MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

76 Mathematical Properties
Figure 2.69: Delahaye Product
Property 87 (Delahaye Product). In his Arithmetica Infinitorum (1655),
John Wallis presented the infinite product representation
π
2
2 2 4 4 6 6 8 8
= · · · · · · · · · · .
1 3 3 5 5 7 7 9
In 1997, Jean-Paul Delahaye [74, p. 205] presented the related infinite product
2π
3 √ 3
3 3 6 6 9 9 12 12
= · · · · · · · · · · .
2 4 5 7 8 10 11 13
The presence of π together with √ 3 suggests that a relationship between the
circle and the equilateral triangle may be hidden within this formula.
We may disentangle these threads as follows. The left-hand-side expression,
pR 2π = p 3 √ , is the ratio of the perimeter of the circumcircle to the perime-
3
ter of the equilateral triangle. Introducing the scaling parameters σk √ :=
(3k−1)(3k+1)
, Delahaye’s product may be rewritten as
3k
pR · σ
lim
k→∞
2 1 · σ2 2 · · ·σ 2 k · · ·
p
= 1.
Thus, if we successively shrink the circumcircle by multiplying its radius by
the factors, σ2 k (k = 1, . . . , ∞), then the resulting circles approach a limiting
position where the perimeter of the circle coincides with that of the equilateral
triangle (Figure 2.69).