Question

Is there such a thing as a 3d fractal?

Sure there are 3D fractals, in the sense that a fractal surface can bend and extend out of a 2 dimensional plane. Take for example the fractal "sponge created by iterations of this method: Starting with a cube divided into 9 congruent smaller cubes, take the 4 cubes that are the center of each face and remove. Now perform the same operation on each of the 5 remaining cubes. Repeat ad infinitum. You are left with an object in a 3D space, but what is its volume? Well, zero, actually, and this is where your question becomes a tricky one. Does an object with zero volume exist in 3D? This is why most mathematicians consider fractals to have intermediate dimension, in this case somewhere between 2 and 3.
Answered by: Rob Landolfi, None, Science Teacher, Washington, DC

Broken glass, mountains and human lungs are some 3d fractal examples. Although someone could say that a fractal can not have an integral dimension as 2 or 3, but something in between.

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