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 👁 pi/4
and 👁 2/pi
, 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 👁 n<9
, and for 👁 n>=9
, the unit 👁 n
-hypercube fits more closely into the 👁 n
-hypersphere than vice versa (Singmaster 1964; Wells 1986,
p. 74). This can be demonstrated by noting that the formulas for the content
👁 V(n)
of the unit 👁 n
-ball, the content 👁 V_c(n)
of its circumscribed hypercube,
and the content 👁 V_i(n)
of its inscribed hypercube are given by
The ratios in question are then
| 👁 R_(round peg) | 👁 = | 👁 (V(n))/(V_c(n))=(pi^(n/2))/(2^nGamma(1/2n+1)) |
(4)
|
| 👁 R_(square peg) | 👁 = | 👁 (V_i(n))/(V_c(n))=(2^nGamma(1/2n+1))/(n^(n/2)pi^(n/2)) |
(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),](https://mathworld.wolfram.com/images/equations/Peg/NumberedEquation1.svg){kind=link}
illustrated above, where the dimension 👁 n
has been treated as a continuous quantity. This ratio crosses
1 at the value 👁 n approx 8.13794
(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 👁 n<9
.
See also
Hole, Hypersphere Packing, Peg Solitaire, Piriform CurveExplore with Wolfram|Alpha
References
Singmaster, D. "On Round Pegs in Square Holes and Square Pegs in Round Holes." Math. Mag. 37, 335-339, 1964.Sloane, N. J. A. Sequence A127454 in "The On-Line Encyclopedia of Integer Sequences."Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 74, 1986.Referenced on Wolfram|Alpha
PegCite this as:
Weisstein, Eric W. "Peg." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Peg.html