MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

100 Applications
(a)
(b) (c) (d)
Figure 3.31: Sphere Wrapping: (a) Mozartkugel. (b) Petals. (c) Square Wrapping.
(d) Equilateral Triangular Wrapping. [75]
Application 31 (Computational Confectionery (Optimal Wrapping)
[75]). Mozartkugel (“Mozart sphere”) is a fine Austrian confectionery composed
of a sphere with a marzipan core (sugar and almond meal), encased in
nougat or praline cream, and coated with dark chocolate (Figure 3.31(a)).
It was invented in 1890 by Paul Fürst in Salzburg (Mozart’s birthplace)
and about 90 million of them are still made and consumed world-wide each
year. Each spherical treat is individually wrapped in a square of aluminum
foil. In order to minimize the amount of wasted material, it is natural to study
the problem of wrapping a sphere by an unfolded shape which will tile the
plane so as to facilitate cutting the pieces of wrapping material from a large
sheet of foil. E. Demaine et al. [75] have considered this problem and shown
that the substitution of an equilateral triangle for the square wrapper leads to
a savings in material. As shown in Figure 3.31(b), they first cut the surface of
the sphere into a number of congruent petals which are then unfolded onto a
plane. The resulting shape may then be enclosed by a square (Figure 3.31(c))
or an equilateral triangle (Figure 3.31(d)). Their analysis shows that the latter
choice results in a material savings of 0.1%. In addition to the direct savings
in material costs, this also indirectly reduces CO2 emissions, thereby partially
alleviating global warming. Way to go Equilateral Triangle!
Application 32 (Triangular Lower and Upper Bounds). Of all triangles
with a given area A, the equilateral triangle has the smallest principal frequency
Λ and the largest torsional rigidity P. Thus for any triangle, we have the lower
bound 2π
4√ 3· √ A ≤ Λ and the upper bound P ≤ √ 3A 2
15 [243].
The principal frequency Λ is the gravest proper tone of a uniform elastic
membrane uniformly stretched and fixed along the boundary of an equilateral
triangle of area A. It has been rendered a purely geometric quantity by dropping
a factor that depends solely on the physical nature of the membrane. P is
the torsional rigidity of the equilateral triangular cross-section, with area A, of
a uniform and isotropic elastic cylinder twisted around an axis perpendicular