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I might be entirely off an my assumptions, but the following has led me to a question.

  1. The Mandelbrot set is contained by an border of infinite length.

  2. Said border is 2-dimensional.

  3. The Hilbert space-filling curve also is 2-dimensional and infinite in length.

  4. The Hilbert curve covers an area of 1 square unit.

Can the Mandelbrot border be assigned an area? Am I justified in saying that the Hilbert curve is has area 1? If it is two dimensional, that seems reasonable. What terminology is used for non-integer dimensions? Also, for integer dimensions. What do you call the measure of the 4D interior of a tesseract? 5D?

I'm in 10th grade, but very good at math (taking Calc BC), so as long as answers don't go to deep into metric spaces, measure theory, topology, et cetera, I should be able to follow them.

Thanks in advance!

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