A mathematician's guided tour through higher dimensions

#67 · 🔥 188 · 💬 59 · 2 years ago · www.quantamagazine.org · Anon84 · 📷
In essence, he proved that it is impossible to put a higher-dimensional object inside one of smaller dimension, or to place one of smaller dimension into one of larger dimension and fill the entire space, without breaking the object into many pieces, as Cantor did, or allowing it to intersect itself, as Peano did. Although Brouwer's work put the notion of dimension on strong mathematical footing, it did not help with our intuition regarding higher-dimensional spaces: Our familiarity with three-dimensional space too easily leads us astray. The surprising realities of high-dimensional space cause problems in statistics and data analysis, known collectively as the "Curse of dimensionality." The number of sample points required for many statistical techniques goes up exponentially with the dimension. An intuitive way to think about Hausdorff dimension is that if we scale, or magnify, a d-dimensional object uniformly by a factor of k, the size of the object increases by a factor of kd. One surprising consequence of Hausdorff's definition is that objects could have non-integer dimensions. Lastly, some readers may be thinking, "Isn't time the fourth dimension?" Indeed, as the inventor said in H.G. Wells' 1895 novel The Time Machine, "There is no difference between Time and any of the three dimensions of Space except that our consciousness moves along it." Time as the fourth dimension exploded in the public imagination in 1919, when a solar eclipse allowed scientists to confirm Albert Einstein's general theory of relativity and the curvature of Hermann Minkowski's flat four-dimensional space-time. Some investigations are geometric, such as Maryna Viazovska's 2016 discovery of the most efficient ways of packing spheres in dimensions eight and 24.
A mathematician's guided tour through higher dimensions



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