Template:Sequence of the Day for May 24

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.