Project 1: The tetrahelix turns out to be a sequence of points H(n)= (r cos(theta n),r sin(theta n), m n) on a helix, where the constants r, theta, m should be determined by the requirement that each edge of the tetrahedron H(n) H(n+1) H(n+2) H(n+3) has length 1. Can we derive the values that Roald Hoffman gets for r, theta, m from the three conditions |H(1)-H(0)|=1, |H(2)-H(0)|=1, |H(3)-H(0)|=1 ?
Project 2: Consider a robot arm linkage made of three unit length edges. Consider the four points p0,p1,p2,p3 along the robot arm. The Cayley configuration coordinates should be the distances a=|p(2)-p(0)| and b=|p(3)-p(0)| and c=|p(3)-p(1)|. Is the set of vectors (a,b,c) a convex set in R^3? How to describe the vector (a,b,c) where the robot arm collides with itself? How are the Cartesian coordinates related to the Cayley coordinates?
Project 3?: Instead of a robot arm, think of two unit line segments [p0,p1] and [q0,q1]. We could consider “Cayley coordinates” d0=|p0-q0| and d1=|p0-q1| and d2=|p1-q1| and see which vectors (d0,d1,d2) correspond to a collision between the two line segments.