3 Dimensional Fractals and
the Search for the 'true' 3D Mandelbrot -
CONTINUED

The Mandelbox Fractal:
A New Landscape of Architectural & Organic Qualities

This is a robust and diverse 3D fractal which often expresses elements of familiar traditional 2D fractals in its intricately textured surface and interior. The Mandelbox is VERY new to the scene, less than a year so far - and the results are breathtaking.

Mandelbox (also known as Amazing Box and tglad's formula) was discovered in 2010 by Tom Lowe, in a flash of insight about applying recursive spherical folding transformations to generate fractals. It was further developed collaboratively 'hive-mind style' at Fractal Forums.

Another great development not by a university lab, but by the genius creative folks at Fractalforums.com. Go there to see what is literally the cutting edge of fractal discovery on a daily or even hourly basis. Really!

Image credits: Top Row: bib from FF, hgjf_radiolaria, & Miqel -
Lower 2 Rows: all by Miqel (Michael Coleman) w/ Mandelbulber

Math of the Mandelbox:
"It is defined in a similar way to the Mandelbrot set as the values of a parameter such that the origin does not escape to infinity under iteration of certain geometrical transformations. It can be defined in any number of dimensions, though 3 dimensions is most common for drawing images of it."
~ (Wiki)

A Quick Zoom into a Mandelbox Mandelboxes come in many shapes and types. One of my favorites is the negative 1.5 box. (scale -1.5)
It seems to have an interesting habit of expressing other familiar fractals within it's complex surfaces.
In this sequence I will focus on the ornament at the corner, which contains a wild variety of shapes & textures.
The other regions of the -1.5 box are quite interesting too, but not as diverse as the corner.
(NOTE: I used Mandelbulber to render all of the following images on this page) .

Scale -1.5 Mandelbox

Zoom into corner and Left Face of the cubic fractal

Continuing zoom into Corner

Following into the edge the corner starts to
assume a more regular form

3D Apollony Fractal

Location: -1.5 Mandelbox

Eventually this corner approximates a Kleinian circle packing fractal, and is also very similar to
the Apollony Gasket
(zoom of previous image, but shown in neutral color to accentuate the form)

the Kleinian circle inversion fractal
& standard Apollony Fractal

- which brings us to the next cool thing about this fractal -

An Interesting Property of the Mandelbox:
Many Familiar
Fractals appear
on it's surface
often in scenic & unique combinations.
koch, menger, sierpinski, apollony, cantor, maskit, etc.
All of these are expressed
in the
unmodified -1.5 box, unless otherwise noted -
submit pix & comparison examples if you have/find others

"also called the conformal disk model, is a model of n-dimensional hyperbolic geometry in which the straight lines of the hyperbolic geometry are segments of circles contained in the disk orthogonal to the boundary of the disk."

The great artist MC Escher popularized the use of
hyperbolic tiling in art and math

Menger Sponge / Sierpinski Carpet Hybrid

Location: -1.5 Mandelbox

Menger Sponge is a cubical fractal created by recursively dividing the cube into equal parts and removing the center.

3D Menger & 2D Sierpinski Fractals

Sierpinski Triangles

Location: -1.5 Mandelbox (and presumably other scales)

Sierpinski Triangles merging into a surface.

One of the simplest fractals, just triangular subdivisions.

Fractal Trees, Sierpenski & Menger Surfaces

Location: -1.5 Mandelbox

A good example of many fractals cohabitating. In the foreground are Branching Trees, the base of the tree is a Sierpinski Carpet & the background
is a slightly jagged Sierpinski triangle surface.

Fractal Tree table by Gernot Oberfell and Jan Wertel
and examples of other fractals seen in image at left

Cantor Dust Treetops Location: -1.5 Mandelbox (and presumably other scales)

this image shows Cantor Dust treetops emerging from the Mandelbox.

This is the 2D square version of the Cantor Set.

IFS & L-System Fractals

Location: Scale -1.3 thru -1.7 Mandelbox

All around the M-box - on surfaces, in textures and at the top of stalks we find IFS and L-System fractals of many different types.
IFS are created by using a starting shape, using several geometric transforms making it into several smaller figures and iterating.
L-systems are kinda the same, they are composed of an angle definition, an axiom, and at least one rule.
The axiom is the initial shape (base) used in the process of generating a fractal.

Dense Fractal Branching Structures

Location: Scale -1.3 thru -1.7 Mandelbox

Densely branched trees of many varieties, connected and disconnected are abundant all around the edges of the M-box.

Drawing and photo of dense Fractal Trees

Von Koch Curve

Location: -1.5 Mandelbox (rotated)

These Koch Curves were found in a certain rotated -1.5 Mandelbox that I call the "Blue Mineral Planet", but are probably present in un-rotated boxes too.

The Koch curve is one of the simplest fractals to make. Starting with a line, keep adding triangles 1/3 the length. Repeat this process x number of times and you have a Koch curve.

Cesaro Curve & Koch Inversions

Location: Scale -1.5 Mandelbox, rotated

i found this variant of a Cesaro Curve in a rotated Mandelbox.

Cesaro is a specific type of Koch Curve.

Koch Snowflake

Location: -1.5 Mandelbox

This is on the side-face of first the corner cube,
I call it "Koch Cove".
On this plateau are many smaller mutated snowflake copies.