From: DGoncz@replikon.net   
      
   There is a matrix equation including a 3x3 matrix which, when solved for   
   its degenerate cases, gives orthonormal surfaces in the various confocal   
   ellipsoidal coordinate systems. There are 11 solutions if memory   
   serves. Adding the toroidal and helical systems, we have 13 systems and   
   so, at least 13 unit volumes. Since in Cartesian coordinates, the axes   
   are fully interchangeable, the unit volume (the unit cube) is fully   
   defined by its volume of 1 L^3. However, in all of the other systems,   
   the axes are not fully interchangeable.   
      
   Let's examine the system I consider next most challenging, the   
   cylindrical coordinate system. I believe it to be a degenerate case of   
   projected ellipsoidal coordinates. (That is not the most popular name   
   for that system)   
      
   With a surface of constant r=1, and surfaces at l (ell) = 0 and l = 1   
   and including a (alpha) from [-pi, ... pi] the unit "slug" has a volume   
   of pi.   
      
   We could normalize this volume to 1 by scaling each axis by pi. But that   
   won't work.   
      
   I am interesting in seeing and counting (enumerating) the most basic   
   unit volumes and their associated surfaces for inclusion in a computer   
   software library of "atomic" features from which "everything" (to second   
   order) may be designed, in a attempt to provide a reasonable and   
   nontrivial basis for para-universal constructors.   
      
   K. Eric Drexler leads the field in attempts to implement an atomic scale   
   additive universal constructor. Adrian Bowyer leads in attempts to   
   implement a human-scale universal constructor. Julian Leland Bell has   
   made significant progress in implementing a subtractive universal   
   constructor--his Swarthmore project was a self-reproducing externally   
   framed milling machine.   
      
    I built a four-axis mill with some self-reproducing features in 1997   
    and sold one of two copies to a hobbyist, advertising it in The Want AD   
    as a "self-reproducing milling machine" for $300. I wrote that up at   
    ESG at MIT. The writeup was mentioned in Kinetic Self-Replicating   
    Machines (KSRM) by Frietas and Merkle in 2004. I recovered the web site   
    mentioned in KSRM using the Wayback Machine maintain the page   
    first.replikon.net to this day, documenting that machine build.   
      
   It seems to me that including advanced math in CAD representations of   
   manufacturable objects would reduce file sizes and eliminate   
   digitization and tiling errors which are becoming a problem as the   
   resolution of additive and subtractive manufacturing machinery   
   increases, which is why I am writing about this here.   
      
   This post would go to sci.math were it not for the ubiquitous use of   
   change of coordinate system in solving the most advanced physics   
   problems. Briefly, when an initial, constraining, or terminal condition   
   of a physics problem is representable most effectively in a coordinate   
   system other than Cartesian, translating the entire problem into that   
   system can provide a solution where no other method will work. The   
   solution to the Navier-Stokes equations with viscosity for the case of   
   flow over a sphere is a famous example--after the change of system the   
   problem is thereby reduced from 3 dimensions to only 1, and is readily   
   solved.   
      
      
   Douglas Goncz   
   Replikon Research FCN 783774974   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)