MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

Mathematical Properties 77
Property 88 (Grunsky-Motzkin-Schoenberg Formula). Suppose that
f(z) is analytic on the equilateral triangle, T, with vertices at 1, w, w 2 where
w := exp (2πı/3). Then [69, p. 129],
� �
f
T
′′ (z) dxdy =
√ 3
2 · [f(1) + wf(w) + w2 f(w 2 )].
While this chapter has certainly made a strong case for the mathematical
richness associated with the equilateral triangle, it runs the risk of leaving the
reader with the impression that it has only theoretical and aesthetic value or,
at best, is useful only within Mathematics itself. Nothing could be further
from the truth! In the next chapter, I will present a sampling of applications
of the equilateral triangle which have been selected to provide a feel for the
diversity of practical uses of the equilateral triangle for comprehending the
world about us that the human race has uncovered (so far).