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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.

Tom Lowe's Mandelbox site:http://sites.google.com/site/mandelbox/
Go here for a technical explanation of the math of this fractal by it's inventor.

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!

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

Mandelbulb 3D - Excellent rendering program. Freeware
Mandelbulber - Bulbs, boxes, IFS and more in 3D. Freeware
• Read the Original thread on Fractalforums wherein the Mandelbox is discovered!

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
full box corner

Continuing zoom into Corner
Following into the edge the corner starts to
assume a more regular form
corneryet another corner ... 5 or 6 zooms

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
Apollony corner mandelbox fractal Apollony Gasket

- 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


Poincare Disks & Hyperbolic Tilings                                Location: -1.5 Mandelbox (and presumably other scales)
"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

Mandelbox hyperbolic tilingQuaternion Julia Sets

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

Mandelbox menger sponge

Sierpinski Triangles 
 Location: -1.5 Mandelbox (and presumably other scales)

Sierpinski Triangles merging into a surface.

One of the simplest fractals, just triangular subdivisions.

Mandelbox sierpinski triangles

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

Mandelbox fractal tree

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.

Mandelbox cantor dust

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.

Mandelbox IFS

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

Mandelbox branching

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.

Mandelbox koch snowflake

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.

Mandelbox cesaro 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.
Construction of Koch 'snowflakes'

Mandelbox koch snowflake


Inspiration and information sources:
Tom Lowe http://sites.google.com/site/mandelbox    -  FractalForums.com
    Bib (Fractal art master!) Gallery at FF  -  Bib's Youtube Mandelbox Videos
Kraftwerk/Mandelwerk (another master artist) on Deviant Art     Skytopia http://www.skytopia.com/project/fractal/mandelbrot.html

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