Sierpinski Square Step Pyramid

The Sierpinski fractal square based pyramid constructed with a cuboid seed (rather than a pyramid) up to level 7. The progression through the levels is also shown. This fractal was generated using the IFS (iterated function system) approach. In this approach, a base feature is recursively copied and transformed. In the present case there are 5 transformations per iteration involving only isotropic scaling and transformation (no rotation or shear). The 4D transformation matrices are given at the start of the video. This particular fractal has, in the limit of infinite recursion (fractals are limit objects), zero volume and infinite surface area.

The purpose of this video was to demonstrate that the limit object is independent of the specific geometry of the base feature. Initially the object looks nothing like a pyramid, but even after 7 iterations the result begins to resemble the perfectly self-similar object obtained by having a pyramid as the base feature.

Now, go make one out of LEGO!! For level 7 you will need 5^7 = 78,125 cube bricks (although longer bricks could be substituted in some places).

Category:

Education

Tags:

• Sierpinski
• Square
• Base
• Step
• Pyramid
• Fractal
• IFS
• Iterated Function System
• Recursion
• Scale
• Translate
• Linear
• Affine
• Transformation
• Self
• Similar

[YesIamJames]

Just made one of these using lego digital designer. My laptop was stuggling at level 5 so that's as big as I'm going to make it. 3125 blocks used (counting 4 by 2  blocks as 2 lots of 2 * 2 blocks

[khyar]

@YesIamJames Cool, I just had a look at Lego Digital Designer. It looks like a nice bit of software! So, are you going to order the bricks? ;-))) Out of interest how much would your level 5 model cost?

[AdmiralMemo]

Next project: Making this in MineCraft

[YesIamJames]

@khyar I think it ended up costing £380 so jabout 2p per brick. So I won't be buying it but I will be buying a chess set I designed when I can afford it. I ended up creating some more fractals in google sketchup instead though. I'll upload them as a video responce when I get time.