The black area is the Mandelbrot set itself, which are the values of the complex number c such that if you repeatedly iterate z ← z ² + c starting at z =0 the value of |z | stays less than 2. The colored areas surrounding the black of the Mandelbrot set are values of c for which z eventually goes to infinity, with the colors indicating how many iterations until |z |>2

You will also notice a subtle shading which seems to show relief.
This is done by considering that the colored bands to be contours on a map
and shading it accordingly to show the slope. ⊕ For a detailed discussion of the algorithm see Hill-Shading the Mandelbrot Set

Out of these simple mathematical operations we get these incredibly complex and fascinating images.

See also monochrome renderings .

Images ⊕ 0 Here we see the whole beetle of the set,
with a large cardioid body and
a circular head with a proboscis.

⊕ -0.19854 +i1.10018 Here we zoom in 250× into a smaller copy of the beetle that's attached
on a filament that extends upward in the positive imaginary
direction from the main cardioid.

⊕ 0.28439 -i0.01359 Here we looking at a 1,000× magnification of a swirling area on the positive real buttocks of the beetle, with many small beetles floating around.

⊕ 0.28443 -i0.01273 This is close by the above image at the same 1000× magnification, focussing on a spiral and a whirlpool.

⊕ -0.79619 -i0.18323 This beautiful and delicate 1,000× magnified structure is in the neck (the gap between the main cardioid and the large circle).

⊕ -0.748986 +i0.055768 Zooming in more to 4000× magnification deep in neck shows a kind of lacework alongside a spiral.

⊕ -0.658448 -i0.466852 Also at 4,000× magnification, on the shoulder of the beetle is this
snowflake-like pattern

⊕ -0.715181 -i0.230028 Higher up on the shoulder at 8,000× magnification we find these two spirals trapping a beetle between them.

⊕ 0.35938 -i0.09109 Just off the buttox of the main beetle at 8,000× there is this square structure.

⊕ -0.745263 -i0.113042 And even further up into the cleft between the head and body, zooming in a bit to 10,000× we find this beautifully symmetric pair of spirals around a beetle.

⊕ -0.48271895 +i0.53096651 Deep in a cleft off the shoulder at 500,000× magnification is this highly textured structure.

⊕ 0.148865892 +i0.642407724 And now at 2,000,000× magnification we first zoom into the hips
of the large beetle and then into a filament coming off one of the small
beetles to get this filament which has an even smaller beetle embedded in
it.

⊕ -0.596360941 +i0.662749640 Here. magnified a bot more than 2,000,000× we have zoomed into a
beetle off the shoulder of the main beetle and then zoomed into the small beetle's anus
to reveal this interesting patchwork, and unusual asymmetric distorted beetle

⊕ 0.250006 Now we are at more than 8,000,000× magnification
deep in the anus of the main beetle, right along the real axis
very close to the asymptotic z =¼ point of the cardioid

⊕ 0.37001085814 +i0.67143543269 This beetle in a radiating crown at 17,000,000× magnification
is embedded in a filament off the main beetle's hips

⊕ -1.99991175020 And now we are down at 130,000,000× magnification at
the very left tip of the beetle's proboscis in the negative real axis.
The curves are generally smooth here, but lo!, here's another beetle

⊕ -0.999208533756 +i0.302364353482 Here we dive into a filament on the head of beetle where we find
a small beetle and dive into a filament on its body toll we get down to
an even smaller beetle at 130,000,000× magnification.

⊕ -1.67440967428 +i0.00004716557 We're now at 1,000,000,000× (billion times!) magnification deep in the neck of a small beetle that is out on the proboscis of the main beetle

⊕ 0.022141191996 +i0.652476712897 We're now just a little beyond 1,000,000,000× magnification off the hips of the main beetle, down through several layers of radiating sunbursts.

⊕ -0.139975337339 -i0.992076239092 Here we are deep (2,000,000,000×) within a filament off the main body passing several levels of beetle as we zoom down.

⊕ -0.749988802386 +i0.006997251234 Here we are at 4,000,000,000× magnification in the narrow
crack of the neck where we see this decorative square.

⊕ 0.432456684114 -i0.340532264460 Now we are at 5,000,000,000× magnification and off
the buttocks we find this lovely rosette.

⊕ -0.1827180644448 +i0.6614075685543 Now we are at 70,000,000,000× magnification in the
hip, deep down a spiral we find this area sandwiched between two spiral areas.

⊕ -1.5367297734418 +i0.0048808425334 Still at 70,000,000,000× magnification in a little spine off the probiscus, we find this square structure.

⊕ 0.44518774607945 -i0.24365489644603 Now we are at 140,000,000,000× magnification off the buttocks of the main beetle, at the very end of the probiscus of a small beetle.

⊕ -1.16706911260479 -i0.29169804375501 Still at 140,000,000,000× magnification on the buttocks of a small beetle that is off the ears of the main beetle.

⊕ -1.25499918723309 +i0.32578646797963 Staying at 140,000,000,000× off the head of a beetle that itself is off the head of the main beetle

⊕ -0.72339816896641 +i0.19406503819411 Yet again at 140,000,000,000× on the shoulder of a beetle that is off the shoulder of the main beetle is this richly textured velvet.

⊕ -0.59636078463178 +i0.66274965378928 Here we we revisit the unusual asymmetric distorted beetle we saw at a bit more
than 2,000,000× (2 million) magnification above and zoom into its neck and further down
to 200,000,000,000× (200 billion) magnification to find this landscape.

⊕ -1.76853740279050 +i0.00054004950144 Here we go deep 1,000,000,000,000× (trillion!) magnification) into the neck of a small
beetle on the proboscis of the main beetle to find this squashed spiral.

⊕ -0.743643887037159 +i0.131825904205312 Here we are at 1,000,000,000,000× magnification
in the neck of a beetle that is in the neck of the top-level beetle

⊕ -1.6744096746527182 +i0.000047165698790702304 Here we go 2,000,000,000,000× magnification) into the
neck of a smaller beetle along the proboscis to find this square structure.

⊕ -0.16089413126577512 +i1.0376191282366105 In the neck of a small beetle on a filament we zoom down
through a succession of spirals till we reach 2,000,000,000,000× magnification.

⊕ -0.7345612674879727 +i0.3601896136089664 This linear feature at 2,000,000,000,000× magnification is
the proboscis of a beetle that's on the filament
of a beetle that is off the shoulder of the main beetle.

⊕ -1.6744096746527197 +i0.0000471656987933374 This lovely structure in the neck of a beetle on the proboscis,
at 5,000,000,000,000× (5 trillion) is about as deep as we could go
using the previous implementation of the almondbread software which
used 64-bit double-precision values. Zooming more starts to reveal
spacial quantization caused by numerical effects.

⊕ -0.24482766477704102 +i0.8132688560198666 Modifying the almondbread software to use 80-bit double-precision values allows zooming to this unsusal short linear feature at 1,000,000,000,000,000× (one quadrillion) magnification.