A Construction That Produces Wallis-Type Formulas

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Generalizations of the geometric construction that repeatedly
attaches rectangles to a square, originally given by Myerson, are presented.
The initial square is replaced with a rectangle, and also the dimensionality of
the construction is increased. By selecting values for the various parameters,
such as the lengths of the sides of the original rectangle or rectangular box
in dimensions more than two and their relationships to the size of the attached
rectangles or rectangular boxes, some interesting formulas are found. Examples
are Wallis-type infinite-product formulas for the areas of *p*-circles with *p** *> 1.

References

[1] S. J. Miller, “A Probabilistic Proof of Wallis’s Formula for π,” The American Mathematical Monthly, Vol. 115, No. 8, 2008, pp. 740-745.

[2] J. Wastlund, “An Elementary Proof of the Wallis Product Formula for Pi,” The American Mathematical Monthly, Vol. 114, No. 10, 2007, pp. 914-917.

[3] G. Myerson, “The Limiting Shape of a Sequence of Rectangles,” The American Mathematical Monthly, Vol. 99, No. 3, 1992, pp. 279-280. doi:10.2307/2325077

[4] L. Short, “Some Generalizations of the Wallis Product,” International Journal of Mathematical Education in Science and Technology, Vol. 23, No. 5, 1992, pp. 695-707.
doi:10.1080/0020739920230508

[5] L. Short and J. P. Melville, “An Unexpected Appearance of Pi,” Mathematical Spectrum, Vol. 25, No. 3, 1993, pp. 65-70.

[6] G. K. Srinivasan, “The Gamma Function: An Eclectic Tour,” The American Mathematical Monthly, Vol. 114, No. 4, 2007, pp. 297-315.

[7] E. T. Whittaker and G. N. Watson, “A Course in Modern Analysis,” 3rd Edition, Cambridge University Press, Cambridge, 1920.

[8] T. M. Apostol, “Mathematical Analysis,” Addison-Wesley, Reading, 1957.

[9] T. J. Bromwich, “An Introduction to the Theory of Infinite Series,” 2nd Edition Revised, Macmillan, London, 1926.

[10] A. C. Thompson, “Minkowski Geometry,” Cambridge University Press, Cambridge, 1996.
doi:10.1017/CBO9781107325845