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This is the third iteration of a Sierpinski triangle. (Since we paper folders only have a finite amount of material to work with, we of course cannot make true fractals. The iterations do have to end somewhere.) The mathematical relationship between the number of iterations in a Sierpinski triangle and the number of units actually works out rather nicely: a single triangular prism, equivalent to a level 0 Sierpinski triangle, contains 1 triangle per side and nine Sonobe units; thereafter, the Nth iteration will contain 3ⁿ triangles per side and 3ⁿ⁺² units.

 

The interconnectedness of individual triangular prisms in this 3D Sierpinski triangle brings to mind the similar connectivity of individual cubes in my "Purinahedron". In both cases, the edge-to-edge connections are illusory; the "joint" between each pair of prisms is actually built from one unit belonging to the first prism and one unit belong to the second. An ant (or some other diminutive creature) could actually travel throughout the interior of this project without much difficulty, since the whole thing is really one contiguous polyhedron and none of the individual prisms are closed.

 

Those of you who try to fold this will quickly discover that Sonobe units do not like being brought together this way! The mathematics may be sound, but the tension between identical sections of this fractal is great, and were it not for glue, it would certainly come apart at the seams. "Pinching" the level 2 and level 1 sub-triangles together was hard; the last unit I put on (at the bottom right in this photograph) was relatively smooth sailing in comparison.

 

And yes, in case you're wondering, the green paper does indeed glow in the dark. That's why I put the ball there.

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Taken on February 4, 2012