Peg
The answer to the question "which fits better, a round peg in a square hole, or a square peg in a round hole?" can be interpreted as asking which is larger,
the ratio of the area of a circle
to its circumscribed square, or the area
of the square to its circumscribed circle?
In two dimensions, the ratios are 
and 
, respectively.
Therefore, a round peg fits better into a square hole than a square peg fits into
a round hole (Wells 1986, p. 74).
However, this result is true only in dimensions 
, and for
, the unit 
-hypercube fits
more closely into the 
-hypersphere than vice versa (Singmaster
1964; Wells 1986, p. 74). This can be demonstrated by noting that the formulas
for the content 
of the unit 
-ball, the content
of its circumscribed hypercube,
and the content 
of its inscribed hypercube
are given by
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
The ratios in question are then
 |  |  |
(4)
|
 |  |  |
(5)
|
(Singmaster 1964). The ratio of these ratios is the transcendental equation
![(R_(round peg))/(R_(square peg))=(pi^nn^(n/2))/(2^(2n)[Gamma(1+1/2n)]^2),](/National_Library/im_/https://mathworld.wolfram.com/images/equations/Peg/NumberedEquation1.gif) |
(6)
|
illustrated above, where the dimension 
has been treated
as a continuous quantity. This ratio crosses 1 at the value 
(OEIS A127454), which must be determined numerically.
As a result, a round peg fits better into a square hole than a square peg fits into
a round hole only for integer dimensions 
.
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