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
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A164102: Decimal expansion of
.
-
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
. The “volume” of the contained hyperball (i.e. a 4-ball) is
. Compare it with the 3-dimensional unit ball: the surface area of a unit sphere in three dimensions (i.e. a 2-sphere) is
. The volume of the contained ball (i.e. a 3-ball) is
. The “volume” of the
-dimensional unit hyperball is given by
-
Vn (1) = (length unit) n, n ≥ 0, |
where
is the
double factorial. (For
, we get a 0-dimensional “volume” of
, i.e. the pure number
1, result of the
empty product.) For a recursion relation, see:
The Volume of a Hypersphere.