveals a seahorse composed of intertwining tendrils, spirals and minibrots, with bulbs anchoring them to the main cardioid.

On the opposite side lies a double spiral in the adjacent Double Spiral Valley (this side is to the left of Seahorse Valley).

Scepter Valley
Figure 8. Scepter Valley.

Scepter Valley cusp: The dark side on the right corresponds to the period-2 bulb left of the main cardioid (The Disk), while the opposite side corresponds to the period-4 bulb left of the period-2 bulb previously mentioned (language, like the Mandelbrot fractal under examination, is recursive too).

Scepter Valley
Figure 9. Scepters at both sides of Scepter Valley.

A closer look at Scepter Valley reveals a cusp that mimics Seahorse Valley, attaching scepter spikes to seahorses (right) and double spirals (left) alike.

Scepter Valley zoom
Figure 10. Enlarged scepter in Scepter Valley.

And here is a deeper zoom at one of the large scepter spikes in the Scepter Valley region.

Double Scepter Valley
Figure 11. Double Scepter Valley.

Double Scepter Valley cusp: The lower dark area belongs to the period-3 bulb atop the period-1 component, also known as the main cardioid, while the upper dark zone corresponds to the period-6 bulb above the period-3 bulb atop the period-1 component, also know as the main cardioid (you’ve got the recursive rhythm now, right?).

Double scepter
Figure 12. Double scepter.

Here is a closer examination of a mirror-image scepter in Double Scepter Valley.

The Antenna
Figure 13. The Antenna.

Using a slightly different color scheme to accentuate the filaments, this image shows the Antenna (or Needle, among other names) on the left side of the Mandelbrot set.

Minibrots
Figure 14. Minibrots.

The Antenna is filled with minibrots —literally, every knot in this section is a miniature Mandelbrot replica.

Minibrot in the Seahorse Valley
Minibrot in the Seahorse Valley
Minibrot in the Elephant Valley
Minibrot in the Elephant Valley
Minibrot in the Scepter Valley
Minibrot in the Scepter Valley

Finally, in this sequence, there are three minibrots in three different areas in the Mandelbrot set; from left to right: distorted minibrot in Seahorse Valley, a similarly skewed minibrot in Elephant Valley, and a normal minibrot in Scepter Valley. Click each one to enlarge.

Cul-de-sac

This concludes the —infinitely detailed— tour of the Mandelbrot set’s most iconic features. From the unmistakable cardioid to the charm of recursive minibrots, each landmark plays a role in the complex geometry of this figure. For further exploration, consider downloading a fractal generator and creating your own photo tour. You’ll soon learn each step into a deeper level contains a universe of its own. For software recommendations, visit the Fractalware section. For more stylish looks at minibrots, visit Forged ad Infinitum in the Passageway. And for an in-depth mathematical description of these features or additional names, refer to the references below.

Though the tour ends here, this fractal never does. Every zoom reveals more patterns, as every detail repeats endlessly. The Mandelbrot set will always have more to show.

References

Picture showcase (top): Mini Mandelbrot picked from within the Elephant Valley region inside another minibrot located at the Antenna. Generated in Fractal Forge. This one is only a recoloring of an old 2000 picture by the same name.

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