A figure such as a Menger Sponge would be practically impossible due to the limited amount of edge on the paper. Same with a 3D checkerboard (assuming you mean cubes connected at the corners) because the total amount of angles adds up to 540 degrees, and paper only allows for a total of 360. That doesn't mean there aren't other tessellations or fractals (such as a Koch Snowflake or a Penrose Triangle) that can't be folded.
Sunburst wrote: triangle tess won't work properly...
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