MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

34 Mathematical Properties
Figure 2.8: Viviani’s Theorem Figure 2.9: Ternary Diagram
With reference to Figure 2.9, each variable is assigned to a vertex as well
as to the clockwise-adjacent edge. These diagrams are employed in petrology,
mineralogy, metallurgy and other physical sciences to portray the composition
of systems composed of three species. In population genetics, it is called the
de Finetti diagram and, in game theory, it is called the simplex plot. Ample
specific instances will be provided in Chapter 3: Applications.
Figure 2.10: Ellipsoidal Shape [243]
Property 10 (Ellipsoidal Shape). The shape of an ellipsoid with semi-axes
a, b and c depends only on the ratios a : b : c. In order to visualize the variety
of these shapes, we may consider a, b and c as homogeneous coordinates ([256,
pp. 16-20], [257, pp. 179-190]) of a point (a, b, c) in a plane and consider
only that part of this plane where a ≥ b ≥ c ≥ 0. These inequalities delimit a
triangle which can be made equilateral by a suitable choice of coordinate system
(Figure 2.10) [243, p. 37].