Based on my intuition of what a fractal is, the Lorenz attractor doesn't fit that category for me. A fractal should have some self similarity, but the attractor seems just like two two-dimensional disks which meet in the middle.
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Welcome to MSE. Who claims that the Lorenz attractor is a fractal? – José Carlos Santos Apr 02 '23 at 18:08
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1The Lorenz attractor is a well known fractal as google could easily illustrate. It seems to me a very fair question. To the point @grevel, first off, the Lorentz attractor exists in a 3D phase space. Presumably the "2D disks" you've seen are just projections of the real object. But I agree it is not obvious how the 3D object presents self-similarity. – kevinkayaks Apr 02 '23 at 18:12
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See https://doi.org/10.1016/j.physd.2003.10.006 – lhf Apr 02 '23 at 18:13
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1I suppose if your view of a fractal is "an extended entity which presents structure at arbitrarily small scales", the Lorenz attractor is obviously a fractal. But if your view is "an entity which exhibits self similarity across scales", it is not clear to me how the Lorenz attractor exhibits self similarity. These notes seem relevant https://www.southampton.ac.uk/~mb1a10/lect7.pdf – kevinkayaks Apr 02 '23 at 18:14
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@kevinkayaks I have actually programmed some attractors and I am familiar with their structure. The attractor seems to me like a 2D manifold in a 3d space. – grevel Apr 03 '23 at 20:08
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Mathematics works with definitions, not intuitions. Whether or not the Lorenz attractor is a fractal depends entirely on how you define the word "fractal". https://math.stackexchange.com/a/2677204/ – Xander Henderson Apr 06 '23 at 17:21