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Peg


Pegs

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).

PegRatio

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

V(n)=(pi^(n/2))/(Gamma(1/2n+1))
(1)
V_c(n)=2^n
(2)
V_i(n)=(2^n)/(n^(n/2)).
(3)

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),
(6)

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 Curve

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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.

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Peg

Cite this as:

Weisstein, Eric W. "Peg." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Peg.html

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