[mod] Fractal World Generation [fractal_worlds]

[Mator]

imgur album - WIP GitHub repository

Introduction
Several years ago, I watched a music video on YouTube of a flight through a mandelbox. The architectural features of the mandelbox inspired me to want to generate fractals in Minecraft. I achieved that goal through a WorldEdit script, though what I truly wanted were massive, world-size fractal objects to explore. Minecraft's 256-height limit and obtuse modding API was a major turnoff to this, so I made my own Voxel Engine in Java. Making this engine taught me a lot. I later made a much faster and cleaner engine in C++ using SDL. I've generated some awesome fractal images with this engine. (1, 2, 3)

However, part of what I wanted was to be able to interact with a fractal world. I recently found out about Minetest, which has brought me full circle back to my original idea - generating fractal worlds to mine, explore, and build in.

The Project
The project will come in two stages. The first stage is using the Lua modding API to make world generation code as a proof of concept. The second stage is native integration into the base game as new world generation modes, including biome and seed support.

Progress

Stage 1 (in progress)

• Menger Sponge ✔
• Cantor Dust ✔
• Mandelbulb ✔
• Mandelbox ✔
• Octahedron Fractal ✔
• Quadratic Koch Surface ✔
• 3d Greek Cross Fractal ✔
• Fractal Pyramid ✔
• Jerusalem Cube The fractal uses an irrational scaling value, and thus has no good volumetric representation.
• 3d Koch Curve Triangles really don't translate well volumetrically. Going to skip this for now.
• Koch Cube ✔
• Koch Snowflake Cube ✔
• Penta Sponge ✔
• H Sponge ✔
• Cube Tree (and variations) ✔
• Hypercomplex Julia Set
• 3d Hilbert Curve ✔
• 3d Peano Curve
• Pythagoras Tree
• Sphere Tree ✔

Stage 2

• Mandelbox: Generates a Mandelbox the size of the entire world.
• Mandelbulb: Generates a Mandelbulb the size of the entire world.
• Fractal biomes: Create fractal biome palettes and support multiple fractal "biomes".
• Parameter seeds: Map seed values to fractal parameters. o
• Fractal garden: Generates a variety of fractals at various sizes throughout the world.

Any help is appreciated!
- Mator

[paramat]

Good to have you active in Minetest and i'm happy to help.

[azekill_DIABLO]

awesome! purely awesome! are the colors random or they represent some kind of patern?

[Mator]

Good to have you active in Minetest and i'm happy to help.

Thank you paramat! :)

awesome! purely awesome! are the colors random or they represent some kind of patern?

The colors are mapped from the iteration that the point exceeded the fallout. When generating fractals such as Mandelboxes and Mandelbulbs you have an algorithm which applies a formula to the point you are testing repeatedly until its magnitude exceeds a certain value, at which point you consider it to be headed to infinity and part of the set. The iteration at which this occurs is mapped to a block in a palette.

Spoiler

E.g. When generating a power 8 mandelbulb I usually use the parameters:

Power: 8
Fallout: 1024
Base iteration: 4
Max Iteration: 16
Palette: [Stone, White Wool, Light Gray Wool, Gray Wool, Black Wool, Red Wool, Orange Wool, Yellow Wool, Green Wool, Dark Green Wool, Blue Wool, Cyan Wool]

For a given point in the mandelbulb, we create a vector with components mapped to the region (1,1,1) to (-1,-1,-1), then apply the mandelbulb formula to it. The first time we apply it is "iteration 0". If the magnitude of the vector exceeds 1024, we stop and would place a block, except our base iteration is 4, so we place nothing. We continue this until we reach iteration 4. If a point exceeds the fallout value at iteration 4 we would place the first block in our "palette", which in this case is Stone. If a point exceeds the fallout value at iteration 5 we would place the second block in our palette, which in this case is White Wool. This continues until we reach the maximum iteration. Anything which doesn't exceed the fallout value by the max iteration is considered not a part of the set.

[azekill_DIABLO]

ok thanks!

[Andrey01]

Interestingly... This is second CaveRealms mod in a screenshot

[Mator]

Interestingly... This is second CaveRealms mod in a screenshot

What do you mean, and which screenshot?

[u19503]

That looks like chaos but in the good way I like it :)

[Desour]

That looks like chaos but in the good way I like it :)

I don't think, that it looks like chaos. It's totally structured. Chaos would be like dust.

[Mator]

Octahedron Fractal generation working. Added images to the imgur album in the OP. Here's a few good ones:

[azekill_DIABLO]

magnificient.

[Mator]

I have three new fractals today.

Fractal Pyramid:





Quadratic Koch Surface:





Greek Cross Fractal:






Additional images have been added to the imgur album linked in the OP as well.

[azekill_DIABLO]

it's just amazing!

[Krock]

These are impressive! It would be lovely if these fractals could be adjusted in a way to use them as a map. Directly thinking of the mandelbrot set.

[Mator]

it's just amazing!

Thanks man. Fractals are fun. :)

These are impressive! It would be lovely if these fractals could be adjusted in a way to use them as a map. Directly thinking of the mandelbrot set.

That's stage 2. I don't have mandelbrot set planned, however, because it's a 2d fractal. I don't really see the point of 2d fractals in a 3d graphics engine. The mandelbulb, however, is similar to the mandelbrot set. It's not a canonical 3d representation of the mandelbrot set, but it has some mathematical similarities.

[paramat]

We already have 3D/4D mandelbrots used as maps in mgfractal.

[Mator]

We already have 3D/4D mandelbrots used as maps in mgfractal.

That's actually a 2d mandelbrot set with height projected based on iteration instead of using iteration to determine color (I think). It is cool though. :)

EDIT: Actually no, I have no idea what that is. o_o

EDIT 2: I think it's some form of Julia Set maybe? Not sure.

[paramat]

2D with height as iteration is too boring for me.
Some are 3D slices of 4D hypercomplex mandelbrot sets, some are 3D hypercomplex mandelbrot sets, each has the corresponding julia option.
All are from http://www.bugman123.com/Hypercomplex/index.html

All parameters are here https://github.com/minetest/minetest/bl ... mple#L1523

Code: Select all

#    Choice of 18 fractals from 9 formulas.
#    1 = 4D "Roundy" mandelbrot set.
#    2 = 4D "Roundy" julia set.
#    3 = 4D "Squarry" mandelbrot set.
#    4 = 4D "Squarry" julia set.
#    5 = 4D "Mandy Cousin" mandelbrot set.
#    6 = 4D "Mandy Cousin" julia set.
#    7 = 4D "Variation" mandelbrot set.
#    8 = 4D "Variation" julia set.
#    9 = 3D "Mandelbrot/Mandelbar" mandelbrot set.
#    10 = 3D "Mandelbrot/Mandelbar" julia set.
#    11 = 3D "Christmas Tree" mandelbrot set.
#    12 = 3D "Christmas Tree" julia set.
#    13 = 3D "Mandelbulb" mandelbrot set.
#    14 = 3D "Mandelbulb" julia set.
#    15 = 3D "Cosine Mandelbulb" mandelbrot set.
#    16 = 3D "Cosine Mandelbulb" julia set.
#    17 = 4D "Mandelbulb" mandelbrot set.
#    18 = 4D "Mandelbulb" julia set.
#    type: int min: 1 max: 18
# mgfractal_fractal = 1

[Mator]

Got four new fractals today, all fractals I hadn't known about before now. Generation algorithms for these fractals are all very similar to the Menger Sponge.

Snowflake Cube





Penta Sponge





H Sponge





Koch Cube

[the_raven_262]

I see you are doing an awesome job here :)
I thought to ask if fractals like the ones here (3d ones) could be made as minetest maps: http://hirnsohle.de/test/fractalLab/(The fractal library section)

[Mator]

I see you are doing an awesome job here :)
I thought to ask if fractals like the ones here (3d ones) could be made as minetest maps: http://hirnsohle.de/test/fractalLab/(The fractal library section)

Hi the_raven,

The only fractal there which I don't already have here is the SphereSponge. I don't know how good it will look volumetrically, but if I can find a description of the algorithm I'll give it a go.

The main thing that site has differently is a bunch of transforms. Some of these are entirely new to me. The biggest difficulty is in figuring out the transform's implementation.

- Mator


EDIT: Found the original posting about the SphereSponge fractal. Will see if I can implement the pseudocode provided there.

[the_raven_262]

Thanks for the answer, I don't know about making a spherical fractal in a cubic world, might seem odd, a lot.
Do you plan on using minetest's ore generation or are you going to use the fractal for it?
Also, I have no idea how does that site add colour to the fractals, but as it colours certain sections of the fractal I can guess colouring has something to do with the fractal itself, would be nice to implement biomes this way.

[Mator]

Thanks for the answer, I don't know about making a spherical fractal in a cubic world, might seem odd, a lot.
Do you plan on using minetest's ore generation or are you going to use the fractal for it?
Also, I have no idea how does that site add colour to the fractals, but as it colours certain sections of the fractal I can guess colouring has something to do with the fractal itself, would be nice to implement biomes this way.

I think the SphereSponge can work in a cubic world so long as I set the minimum sphere radius sufficiently high.

I'm planning on taking both a "natural" approach to biome/ore generation (via MineTest algorithms or similar) and potentially a fractal approach as well. We'll see how it goes once I get there. :)

[Mator]

Added cube tree fractal after taking a bit of a break.

[the_raven_262]

Awesome! I really hoped you didn't abandon this project :)