MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

Applications 101
(a)
Figure 3.32: Triangular Lower and Upper Bounds: (a) Vibrating Membrane.
(b) Cylinder Under Torsion. [243]
to the cross-section. The couple resisting such torsion is equal to θµP where
θ is the twist or angle of rotation per unit length and µ is the shear modulus.
So defined, P is a purely geometric quantity depending on the shape and
size of the cross-section. Figure 3.32(a) shows one of the vibrational modes
of a triangular membrane while Figure 3.32(b) shows the shear stress in the
cross-section of an equilateral triangular prism under torsion.
T
3
2.5
2
1.5
1
0.5
0
1
0.8
0.6
y
0.4
0.2
0
0
Figure 3.33: Fundamental Mode [219]
Application 33 (Laplacian Eigenstructure). The eigenvalues and eigenfunctions
of the Laplace operator, ∆ := ∂2
∂x2 + ∂2
∂y2, on the equilateral triangle
play an important role in Applied Mathematics.
0.2
0.4
x
0.6
0.8
1
(b)