12/06/2019

13. Squaring the Circle in n-Dimensions


In the previous post we saw how, starting from the Wallis sieve, we can arrive at a 3D-cubic fractal-like and by assembling 8 of these cubes we have the same volume as a sphere of unitary radius. Below we will explain in an analytical way how to build a sieve in 2D, 3D, etc.

For 2D we have a square divided into 9 equal parts:


The number of elements (or cardinality) of the Cartesian product of 2 sets is the product of the number of elements of the 2 sets. Generalizing, the number of elements of the Cartesian product of n sets is the product of the number of elements of each set.

We can give a tabular representation, which consists in writing all the possible pairs enclosed in braces:

{(1, 1); (1, 2); (1, 3); (2, 1); (2, 2); (2, 3); (3, 1); (3, 2); (3, 3)}

or by means of a double entry table, obtained with the elements of the first set placed in the column and those of the second in the first row; the boxes contain the various pairs that are obtained:


As can be seen, the only square removed is (2, 2).

In the case of 3D fractal-like we can do a similar reasoning and the 7 cubes removed will be:

(1, 2, 2); (3, 2, 2); (2, 1, 2); (2, 2, 2); (2, 3, 2); (2, 2, 1); (2, 2, 3)

The following table summarizes (for each dimension) how many groups of numbers 2 we can find; eg in 3D we have 8 triples with zero 2, 12 with only 2, 6 with 2 elements equal to 2 and 1 with all the numbers equal to 2 (which corresponds to the central cube):


the combinations with at least 2 elements equal to 2 are 6 + 1 = 7, on a total of 27 cubes (shown in the last column) and for a cube of side 3.

We can do the same reasoning for a cube of side 5:


last column shows the number of cubes remaining.

Note: the coefficients that appear in the table can be easily calculated using the formula of the Newton binomial - in this case (4 + 1) n -


Now, in the previous post we saw that in 2D for an initial value of 4, multiplied by 8/9, 24/25, 48/49, 80/81, ..., (n2 - 1) / n2
we have the formula derived from the Wallis product:


which can be summarized as follows:


While in 3D we have this result:


where the numbers in gray represent the various denominators.


I omit the passages and report what I calculated with Wolfram|Alpha:


which can be summarized:


In the formulas the Euler Gamma function appears which extends the factorial concept (even to complex numbers). At the end of the post I report the chart and some notable formulas concerning it.

Here I am only interested in noting that the right part of the formulas containing the index n tends to 1 when tending n to infinity.

So what remains of the 3 formulas is pi and the number is in the denominator. Remembering that each formula must be multiplied by the number of "quadrants", the first (2D) 22 = 4, the second (3D) for 23 = 8 and the third (4D) for 16.

Thus obtaining the "Volumes" of the n-dimensional spheres:


This is not yet a real demonstration, but it contains all the elements and could also be used as an alternative method to calculate the volume of a hyper-sphere.















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