What would a fourth dimension—somehow perpendicular to our three—look like? Via QuantaMagazine
. This objective became known as the “invariance of dimension” problem.
Finally, in 1912, almost half a century after Cantor’s discovery, and after many failed attempts to prove the invariance of dimension, L.E.J. Brouwer succeeded by employing some methods of his own creation.
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. Also, as dimensions increase, points will cluster together less often. Thus, it’s often important to find ways to reduce the dimension of high-dimensional data.
The story of dimension didn’t end with Brouwer. Just a few years afterward, Felix Hausdorff developed a definition of dimension that—generations later—proved essential for modern math. An intuitive way to think about Hausdorff dimension is that if we scale, or magnify, a, the size of the object increases by a factor of. Suppose we scale a point, a line segment, a square and a cube by a factor of 3.
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