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Template:Sequence of the Day for May 24

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Intended for: May 24, 2013

Timetable

  • First draft entered by Alonso del Arte on April 6, 2012
  • Draft reviewed by Daniel Forgues on May 13, 2012; May 24, 2015; May 24, 2016
  • Draft to be approved by April 24, 2013
Yesterday's SOTD * Tomorrow's SOTD

The line below marks the end of the <noinclude> ... </noinclude> section.



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.