```Date: Jan 10, 2013 11:56 AM
Author: kirby urner
Subject: More STEM math (re the maths in STEM)

Geometry:We were recently looking at an algorithm for dividing the sphere intomore and more triangles.http://mathforum.org/kb/message.jspa?messageID=7937486#reply-tree(discussion on math-teach)The number of vertexes, by this algorithm, goes through the"icosahedral numbers":  1, 12, 42, 92, 162... or 10 * L^2 + 2 where L= layer number.By means of a visual transformation that shifts the balls in anunderstandable manner, these also become the "cuboctahedral numbers"as a proof will show.http://mathforum.org/kb/message.jspa?messageID=7945392Biology: "All of these numbers are in fact found in actual viruses, 12 forcertain bacteriophages, 42 for wart viruses, 92 for reovirus, 162 forherpesvirus, 252 for adenovirus and 812 for a virus attackingcrane-flies (Tipula or daddy-long-legs)" - The Natural History ofViruses by C.H. Andrews (W.W. Norton R Co., 1967).On other lists, we've been discussing convergent ratios, namely Pi and Phi.The Python generator for Pi still hasn't been traced to a source (itcame up in that high school in Beverly Hills), but appears toaccurately spit out successive digits.  We may need more professionalhelp with it.  We have a closed form expression for Phi that puts theburden of finding more digits on the SQRT(5) algorithm.http://mail.python.org/pipermail/edu-sig/2012-December/010729.html(discussing the digits-of-PI generator)http://wikieducator.org/PYTHON_TUTORIALS#Generators  (more in STEMmaths re Python generators)Phi may be approached geometrically through the ratio 5:d where d isthe "length diagonal of a regular pentagon P".  5:d is not PHI but ismore PI-like in measuring a perimeter all the way around, but aschords of a circle (central angles subtend chordal lengths).  If theedges of P are all 1, then d == PHI.Biology:Lets remember that the regular icosahedron may be constructed usingthree mutually orthogonal golden rectangles.Notes for teachers:This lesson usually features short computer programs (just a few linesin an interactive environment) generate (a) figurate numbers (b)polyhedral numbers (c) successive approximations of irrational ratios.Excitement over the discovery of the 5-fold symmetric structure of thetypical virus was matched decades later by excitement over thediscovery of quasi-crystals, as 5-fold symmetry within crystalstructures had earlier been deemed impossible.Segues / followup:  as tiles may be used to tile a plane with no gaps,so may various sets of polyhedrons fill space.  The sphere dividingalgorithm used triangles that were not all similar to one another.Virus protein shells come with other numbers of capsomeres expressedby another formula by Michael Goldberg.  See H.S.M. Coxeter's articleon the micro-architecture of the virus.OEIS:  http://oeis.org/A005901Kirby
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