What Are the Practical Applications of the Fourth Dimension in Analysis?

JaredPM
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How do you calculate an object in 4 dimensions? Like the 4 dimensional cube. I understand that a point is the beginning of a line and a line is the beginning of a plane. From there a plane translates into a 3 dimensional object. A 3 dimensional object translates into a 4 dimensional thing... I am confused at what the fourth dimension represents and what it offers in the field of applied mathematics. So what are some of the uses of the 4th dimension in analysis?
 
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There is no "the" fourth dimension; dimension merely describes the number of coordinates needed to specify a point. I'm doing statistics with a 74 dimensional dataset as we speak; this isn't particularly unusual. There's no need to attach any sort of "real world" significance to it.
 
Although in physics, 4th dimension is usually regarded to be 'time'. Therefore, your representation would be the cube's shape and position as time passes (so, a bunch of different plots of cubes).
 
Hello! There is a simple line in the textbook. If ##S## is a manifold, an injectively immersed submanifold ##M## of ##S## is embedded if and only if ##M## is locally closed in ##S##. Recall the definition. M is locally closed if for each point ##x\in M## there open ##U\subset S## such that ##M\cap U## is closed in ##U##. Embedding to injective immesion is simple. The opposite direction is hard. Suppose I have ##N## as source manifold and ##f:N\rightarrow S## is the injective...
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