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 into

more 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 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

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.

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 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.

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 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.

OEIS: http://oeis.org/A005901

Kirby