Date: May 13, 2001 11:44 AM
Author: Kirby Urner
Subject: [math-learn] Geometry Lab (Part 2 of 2)


A Math Computer Lab
Part 2
by Kirby Urner
Oregon Curriculum Network
May 13, 2001

[ Part 1 of this Geometry Lab is archived at:
http://groups.yahoo.com/group/math-learn/message/744
]

Firstly, let's go over finding that mystery scale factor, which
grows the volume 6 rhombic docahedron such that it has an 81:80
volume relationship with the cuboctahedron of volume 20.

c:newr = 20:newr = 80:81

So c/newr = 80/81, and 81*c/80 = newr = 20.25, or 81/4.

r.volume = 6, but we want it to become 81/4.

6*x = 81/4
x = 81/24 = 27/8

But it's the 3rd root of x that will be our linear scale factor,
since volume changes as a 3rd power of linear change. So what's
the 3rd root of 27/8? Answer: 3/2.

Secondly, we needed to get our coordinates listed in the format
Qhull understands. One method might involve cutting and pasting.
Open a new, blank window and write:

3 #3D file
26

at the top. Then we need lists of XYZ coordinates without braces.
Using the clues I gave you last time, you could do something like
this:

>>> from rbf import *
>>> r = Rhdodeca()
>>> r = r*(3.0/2.0)
>>> xyzlist = [getattr(x,'xyz') for x in r.vertices.values()]
>>> for point in xyzlist:

print "%s %s %s" % point


0.0 0.0 -1.06066017178
-1.06066017178 0.0 0.0
0.0 -1.06066017178 0.0
0.0 1.06066017178 0.0
1.06066017178 0.0 0.0
-0.53033008589 0.53033008589 0.53033008589
0.0 0.0 1.06066017178
0.53033008589 0.53033008589 -0.53033008589
0.53033008589 -0.53033008589 0.53033008589
0.53033008589 -0.53033008589 -0.53033008589
-0.53033008589 -0.53033008589 -0.53033008589
-0.53033008589 -0.53033008589 0.53033008589
-0.53033008589 0.53033008589 -0.53033008589
0.53033008589 0.53033008589 0.53033008589

Then just cut and past that chunk of information to your new file.
Do the same for the cuboctahedron. Save as rcdual.txt or some such,
and you're done. If you had a more elaborate solution, that's great.

Anyway, take about 15 minutes to create this text file if you
haven't already, or move directly to using Qhull below. If you
finish early, you'll have more time to tweak Povray and develop
fancier renderings.

Now we're ready to run Qhull.

The command line format is:

> qhull < input.txt o E0.000001> output.txt

To avoid path issues, the easiest thing will be to copy your
rcdual.txt right into qhull's working directory.

The E parameter is necessary to keep qhull from trying to be
*too* precise when it figures out which points belong to the
same face. Our operating level of precision isn't really out
to the final decimal place, and if we let qhull take its
analysis to that level, it will enforce standards of
coplanarity that exceed our requirements, and give us an
all-triangle-faceted shape.

Povray won't draw a face if the points given it aren't coplanar
within a very narrow range. If Povray still gives us faces, vs.
warnings that it has to ignore our polygon instructions, this
is evidence that our resultant shape is flat-faced.

Of course the best proof would be algebraic/symbolic, so lets
put a bookmark here for a future project. We also need to do
some more algebra to show why the 80:81 volume ratio, between
the cuboctahedron and its structural dual, is really correct.
Remember I gave that to you as a starting place, and you used
that to derive a scale factor.

The file you get back from Qhull will look like this:

3
26 24 48
0.707106781187 0.707106781187 0
-0.707106781187 -0.707106781187 0
0.707106781187 -0.707106781187 0

<< snip -- repeat of the vertices >>

-0.53033008589 0.53033008589 -0.53033008589
0.53033008589 0.53033008589 0.53033008589
4 11 18 10 17
4 25 9 18 11
4 4 15 11 17
4 13 4 17 10
4 6 24 4 13
4 0 25 11 15

<< snip -- a lot of quadrilateral faces >>

So now we're ready to convert this to a povray file for
rendering. I've already supplied a conversion utility called
qhull.py, inside of which is the convert method. You run it like
this:

>>> from qhull import convert
>>> convert('input.txt','output.pov')


I've preset some textures to make your rendering look more like a
museum piece. Here's what I get:

http://www.inetarena.com/~pdx4d/ocn/graphics/trapez.gif
(plus I added the caption in PaintShop).

If you have extra time at the end of the lab, feel free to
play around with these settings by popping open qhull.py
and tweaking the source code. You could try using a wood
surface.

For those of you less familiar with Povray (I know the art
teacher has been using it), I've put some information in a
handout.

The trapezoidal icositetrahedron has its own dual of course:
the small rhombicuboctahedron. You'll find these listed on
pp 80-81 (heh) of Robert Williams' 'The Geometrical Foundation
of Natural Structure', one of our source books for this class.




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