User:Jean-Bernard François

$whoami?

I'm Jean-Bernard François (aka infofiltrage) born in France the 03/05/69 & DP Engineer (Engineering Graphics Software).

My current blog is Bondissant (almost art from geometry).

I love number theory, mathematics, and of course OEIS.

I only worked under Debian Linux System with LibreOffice, Inkscape and a LiberKey.

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A099730 | A000302

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Collatz | Diophantine | Primes | Number Theory

Sequence of the Day

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Sequence of the Day for May 24

A164102: Decimal expansion of

2 π 2

.

19.739208802...

A lot of us have quite enough trouble with just three dimensions. The hypersurface “area” of a unit hypersphere in four dimensions (i.e. a 3-sphere) is

2 π 2 (length unit) 3

. The “volume” of the contained hyperball (i.e. a 4-ball) is

π 2 2 (length unit) 4

. Compare it with the 3-dimensional unit ball: the surface area of a unit sphere in three dimensions (i.e. a 2-sphere) is

4 π (length unit) 2

. The volume of the contained ball (i.e. a 3-ball) is

4 π 3 (length unit) 3

. The “volume” of the

n

-dimensional unit hyperball is given by

Vn (1) = 2  ⌈ n / 2⌉ π  ⌊  n / 2⌋ n!! (length unit) n, n ≥ 0,

where

n!!

is the double factorial. (For

n = 0

, we get a 0-dimensional “volume” of

1 (length unit) 0

, i.e. the pure number 1, result of the empty product.) For a recursion relation, see: The Volume of a Hypersphere.