4.1 Higher-Frequency Spheres: Introduction
Now some concrete applications of the methods discussed in Chapter 3 can be made. They are applied to "higher-frequency" versions of the simple spherical structures discussed in Chapter 2. "Higher-frequency" in this context means that the spherical structures are composed of a greater number of members. If the members used are about the same size as before, this means the sphere will grow in size. If instead the radius of the sphere stays the same size, the surface now has a finer texture.
As in the model for the t-tetrahedron of Section 2.4, the tensegrities are considered to be a collection of tendon triangles lying approximately on a sphere interconnected with adjacent tendon triangles via struts and tendons. The lengths of the struts as well as of the lengths of the tendons making up the tendon triangles are fixed, and thus appear as parameters in the mathematical programming problem, while the second powers of the lengths of the tendons interconnecting adjacent tendon triangles appear in the objective function and are collectively minimized.
Although at least one member's length must appear as a constraint for the problem to be mathematically determinant, there is nothing hard and fast about the classification of members as minimands or constraints. For different applications, different classifications might be useful. The classification selected here is convenient because it allows a good number of the tendon and strut lengths to be constrained, and still enough degrees of freedom are left in the minimization process that tendons of the same class aren't wildly asymmetric. Having a good number of the member lengths constrained is convenient because it means their lengths can be specified precisely; all the tendons or struts of a certain class can be constrained to have the same lengths.