I've currently moved over to blogging at http://mathcraft.wonderhowto.com/

Here I will be posting about 4 times per week and will have a project every week with more in depth information and a how to component and templates. There will also be a user forum for people to submit pictures and ideas.

Thanks! I hope to see you there!

Fractal Papercraft

This post combines two of my hobbies, making things out of paper (papercrafting) and exploring and building fractals. For those of you that are not familiar with fractals, they are objects that have repeating patterns at different sizes. In other words when you zoom in on a part of the object it looks like a smaller copy of the entire object. Here are 3 different fractals that I have built out of paper.

Sierpinski Tetrahedron (Tetrix). This is the 3D analog of the Sierpinski Triangle. It is formed by taking 4 tetrahedrons and gluing them together vertex to vertex forming a single hollow tetrahedon. You then repeat with 4 or these units, and then repeat with 4 of those units and so on forever. This tetrahedron is iteration 3 with 64 individual tetrahedrons and is about 9 inches tall. One interesting thing about this fractal is that if you look at it from the correct angle it looks like the Sierpinski Triangle. Another is that if you look at it from 2 certain angles it actually looks like a completely filled in 2D surface. (I actually want to put an image on it so that you only see it from the proper angle)

3D fractal tree. This is a 3D fractal tree of my own design. Every branch is a triangular prism and each branches into 3 smaller branches at a defined angle. It has 4 stages of branches meaning there are 81 of the smallest branches, 27 of the next smallest, 9 of the next, 3 of the next, and finally 1 trunk. If it went another iteration then it would intersect itself...meaning I didn't design it properly.

Koch Surface. This is a 2D analog of a variation of a 1D Koch curve or Koch Snowflake. Each main box has 5 boxes that are 1/3 the length, width, and height placed in the center of each face. It is also surrounded by 8 other boxes 1/3 the length, width, and height. This process is then repeated. The one I made has an initial box that is about 4 inches on a side, 5 more boxes 1/3 this size (I didn't surround it by the 8 other boxes), 5x13=65 1/9 this size and finally 65x13= 845 1/27 this size. The smalles boxes are actually made from 1/8 bass wood and painted.


Words of Life said...

Thanks for sharing these! Fractals are fascinating!

JonBoy said...

Awesome work!

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