The complete set of Jitterbug transformers and the analysis of their motion (Q748881)

For A a rotation (in \({\mathbb{R}}^ 3)\), gen(A) denotes the cyclic group of rotations generated by A and \(r_ A\) the axis of A; if \(P\in {\mathbb{R}}^ 3\), then the image of the segment [P,A(P)] over gen(A) is the regular polygon produced by A in P. If gen(A) is finite, say s, there are rotations R and \(R^{-1}\) in gen(A) whose central angle is the smallest one (besides 0) and is equal to \(2\pi\) /s. R and \(R^{-1}\) are generators of gen(A). Let \(A=R^ d\) \((1\leq d<s)\to A=(R^{-1})^{s- d}\), where \(d\leq s-d \to d\leq s/2\). d is called the density of the polygon and \(m=s/d\) the polygonal value of A. The polygon is denoted by \(\{\) \(m\}\). Let A and B be two rotations with \(r_ A\cap r_ B=\{0\}\). A produces \(\{\) \(m\}\) in P and B produces \(\{\) \(n\}\) in P. This pair of polygons is called a dipolygon produced by the base \(\{\) A,B\(\}\) in P. Let G be the group generated by A and B. The image of \(\{\) A,B\(\}\) under G is called the dipolygonid produced by the base \(\{\) A,B\(\}\) in P. The author gives a classification of dipolygonids and applications in architecture, engineering art and mathematics.