Going back to this thread: http://groups.yahoo.com/group/WittrsAMR/message/1215 ... I'm recommending this archive for readings into the underlying philosophy, regarding how "non-traditional" triangles and tetrahedra may serve as units of area and volume respectively. Remarks on the Foundations of Mathematics deals with precisely these low level ethnographic or "form of life" questions, regarding our most logical grammars, such as Principia Mathematica. That's the level at which we must deal with the relationship between right angles and 1st / 2nd / 3rd etc. powering. Moving from a more 90-degree to more 60-degree aesthetic will be smoother if we preserve this philosophical context, i.e. if we accomplish our "gear shifting" in terms of "language games" (or "namespaces" as the case may be), combined with new ways of looking. The switch in question requires only a high school level of computational ability, but the gestalt changes could be more challenging, which is where Wittgenstein's philosophy proves a help. It's a two way switch we're talking about, i.e. going back and forth, very like the duckrabbit in that way. Links: http://bfi.org/our_programs/bfi_community/synergetics/synergetics/nctm_embraces_synergetics http://mathforum.org/kb/thread.jspa?threadID=2036171&tstart=0 http://coffeeshopsnet.blogspot.com/2009/12/duck-rabbit.html Kirby Urner 4Dsolutions.net PS: recall that H.S.M. Coxeter was Wittgenstein's student for awhile. Coxeter went on to become the great geometer of the 20th century. He partially overlapped with Bucky Fuller and his philosophy (metaphysics) on numerous occasions, an embedded example in this blog entry: http://worldgame.blogspot.com/2010/02/small-steps.html