User blog comment:Holomanga/A Thought On Dimensionality/@comment-26335203-20160108200100/@comment-39162-20160209224912

An entry in a collection can be described by a set of numbers (x,y,z...), which are coordinates. For example, if you have a 10x10 square, different places in the square can be defined by an x and a y coordinate. One corner is at (0,0), the centre is at (5,5), etc.

Each of these coordinates is also a member of a collection. In the above example, the coordinates are numbers from 0-10.

If you are naming positions in one collection, using numbers chosen from another collection, my equation gives you the smallest number of numbers you need to make it work. This is equal to the number of dimensions that the collection has.

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 * A| and |B| are the number of items in the collections A and B. For example, in the 10x10 square example, |A| = 11 (11 numbers from 0 to 10) and |B| = 121 (121 points inside the square or on its edges).

The log means that you get the number X for which |A| multiplied by itself X times is B. In this case, it's 2, because 11*11 is 121. That matches what we have earlier - two numbers to describe points inside the square.