Fractal Origami

Yes that’s right, believe it or not some folks have so much time to spare while rendering fractals, they resort to even more inventive ways to spend their precious fractaltime. Since fractals are all about repetitive forms and the japanese art of Origami is all about repetitive folds, it’s only natural that the two disciplines meet up to form a paper “Menger sponge“, the first 3 dimensional fractal ever discovered.

Mathematical Origins

A “Menger sponge” is mathematical shape first described (or discovered) by Karl Menger in 1928. Basically it’s a cube that repeats its form within itself. It is the 3D form of the 2D mathematical shape called “Sierpinski carpet“, first described by Wacław Sierpiński in 1916. And the “Sierpinski Carpet” in turn is the 2D form of the 1D mathematical shapes of repeating points on a single line called the “Cantor set“, first described by Georg Cantor in 1883. There is a nice article on the history of the “Menger Sponge” and you can read up on just about everything about Karl Menger on the website of the Illinois Institue of Technology (ITT).

Wacław Franciszek SierpińskiKarl Menger

Fractals in paper

In 2006 a level 3 “Menger sponge” made entirely out of business cards made by Dr. Jeannine Mosely was exhibited by the Institute For Figuring (IFF), an organization dedicated to the promotion of “the poetic and aesthetic dimensions of science, mathematics and the technical arts”. More on the IFF on Wikipedia and you can read all about the business card Menger sponge exhibition on the IFF hosted website, which also contains a tutorial on how to make your own and has also has some great other links. And you can see more photos of the contruction of the cube on Flickr or read up on the history of the “Menger Sponge” up to the modern business card version.

More Origami (Morigami?)

Nicholas Rougeux web developer and artist, after being inspired by seeing the exhibition, also made some origami Menger sponges and also made an attempt at a level 4 Menger sponge, an idea which he gave up on in 2009, probably after realizing the amount of work involved no doubt. But still being bitten by the folding bug he is currently attempting a mini Post-It level 3 Menger sponge, you can watch the progress on Flickr. He also wrote a great online tutorial and even dedicated an entire website called “Mengermania” to it with lots of links on the subject.

Another website called “Fractigami” also has some nice photos of Menger sponges as they are being built and you can even buy your own pre-built paper Menger sponge. And Dutchpapergirl posted a nice video (look no hands) of her origami Menger sponge on Youtube and on her website.

“We are the Borigami, resistance to folding is futile” (Saquedon, 2009)

Links:
http://en.wikipedia.org/wiki/Origami

http://en.wikipedia.org/wiki/Menger_sponge
http://en.wikipedia.org/wiki/Karl_Menger
http://www.iit.edu/csl/am/about/menger/about.shtml

http://en.wikipedia.org/wiki/Sierpinski_carpet
http://en.wikipedia.org/wiki/Wacław_Sierpiński

http://en.wikipedia.org/wiki/Cantor_set
http://en.wikipedia.org/wiki/Georg_Cantor

http://www.iit.edu/
http://www.theiff.org/
http://en.wikipedia.org/wiki/Institute_For_Figuring

http://www.theiff.org/oexhibits/menger02.html

http://www.theiff.org/images/menger/sponge%20cube%20instructions.pdf
http://mengermania.c82.net/instructions.php

http://www.c82.net/about.php (Nicholas Rougeux)
http://mengermania.c82.net/index.php
http://www.flickr.com/photos/rougeux/sets/72157621702780335/

http://www.theiff.org/oexhibits/paper07.html (Links)

http://barcodebattler.co.uk/origami/index.htm
http://barcodebattler.co.uk/origami/order.htm

http://www.youtube.com/watch?v=2vuD4PbOBSI
http://www.youtube.com/user/dutchpapergirl
http://www.l.van.breemen.scarlet.nl/origami/origamieng2.html#origamifractaleng

Fractal Food Romanesco

John Walker wrote a great article called “Fractal Food, Self-Similarity on the supermarket shelf” about fractals in nature in general and specifically one of the most recognizable fractal shapes in nature, namely the shape of the “Romanesco (broccoli)“. The article explains the basics on how the shape can be replicated with a mathematical system called “Cellular Automata” and it also has some great photos.

Scientific Paper

Sang-Hoon Kim wrote a related scientific paper called “Fractal dimensions of a green broccoli and a white cauliflower” on the subject in 2004 at the Mokpo National Maritime University (MNMU) in Korea.

 

3D Broccoli

Aleksandar Rodic made a great 3D animation using a technique called procedural modeling to build the similar mathematical shapes of the Romanesco. He even wrote an article on how the 3D broccoli animation was made.

Links:
http://www.fourmilab.ch/images/Romanesco/
http://en.wikipedia.org/wiki/Romanesco_broccoli

http://en.wikipedia.org/wiki/John_Walker_(programmer)
http://aleksandarrodic.com/

http://www.fourmilab.ch/cellab/
http://en.wikipedia.org/wiki/Cellular_automata

http://aleksandarrodic.com/?page=broccoli
http://en.wikipedia.org/wiki/Procedural_modeling

http://www.iamu-edu.org/members/mnmu.php
http://www.mmu.ac.kr/
https://fractuality.files.wordpress.com/2009/10/fractal-dimensions-of-a-green-broccoli-and-a-white-cauliflower.pdf

Sierpinski shapes in shells