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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.