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More STEM math (re the maths in STEM)
Posted:
Jan 10, 2013 11:56 AM


Geometry:
We were recently looking at an algorithm for dividing the sphere into more and more triangles.
http://mathforum.org/kb/message.jspa?messageID=7937486#replytree (discussion on mathteach)
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 an understandable manner, these also become the "cuboctahedral numbers" as a proof will show.
http://mathforum.org/kb/message.jspa?messageID=7945392
Biology:
"All of these numbers are in fact found in actual viruses, 12 for certain bacteriophages, 42 for wart viruses, 92 for reovirus, 162 for herpesvirus, 252 for adenovirus and 812 for a virus attacking craneflies (Tipula or daddylonglegs)"  The Natural History of Viruses 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 (it came up in that high school in Beverly Hills), but appears to accurately spit out successive digits. We may need more professional help with it. We have a closed form expression for Phi that puts the burden of finding more digits on the SQRT(5) algorithm.
http://mail.python.org/pipermail/edusig/2012December/010729.html (discussing the digitsofPI generator)
http://wikieducator.org/PYTHON_TUTORIALS#Generators (more in STEM maths re Python generators)
Phi may be approached geometrically through the ratio 5:d where d is the "length diagonal of a regular pentagon P". 5:d is not PHI but is more PIlike in measuring a perimeter all the way around, but as chords of a circle (central angles subtend chordal lengths). If the edges of P are all 1, then d == PHI.
Biology:
Lets remember that the regular icosahedron may be constructed using three mutually orthogonal golden rectangles.
Notes for teachers:
This lesson usually features short computer programs (just a few lines in an interactive environment) generate (a) figurate numbers (b) polyhedral numbers (c) successive approximations of irrational ratios.
Excitement over the discovery of the 5fold symmetric structure of the typical virus was matched decades later by excitement over the discovery of quasicrystals, as 5fold symmetry within crystal structures 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 dividing algorithm used triangles that were not all similar to one another. Virus protein shells come with other numbers of capsomeres expressed by another formula by Michael Goldberg. See H.S.M. Coxeter's article on the microarchitecture of the virus.
OEIS: http://oeis.org/A005901
Kirby



