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.
https://math.stackexchange.com/questions/1783732/is-a-hypersphere-of-non-integer-dimension-a-fractal
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