# Fractal Types

## Contents

- 1 Introduction
- 2 IFS
- 3 3D Inverse Tree
- 4 Fractal Flames
- 5 Kleinian sets
- 6 PopCorn City
- 7 Conditional Quaternion Julia
- 8 Quaternion Julia
- 9 2.5D IFS
- 10 Julia Set Z^2+C
- 11 Julia Set (Z^2+C)/(Z^2+C1)
- 12 Julia Set Z^3+C vs. IFS
- 13 Quaternion IFS
- 14 Apollonian Set
- 15 Spherical Julia Set Z^2+C
- 16 Spherical Julia Set (Z^2+C)/(Z^2+C1)
- 17 2.7D IFS
- 18 Julia Set II
- 19 Julia Set III
- 20 Gumowski-MiraAttractor
- 21 Spherical Kleinians
- 22 3D Kleinians
- 23 Hyperbolic Tessellations
- 24 Incendia Script
- 25 Fractal Flames 2
- 26 IFS Reflection
- 27 Julia Set IV
- 28 Julia Set V
- 29 Julia Set VI
- 30 Scroll and Double Scroll IFS
- 31 Spiky Julia Variations
- 32 Radiolaria Fractals
- 33 Hyperbolic Tesselations II
- 34 Fractal Pagoda
- 35 Quaternion Julia II

# Introduction

Incendia has 28 predefined fractal types, also it includes an scripting service that allow the user to create more fractals.

# IFS

This fractal type consists in a series of affine transformations (scaling, rotation, shear and translation) that are iterater thought the chaos game. Use the transformation editor if you want to edit the transformation parameters.

# 3D Inverse Tree

This fractal is a 3D version of the classic Inverse Tree created by Paul Bourke.
For more information on this, visit the original source at Paul web site:Inverse Tree

# Fractal Flames

The fractal flames are a variation of the standard IFS algorithm, that includes non-linear functions.
Consult the wikipedia article for more information about Fractal Flames

# Kleinian sets

This Fractal is a 2.5D version of the Kleinian Fractals that appears on the Indra Pearl's book.

Special Thanks to Thomas Ludwig for the C version of the Granny Recipe that appears on the Indra book.

The 3D extrusion is based on the orbit divergence determined by two nearly equal sets.

# PopCorn City

This fractal is a 3D extrusion of the classic Pickover Popcorn Fractal. You can find more information about this fractal type on PopCorn Fractal

# Conditional Quaternion Julia

This fractal type consist basically in a boolean orbit iteration of the Quaternion Julia set.
The boolean iteration alternates two different quaternion julia parameters depending if the orbit is in the positive or the negative real axis.
Hint: Since the orbit is alternated during the iteration, when a baseshape is applied, it appears to be cut.

# Quaternion Julia

This is the basic inverse quaternion julia set.

# 2.5D IFS

This fractal consist in a set of 2D IFS transformations extruded into 3D along the Y axis.

# Julia Set Z^2+C

Classic Inverse Julia Set algorithm for √(*z*
- *c*)

# Julia Set (Z^2+C)/(Z^2+C1)

Inverse Julia Set calculation for

√(z
- c) |

√(z
- d) |

on the (x,z) plane and extruded among the Y axis.

# Julia Set Z^3+C vs. IFS

This fractal consists in a combination of the Inverse Julia Set for ^{3}√(*z*
- *c*)

# Quaternion IFS

A variation of the standard IFS fractal using a quaternion root.

This fractal was selected due to the large amount of spirals that it produces.

# Apollonian Set

This fractal consist in a IFS made of Spherical Inversion transformations.

for more information consult the Wikipedia article Apollonian Gasket

# Spherical Julia Set Z^2+C

2.5D Julia set mapped over a sphere.

# Spherical Julia Set (Z^2+C)/(Z^2+C1)

Spherical version of the 2.5D Julia Set forz^2+c |

z^2+c1 |

# 2.7D IFS

This fractal consists in a extruded 2D flame fractal on the Y axis, on where the flame variations affects also the Y extrusion.

# Julia Set II

A variation on the Inverse Julia Set algorithm for √(*z*
- *c*)

# Julia Set III

This fractal consists in a combination (using complex division) of two Julia II fractals.

# Gumowski-MiraAttractor

This fractal consists in a 2.5D variation of the Gumowski Mira Transformation (often called The Planckton Attractor).
You can get more information from : Computer Simulation on the Gumowski Mira Transformation

# Spherical Kleinians

An Spherical variation of the Kleinian Sets.

# 3D Kleinians

This fractal is a quaternionic variation of the mobius transformations. Since the Stellarnions fractal was deprecated, this type actually uses the slot (#20) that the Stellarnions used.

# Hyperbolic Tessellations

This fractal consists in a series of Spherical transformations that generates an Hyperbolic Tiling.
The Spherical Transformations are specially configured for generating the Hyperbolic Tiling. If you move them (with the transformation editor) you will get a simple Spherical Inversion fractal, like in the Apollonian Fractal.

No Mobius Transformations are used on this fractal (There's a lot of confusion between the 2D mobius transformations and the 3D spherical inversion transformations, as I have noticed recently).

# Incendia Script

Incendia Script consists in a series of scripted commands that are used to define fractals and baseshapes.

When this fractal type is selected, a "Load Script" button becomes visible and it lets you to select an script for Incendia.

Please, visit the Incendia Script section for more information.

# Fractal Flames 2

A variation of the classic Fractal Flame algorithm:

**Classic Algorithm:**

X_{new}=weight1*Function1(x)+weight2*Function2(x)

**Flame2 Algorithm:**

S=weight1*Function1(x)

X_{new}=S+weight2*Function2(S)

# IFS Reflection

This fractal is a variation on the IFS fractals on where the transformations are mirrored in one axis.

# Julia Set IV

# Julia Set V

# Julia Set VI

# Scroll and Double Scroll IFS

This new fractal type is the combination of an standard IFS equation with an Strange Attractor.
The result of this combination is an strange paper like structure (not present in all parameters) that its intersting for volumetric rendering.
Two variations:

- Scroll, Interesting paper like trees.
- Double Scroll, Produces a wide array of fractal papyrus.

# Spiky Julia Variations

This variation of the common Julia set produces a large array of julia like shapes surrounded by long spikes. Two versions are presented,

- Spiky Julia 3D, A full 3D julia set.
- Spiky Julia 2.5D, The standard 2.5D formula.

An interesting feature of this equation is the fact that they rarely present the "infinty hole" problem that are presented in most of the inverse julia sets.

# Radiolaria Fractals

The Radiolarian Fractals are an special case of spherical functions that with the combination of a boolean operator.
They are included on this release of Incendia EX mainly for producing Radiolaria Shells by exporting meshes from the Volumetric Mesh Generator.
Five configurations are presented (each corresponding to a platonic solid):

- Tetrahedral, It often produces interesting Sierpinki Shells.
- Octahedral, Interesting pyramidal shells.
- Hexahedral, Nice for creating distorted builidings.
- Dodecahedral, Some of them produces interesting star like patterns.
- Icosahedral, The most beautiful of the whole group, it produces very intrincated spherical shells.

If you use baseshapes with them, you will notice that in some points (particulary near the center) the baseshapes aren't iterated. Thats due to the boolean operator.

# Hyperbolic Tesselations II

This Fractal type consists in a variation of the standard Hyperbolic Tessellation on where several transformation are added to the upper and lower plane of the tessellation disc.

This approach extrudes the tessellation pattern extending it toward the 3D space.

# Fractal Pagoda

This fractal is a variation of the Castle Fractal.

Instead of using a a circular base, it uses an square one, creating a pagoda-like structure.

# Quaternion Julia II

This variation of the Quaternion Julia fractal adds an hyperbolic transformation to the iteration procedure, creating several spirals on the julia set.

Due to the Hyperbolic transformation, some quaternionic C values produce unstable sets. In those cases, the iteration just returns to the origin