A Construction That Produces Wallis-Type Formulas

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.


Introduction
Wallis's product formula for /2 is 1 2 2 2 2 4 4 6 6 2 1 2 1 1 3 3 5 5 7 2 Two more-or-less elementary proofs are given in [1,2].An interesting geometric construction, which first appeared in [3], produces this infinite product.The construction is somewhat generalized in [4,5].The purpose of this paper is to further generalize the construction in [3].Among the naturally occurring special cases of the generalization are infinite-product representations of areas of p-circles.In Section 2, we give an account of the construction and some generalizations.In Section 3, we review the gamma function, since our results are written in terms of that function.Section 4 describes unit super-circles or p-circles 1 p p x y   for p ≥ 1, since their areas are produced by certain generalizations of the construction.Section 5 contains several interesting outcomes of some new generalizations of the geometric construction, including the Wallis formula for p-circles.

The Construction
The following construction produces the Wallis product (1) [3][4][5].See Figure 1.Let w j be the width and h i be the height of the current rectangle at the current step for the appropriate values of i and j.
The initial square and the first few steps in the construction are:  The initial square in Figure 1(a) has sides w 0 = 1 and h 0 = 1 and area 1.  The first step is to attach to the right a square with sides w 1w 0 = 1 and h 0 = 1, so that the current rectangle in Figure 1(b) has sides w 1 = 2 and h 0 = 1 and area 2.  The second step is to attach to the top a rectangle of area 1 with sides w 1 = 2 and h 1h 0 = 1/2, so that the current rectangle in Figure 1(c) has sides w 1 = 2 and h 1 = 3/2 and area 3.  The third step is to attach to the right a rectangle of area 1 with sides w 2w 1 = 2/3 and h 1 = 3/2, so that the current rectangle in Figure 1(d) has sides w 2 = 8/3 and h 1 = 3/2 and area 4.  The fourth step is to attach to the top a rectangle of area 1 with sides w 2 = 8/3 and h 2h 1 = 3/8, so that the current rectangle in Figure 1(e) has sides w 2 = 8/3 and h 2 = 15/8 and area 5.The construction continues indefinitely in this way.
In [3][4][5], it is shown that which is /2 by Wallis's product formula (1).Short [4] and Short and Melville [5] generalize the construction with all the rectangles adjoined to the right having area A, all the rectangles adjoined to the top having area B, and the initial square having area .They show that: For the last case, they demonstrate how averaging methods can be employed to obtain Limit n n n w h  numerically to a desired precision.
These results are examples in the further generalizations in Section 5, so we do not discuss their details.The generalizations in Section 5 include allowing the initial figure to be a rectangle and increasing the dimensionality of the construction.Properties of the gamma function are very useful and are reviewed in the next section.

The Gamma Function
The purpose of this section is to remind readers about some properties of the gamma function, including infinite-product representations.A source that is relatively complete and takes a historical perspective is [6].We restrict the domain of the gamma function to the positive real numbers, since only those values concern us.The most familiar definition of the gamma function is the convergent, improper integral For positive integer , which can be derived by the change of variable x = ny in (7) which is valid for [7, pp.238-239, 9, p. 115], greatly simplifies many derivations in Section 5.The infinite product on the left-hand side of (8) would have diverged if there were not the same number of terms in its numerator and denominator or if (9) were not satisfied.Taking a 1 = a 2 = 0 and b 1 = -b 2 = -1/2, we obtain the Wallis product formula (1), since Euler's Integral of the First Kind, also known as the beta function, is [7, pp.253-256].

Super Circles
In the same way that Euclidean geometry is based on the unit circle x 2 + y 2 = 1, l p geometries, p ≥ 1, are based on the unit super-circle or p-circle  .These are Minkowski geometries, which are characterized by their unit circles that enclose a convex, symmetric region [10].See Figure 2 for graphs of the l p unit circles for p = 1, 3/2, 2, and 4. The enclosed area of the l p super circle is using ( 4) and (10), where t = x p .The enclosed region is not convex for 0 < p < 1, so the circle does not give a Minkowski geometry; however, (11) gives the areas of those regions.For p = 2, (11) gives , using (4), ( 5), and (6).The enclosed area of the upper half of the l p super circle is

Generalizations of the Construction
ing the

Starting with a Rectangle in Two
The fi f the construction is to begin nd B, as described in Section 2.
The iterativ ined by and for n ≥ 0, giving the next values of w n+1 n+1 and h in turn.After 2n steps, the area of the whole figure, which is a rectangle, is and after 2n + 1 steps, the area of the re (17) We find recursion relationships between n+1 n stituting ( 16) into (14) gives ctangle is w and w and between h n+1 and h n .Sub for n ≥ 0. Substituting (17) into (15) gives for n ≥ 0. Dividing ( 18) by ( 19) gives From ( 13) and (20) In order to apply (8), the index in the infinite product must start at 1, not 0. Changing the index to m = n + 1 and reverting back to n give

if and only if ab B ab B ab A B ab A B A B
from ( 8) and (9).Equation ( 22) implies that A = values of a and b and, using (4), B for any 2 For the special case a = b = A = B = 1, ( 21) and ( 23) is the Wallis product (1) with (2).Also, this is U( 2) from ( 1There is a large variety of ratios and limits to investigate s increased.In three dimenjoining rectangular boxes, instead of rectangles.The iterative steps of adjoining recta 2).

Three Dimensions
when the dimensionality i sions, the process is determined by the initial box with sides 0 0 0 , , and and ad ngular boxes of volumes A, B, and C are determined by and After 3n steps, the volume of the whole rectangular box is + 1 steps, the volume is   and after 3n + 2 steps, the volume is Solving (25) for w  and using ( Similarly, ( 26) and (29) give and ( 27) and (30) give Limits of ratios of (31), (32), and (33) give a va interesting expressions.We present three exam w n /d n .Dividing (31) by (33) gives riety of ples for From ( 24) and ( 34) Changing the index to m = n + 1 and reverting back n give Then, from ( 8) and (9).Equation (36) implies any values of a, b, c, and B and, using (4), Consider three special cases of (37).For the first one, set a = b =c = A = B = C = 1, then (12), (35), and (37) give   For the second special case, set     for p > 2. The domain of U(p) is restricted by For the third case, set     Using this construction, Wallis-type product formulas .have been obtained for half the areas of p-c 1. dimensions and (25), ( three dimensions, the iterative steps of adjoining rectangular boxes are determined by

N Dimensions
for the (Nn + j) th step with 1 < j < N, and .Analogous to ( 16) and (17) in Subsection 5.1 for two dimensions and (28), (29), and (30) in Subsection 5.2 for three dimensions, the vo es are

Summary
We have generalized the infinite geometric construction of attaching rectangles to a square, which was originally presented in [3], by allowing the initial square to be replaced with a the lengths of the sides of the origiectangular box in dimensions more than lationships to the size of the attached For N = 2, (41) is the ula (1) with (2), and rectangle and by increasing the dimensionality of the construction.Selecting values for the various parameters, such as nal rectangle or r two and their re rectangles or rectangular boxes, gives some interesting formulas.Examples are Wallis-type formulas (38) through

Figure 1 .
Figure 1.The initial unit square and the first four steps of the construction.

( 13 )
The adjoined rectangles have areas A a e steps of adjoining rectangles are determThe generalizations in this section include allow initial figure to be a rectangle and increasing the dimensionality of the construction.Selecting various values for the parameters gives interesting formulas, including Wallis-type formulas for half the areas of p-circles.Dimensionsrst generalization o with a rectangle with width a and height b, instead of a square, mit the initial shape to be a unit hypercube and each adjoined shape to be i) = 1 with Analogous to (14) and (15) in Subsection 5.1 for two 26), and (27) in Subsection 5.2 for ) th step, ircles for p > For the construction in N dimensions, we li rectangular of unit hyper volume.The n th value of the i th side's length is s n (i), and the initial sides' lengths are s 0 + j) th step with 1 ≤ j ≤ N -1.Using an analysis paralleling Subsections 5.1 and 5N = 3, it is (38).