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Hm... What about a fractal shape with no right angles? Is that feasible? Or do you just approximate it with a fractal arrangement of rectangles?
by mannykannot 3y ago
Hm... What about a fractal shape with no right angles? Is that feasible? Or do you just approximate it with a fractal arrangement of rectangles?
- deleted 3y ago[deleted]
- calamari4065 3y agoI can't say I've ever done calculus on a fractal, but my guess is that most fractals have a fixed surface area ratio. Since fractals have infinite textural depth, there's a hard limit to their surface area compared to the overall area. But what you're asking about is the inaccuracies in the Reimann Integral https://en.m.wikipedia.org/wiki/Riemann_integral https://en.m.wikipedia.org/wiki/Riemann_integral The short version is that you pick a reasonably small size for your rectangles, and it mostly averages out. The discrepancy is so small that it can largely be ignored. But for argument's sake, you can simply imagine infinitely many infinitely small rectangles. That will fit the curve perfectly, but is impractical to compute. So we accept the slightly less accurate value.
- fspeech 3y agoA fractal has measure zero in the space it's embedded in. It obviously can not contain any object of nonzero measure of that space.
- Filligree 3y agoDoes that include space-filling curves? …do those count as fractals, if the fractal dimension would be an integer?