Date: Jan 10, 2013 11:56 AM
Author: kirby urner
Subject: More STEM math (re the maths in STEM)
We were recently looking at an algorithm for dividing the sphere into
more and more triangles.
(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 an
understandable manner, these also become the "cuboctahedral numbers"
as a proof will show.
"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
crane-flies (Tipula or daddy-long-legs)" - 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.
(discussing the digits-of-PI 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 PI-like 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.
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 5-fold symmetric structure of the
typical virus was matched decades later by excitement over the
discovery of quasi-crystals, as 5-fold 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 micro-architecture of the virus.