Saturday, June 2, 2007

Spierpinski icosahedron and other fracal objects

I found the following site, in which they have created amazing artworks of Sierpinski icosahedron with modular paper folding.
However this site is in German. I am not sure how they did it. It seems to me they have a summer camp for high school or primary school kids to create this model collaboratively.

Here is the picture of the model they created:

Alternatively, Wikipedia has a long list of fractal dimensions for many different objects:
List of fractals by Hausdorff dimension

Many natural objects exhibit fractal structure too:
Check this amazing Fractal food in supermarket out.

Friday, June 1, 2007

Regular Sierpinski Polyhedra

I found an interesting article on the Sierpinski Polyhedra at "" entitled
"Regular Sierpinski Polyhedra1" by Aimee Kunnen and Steven Schlicker.

They made a detailed study on the Spierpinski polyhedra in general, and also report he fractal dimensions for five regular polyhedra:

1. Sierpinski tetrahedron: log(4)/log(2) = 2
2. Sierpinski hexahedron: log(8)/log(2) = 3
3. regular octahedron: log(6)/log(2) ≈ 2 585
4. Sierpinski dodecahedron: log(2)/log(d/d1) ≈ 2 32958
5. Sierpinski icosahedron: log(12)/log(d/d1) ≈ 2.581926
The meaning of d and d1 is basically the scaling factor I mentioned in the previous post, but for details, please check the original paper. The calculation of this ratio is a geometric problem, which can be solved in principle. Instead of doing complicated calculation, I just performed an estimate by straightforward inspection on the picture I obtained from the scanner.

Now the problem is what the value of log(d/d1) for the Spierpinski buckyball (Spierpinski truncated icosahedron) Chuang and his classmates made is. Or more generally, find out the contruction rules and the corresponding fractal structures, and dimension for Sierpinski Achimedean solids, etc.

Sierpinski buckyball

After a short discussion with Chuang, I now think the Sierpinski buckyball is a better name for this system than the Escher buckyball.

Interestingly, it is not hard to find out the Hausdorff (fractal) dimension of this system:

D=log 90/log scaling factor = log(90)/log(8) ~ log(90)/log(7) = 2.16 ~ 2.31

(More careful measurement indicates the scaling factor is 16 cm/2.5cm =6.5, thus D = 2.4)

where I made a rough estimation for the magnification in order to get the scaling factor from the small buckyball to the large buckyball. Detailed calculation is more involved for the 3-D geometry of truncated icosahedron is needed.

Sierpinski's pyramid from Wikipedia (Fractal dimension = log 4 / log 2 =2):

Escher's buckyball

Chern Chuang and his classmates created this amazing beaded buckyball made from 90 small beaded buckyballs. They are going to present this beautiful artwork as a gift to our chemistry department in the graduation ceremony for NTUCHEM class 2007 this coming weekend. I wish I can have a picture for this event.

As to the name of this design, we call this ball as a buckybuckyball, or level-2 buckyball. Since we can again use this buckyball as a new type of beads to create the next level of buckybuckybuckyball (level-3 buckyball), continuing in this direction recursively, we can have a very complicated fractal structure of Sierpinski type. Also, this kind of artwork containing hidden recursive structures has been first used Escher in his amazing artworks, so it is not a bad idea to call this kind of buckyball as Escher's buckyball, or Escher's ball.

The First Beaded T240 (D5d)

This is the first beaded T240 I made right after I bought the materials from the Little Bear's Mother and made my first C60 last year. This beaded torus was made from 360 10mm faceted beads. It is easy to see that there are some loose ends of threads in this beaded torus since I was not very good at handling these remaining threads at that time. It is not very supprising that I chose monochromic beads for weaving this torus.

At that time, I am still not familiar with the best algorithm for weaving a torus, not to mention the possibility to put beads with different colors on the pentagons and heptagons. Therefore, this first T240 has only one color.