On Jul 31, 2006, at 10:10 PM, C Goodman-Strauss wrote:

>
>  Hello, in the Advanced Applied class I'm teaching (aka A First Look at
>  Fourier Series Stuff), we are going over solving various physically
>  motivated boundary value problems. A problem came up today, describing
>  the motion of a rod fixed at one end, given a vigorous twist and
>  released. The solution seemed a little weird looking, so I made a GC
>  file to take a look. I dunno, does this look realistic?

This looks great!! It certainly looks plausible. There are interesting
little sub-twists going on within the twists.  This brings up the whole
fascinating question of node formation. Seems to me that given the
simplest initial conditions --- setting up a lot of coupled springs (a
discrete model of a spring, that is) so the profile is a simple sine
wave --- you're not going to get any nodes except at the ends of the
string. Here, if you could somehow separately twist the pieces of the
rod so that the amount of twist was a pure sinusoidal function of the
distance, you'd get only the base harmonic, I believe.

Whoops; I just read you're comment about the wobble being an artifact.

If anybody has the time and patience, some kind of vibration demo where
the initial amounts of stretch could be set on-screen would seem like a
worthwhile thing to make.


Would be great if you uploaded this file to the physics section? We
don't have too many files there yet.

>  (Unfortunately, since GC doesn't support mathematical formatting in
> the
>  text environment, I didn't have a quick way of actually typing in the
>  equation for which this is a solution...)

Aligning the wedges on the end cap with the striped on the rod makes
the shape even more vivid.

On Jul 31, 2006, at 7:10 PM, C Goodman-Strauss wrote:
Hello, in the Advanced Applied class I'm teaching (aka A First Look at
Fourier Series Stuff), we are going over solving various physically
motivated boundary value problems. A problem came up today, describing
the motion of a rod fixed at one end, given a vigorous twist and
released. The solution seemed a little weird looking, so I made a GC
file to take a look.  I dunno, does this look realistic?
Have fun,
Chaim

----------
No!
At first glance it didn't seem right
Then looking closer I expected the twist to start at the moving end and
travel downward to the fixed end

So I would say it looks OK with n
going  0 to 1, 2 to 3, 4 to 3, and 2 to 1

But to me it doesn't look right with n
going 1 to 2, 3 to 4, 3 to 2, and 1 to 0

a good distraction to my busy day
Arne

Trying to figure out exactly what goes on with parametric graphing of a
sphere. GC's built-in color-grids, striping and checkering change their
order if you interchange the u and v parameter.

"p_1" and "a_1" sliders can be adusted to determine how large a polar
angle and how large an azimuthal angle of the sphere to draw.

Also, there's a "coordinate box" to help visualize the position of the
endpoint more easily.

Here's a fun variation on the twisty rod file: Simply plot z=A(x,y);
this shows, at x,y, the amount twisted at position x along the rod, at
time y. However, really, for the rod, 0<x<1 and y>0 (shown in red). If
we look at a larger domain, we see that the series sums up to an
egg-carton shape, with square pyramids, up and down, for the dimples.
(Alternatively, this could be seen as the top of a layer of packed
rhombic dodecahedra)

As far as I can tell, the edges are actually sharp, at least as the
number N of terms increases. (Even around n=10 or so, they would appear
perfectly sharp, I think, if rendered exactly, but GC is assembling the
picture out of little squares)

Have fun,

Building on Ron's suggestion, I'm using a sphere to graph the points of
a polyhedron. In this case, it's an octahedron, but if you just plug in
the right angles, you can graph any spherically inscribed polyhedron.

The following puts all the points into one function, O_H(…, …) :

Then the following equation will plot them all at once:

Almost have one function drawing all the faces of an octahedron. The
trick is getting the plotting to "wrap around" and plot the face from
vertex 5 back to vertex 2

By transforming the sphere it's defined on, you can transform the
points of the octahedron.

Here are a pair of files illustrating the explanation in the Wikipedia
article on the "Roman surface",
http://en.wikipedia.org/wiki/Roman_surface :
If you open them together, they should be in position for 3D viewing.
Just cross your eyes slightly so that a third image appears in the
middle, and then focus on this middle image.

Here's an interesting progressive plot where we start out with three
flat squares and they're bent into saddle shapes (hyperbolic
paraboloids).

I'm just guessing here, but it seems as though there must be some
connection here to the fact that in the projective plane every point
has three coordinates, since we use homogeneous coordinates.

Might be possible to do some interesting visualizations of tilings and
of theorems in projective geometry using these files.

Having the option to draw lines as 3D tubes (coming in the next
release) should make this kind of thing even clearer.

Here's a really beautiful but, as far as I know, meaningless, animation
that looks like unfolding butterfly wings, or something.

I seem to have the right formula in cylindrical coordinates. I want to
be able to have these grid lines, but be able to rotate and translate
these shapes around. I thought I could just add something to the angle
to rotate it, but am too lazy to try to figure this out now.

Excuse me for some stuff that's mostly of aesthetic interest, but
here's a really nice symmetrical view of three wings consisting of
hyperbolic paraboloids unfolding:

Here's the yellow wing (at top above) alone:

And here it is from the side:

As n gets smaller, it gets smaller and steeper:

The ruled-surface graphing seems to be harder to grasp than the
cylindrical contour lines or the curving gridlines.

Start the animation by clicking the "Play" arrow.

It helps to visualize this if you start the surface rotating around the
z-axis (by pressing Option-Control-z, at least on Macs) and then press
the "Play" arrow.

Although the hyperbolic paraboloids on each face of the tetrahedron are
curved, the intersections are straight lines.

It might be interesting to find and put dots on the singular points, as
mentioned in the Wikipedia article.

The drawings make clear how a model could be built with colored yarn,
for example. You just need a tetrahedral frame with regularaly spaced
holes for the yarn.

Here's a spinning model with just the grid lines. Making a two-pane
synchronized 3D view would really help. Also, with the 3D tubes in GC's
next release, this should really be stupendous.

Making use of the helpful "Moebius Band" file by Rodney Topor at
http://www.pacifict.com/today/moebius.gcf, I've been learning a lot
about rotations in cylindrical coordinates. Here are some efforts to
visualize in detail exactly what's going on:

The Möbius strip has one-half spin:

http://mathworld.wolfram.com/MoebiusStrip.html :
> The Möbius strip, also called the twisted cylinder (Henle 1994, p.
> 110), is a one-sided nonorientable  surface obtained by cutting a
> closed band into a single strip, giving one of  the two ends thus
> produced a half twist, and then reattaching the two ends (right
> figure; Gray 1997, pp. 322-323). The strip bearing his name was
> invented by Möbius  in 1858, although it was independently discovered
> by Listing, who published it, while  Möbius did not (Derbyshire 2004,
> p. 381). Like the cylinder, it is not a true surface, but rather a
> surface with boundary (Henle 1994, p. 110).

Here's one spin per revolution:

And here are 4 spins per revolution:

In effect, the r and z coordinates when you're in cylindrical
coordinates can be thought of as a fixed plane at angle theta, so you
can revolve any object drawn with fixed theta around the z-axis by just
changing theta.

These definitions show how the more complicated situations are built up
using the simpler ones:

For the last definition, note that

are the cylindrical coordinates, so the matrix

spins around the ellipse in the rz plane for a given angle theta.

To apply the above defintions, we just carefully use different
combinations of parameters:

You can examine just what each equation does by varying the
sector-tracing parameter "S".

For ellipses, I really shouldn't have had an "r" parameter and an "h"
parameter. Just needed one angle parameter. But as a fringe benefit,
you can get nice Lissajous type cross-sections to spin around.

This could be a little clearer, and I need to add something for the
up/down reversal. Also, it might be more accurate to actually use a
single twisted surface, although that would be harder to visualize.

It's a good idea to option-click on the animation arrow, so that the
pointing finger will keep on going around in the same direction,
instead of confusingly reversing direction.

I think this is the best version yet. The whole graph rotates as the
finger goes around the strip.

Following is a full-screen version. Best yet, but picture to big to
show.

On second thought, all the colored stripes were kind of distracting,
and prevented the surfaces from being at top resolution.  I think this
is clearer, and focuses attention on the important thing, the
travelling fickle finger of fate.  (Any Laugh-In fans out there?)  Any
preferences?

A final effort.  I didn't have both patches transparent above, as I do
here. This makes them lighter and easier to see.

Making the torus opaque may be even clearer:

Hello everyone. I have recently joined the group and would like to access
the files that are posted there - going to members/files doesn't seem to
work. Also I need to know a simple way of graphing 3d polygons by their
vertices. I am using the PC version of the program. Can anyone help?
Thanks James

Last night I uploaded a dozen or so files to the physics section, mostly
concerned with waves and oscillations.  They're not the works of art
that Chris produces, but I have found them very useful for pedagogical
purposes as classroom demonstration tools.  My favorites are the Doppler
shift demos and the damped, driven pendulum phase trajectory plots.

David Craig


<http://www.panix.com/~dac/>

Probably a sign of my spotty mat education, but I was amazed to
discover the following algebraic properties of the Golden Mean (aka
Golden Ratio, Golden Section, Extreme-Mean Ratio, etc.)

Wikipedia has a good article at
http://en.wikipedia.org/wiki/Golden_ratio

>

Also Mathworld, at http://mathworld.wolfram.com/GoldenRatio.html :

>  has surprising connections with continued fractions and the Euclidean
> algorithm for computing the greatest common divisor of two integers.
>

>
>
> Given a rectangle having sides in the ratio

> ,

>  is defined such  that partitioning the original rectangle  into a
> square and new rectangle results in a new rectangle  having sides with
> a ratio

> . Such a rectangle is called a golden rectangle, and successive points
> dividing a golden  rectangle into squares lie on a logarithmic spiral.
> This  figure is known as a whirling square.

It seems to me that the "Pringles Chip" view of hyperbolic paraboloids
that you get in cylindrical coordinates is the clearest.

But I can't seem to rotate these into position using their
cylindrical-coordinates form:

The idea is to end up with a similar situation to the following, which
is a tetrahedral analog to the the "Roman surface".

In these two views, even though the three saddle-shapes that make up
the surface have only half-filled in, one of the "dimpled faces" of the
tetrahedron has been completely filled in already.

On Aug 5, 2006, at 2:13 AM, James Taylor wrote:

> Hello everyone. I have recently joined the group and would like to
> access
>  the files that are posted there - going to members/files doesn't seem
> to
>  work. Also I need to know a simple way of graphing 3d polygons by
> their
>  vertices. I am using the PC version of the program. Can anyone help?
>  Thanks James

Here is a no-frills version, with no functions, just the points and the
minimum things you need to do to get lines and surfaces.

I couldn't resist trying out a few fancy graphics effects, but you can
get good results just with plane colors for the faces.

I hope this helps.  If there are any questions, please feel free to ask.
I've sent some e-mails showing how you can speed up the process using
variables and functions, and will be posting them soon. I'd be glad to
send them out again.

Edges get drawn like this.

If you put a transparent drawing on top of another one, you can get
some interesting effects.  The equation that comes later is drawn on
top. At least, that's the way it seems to work most of the time, but
sometimes things may get reversed.

Just experimenting a little more with overlaying surfaces. I may have
been wrong; it looks like the surfaces may get drawn with the ones
listed sooner being on top. I'm confused about this.

Chris
Thank you very much for your explanation, although I am still thinking as to
why it works. As it happens, I'd figured out a similar approach in the
meantime, but it was mainly trial and error.
By perseverance I discovered that uv(p-q)+u(q-s)+ s does the job, which I
then re-wrote as
uv(p)-u(v-1)q -(u-1)s

By sheer coincidence, my project involves the golden ratio. Some years ago I
came across the interesting fact that the parallel edges of an icosahedron
form the opposite sides of a golden rectangle, the other side of which is a
space diagonal. Further, the golden rectangles occur in three mutually
perpendicular groups. I proceeded to make a model of this, which I have to
this day.
I recently decided to make a computer model of it, hence my request. My
latest version is attached, showing several faces removed so that the
rectangles can be seen.  I used the fact that the ratio of succeeding terms
of the Fibonacci sequence approaches the G.R. and so I used the successive
terms 21 and 34 as the sides of the rectangles.)

James

-----Original Message-----
From: Chris Young [mailto:c1572young@...]
Sent: Monday, 7 August 2006 12:24 AM
To: James Taylor
Cc: GraphingCalcUsers group
Subject: Re: [GraphingCalcUsers] polygons and files


On Aug 5, 2006, at 2:13 AM, James Taylor wrote:

> Hello everyone. I have recently joined the group and would like to
> access
>  the files that are posted there - going to members/files doesn't seem
> to
>  work. Also I need to know a simple way of graphing 3d polygons by
> their
>  vertices. I am using the PC version of the program. Can anyone help?
>  Thanks James

Here is a no-frills version, with no functions, just the points and the
minimum things you need to do to get lines and surfaces.

I couldn't resist trying out a few fancy graphics effects, but you can
get good results just with plane colors for the faces.

I hope this helps.  If there are any questions, please feel free to ask.
I've sent some e-mails showing how you can speed up the process using
variables and functions, and will be posting them soon. I'd be glad to
send them out again.

Hi,

I thought I would say something about linear interpolation. This is a
simple but useful way to connect all sorts of things, and these
triangular faces are a special example.

Suppose we have object A that we want to morph to object B, by just
pairing spots on each, and then tracking straight from a spot on A to
the corresponding spot on B. All you have to do is

A*(1-s) + B*s

where s runs from 0 to 1. When s=0, we have "all A"; when s = 1, we
have "all B", and in between, we have a weighted avg of each.

In particular, the line segment between two points p and q is just
p*(1-s) + q*(s)  [to actually plot this in GC, use t, since this is a
curve]

But A and B could be anything at all! Suppose we have two curves A(t)
and B(t); then for each t0, we will connect A(t0) to B(t0); since we
want a surface, and GC requires u's and v's, we plot

A(u) * (1-v) + B(u) * v

Now to put it all together: If A is the line segment between p and q,
and B is the point r, we get

(   p*(1-u) + q*(u)  )  *(1-v)    +    r * v

    ^^^^^
line segment

This simplifies, and (switching all the points around) you get James'
expression.

BTW, the reg dodeca nicely inscribes a cube and three "double golden"
rectangles (phi:1/phi)







On Aug 7, 2006, at 6:00 AM, James Taylor wrote:

> Chris
>  Thank you very much for your explanation, although I am still
> thinking as to
>  why it works. As it happens, I'd figured out a similar approach in the
>  meantime, but it was mainly trial and error.
>  By perseverance I discovered that uv(p-q)+u(q-s)+ s does the job,
> which I
>  then re-wrote as
>  uv(p)-u(v-1)q -(u-1)s
>
>  By sheer coincidence, my project involves the golden ratio. Some
> years ago I
>  came across the interesting fact that the parallel edges of an
> icosahedron
>  form the opposite sides of a golden rectangle, the other side of
> which is a
>  space diagonal. Further, the golden rectangles occur in three mutually
>  perpendicular groups. I proceeded to make a model of this, which I
> have to
>  this day.
>  I recently decided to make a computer model of it, hence my request.
> My
>  latest version is attached, showing several faces removed so that the
>  rectangles can be seen. I used the fact that the ratio of succeeding
> terms
>  of the Fibonacci sequence approaches the G.R. and so I used the
> successive
>  terms 21 and 34 as the sides of the rectangles.)
>
>  James
>
>  -----Original Message-----
>  From: Chris Young [mailto:c1572young@...]
>  Sent: Monday, 7 August 2006 12:24 AM
>  To: James Taylor
>  Cc: GraphingCalcUsers group
>  Subject: Re: [GraphingCalcUsers] polygons and files
>
>  On Aug 5, 2006, at 2:13 AM, James Taylor wrote:
>
>  > Hello everyone. I have recently joined the group and would like to
>  > access
>  > the files that are posted there - going to members/files doesn't
> seem
>  > to
>  > work. Also I need to know a simple way of graphing 3d polygons by
>  > their
>  > vertices. I am using the PC version of the program. Can anyone help?
>  > Thanks James
>
>  Here is a no-frills version, with no functions, just the points and
> the
>  minimum things you need to do to get lines and surfaces.
>
>  I couldn't resist trying out a few fancy graphics effects, but you can
>  get good results just with plane colors for the faces.
>
>  I hope this helps. If there are any questions, please feel free to
> ask.
>  I've sent some e-mails showing how you can speed up the process using
>  variables and functions, and will be posting them soon. I'd be glad to
>  send them out again.
>
>    <goldenrecicos.gcf>

Very nice job! I colored things symmetrically to try to make things a
little clearer.

Would you feel like uploading one of these versions to the Geometry
section, or could I, giving you credit for it?


On Aug 7, 2006, at 7:00 AM, James Taylor wrote:

> Thank you very much for your explanation, although I am still thinking
> as to
> why it works. As it happens, I'd figured out a similar approach in the
> meantime, but it was mainly trial and error.
> By perseverance I discovered that uv(p-q)+u(q-s)+ s does the job,
> which I
> then re-wrote as
> uv(p)-u(v-1)q -(u-1)s
>
> By sheer coincidence, my project involves the golden ratio. Some years
> ago I
> came across the interesting fact that the parallel edges of an
> icosahedron
> form the opposite sides of a golden rectangle, the other side of which
> is a
> space diagonal. Further, the golden rectangles occur in three mutually
> perpendicular groups. I proceeded to make a model of this, which I
> have to
> this day.

> I recently decided to make a computer model of it, hence my request. My
> latest version is attached, showing several faces removed so that the
> rectangles can be seen.  I used the fact that the ratio of succeeding
> terms
> of the Fibonacci sequence approaches the G.R. and so I used the
> successive
> terms 21 and 34 as the sides of the rectangles.)

Here are a couple attempts to visualize the rectangles inside. Also, I
just grabbed the coordinates from Wikipedia, although I'd still like to
change the orientation so that there's rectangle lengthise along the
x-axis. I'm trying to make out if there are any relationships between
the order of the corners and the pattern of connections. Is there some
way we could show a morph from 3 stacked Golden Rectangles to an
icosahedron?

> http://en.wikipedia.org/wiki/Icosahedron#Cartesian_coordinates :
>
> ”ČThe following Cartesian coordinates define the vertices of an
> icosahedron centered at the origin:
>
> (”Ž1, ”Ž¦Õ, 0), (”Ž¦Õ, 0, ”Ž1), (0, ”Ž1,
”Ž¦Õ)
>
> where  ¦Õ = (1+¢å5)/2 is the golden ratio (also written ¦Ó).
>
> Note that these vertices form sets of three mutually orthogonal golden
> rectangles.Ӄ

I wish GC's contrast wasn't so intense (or that it could be
adjusted)—it's often hard to make out all the colors in opaque mode.

Actually, I was just after a black-and-white checkered grid, but with
square checkers, which isn't what you get with the usual
paratermizations with u and v.

Here are views with "Taxicab distances" -- |x| + |y|. Not sure what
it's could for, just fishing around for connections. ... Just checked
my "Platonic and Archimedean Solids" book and notice that the dual to
the icosaahedron is the dodecahedron. I.e., that's the regular
polyhedron that just fits inside and outside with vertices on face
centers and vice versa.

It turns out that if you connect the midpoints of all the short ends of
the rectangles, you get an octahedron nested inside the icosahedron. I
got this from page 54 of the excellent book "Platonic & Archimedean
Solids: The Geometry of Space" by Daud Sutton (Wooden Books, 2002),
which has pictures (a little hard to interpret) showing "Each [regular
polyhedron] Embracing Every Other".

What it all means, I'm not sure. Given the connections between the
symmetries of the icosahedron and the solvability of the 5th degree and
higher equations, it might have some real significance. I understand
virtually nothing about this, despite having attempted to fathom Gabor
Toth's explanation in "Glimpses of Algebra and Geometry" and similar
texts.

For any algebraic geometry experts out there: Is there some way we can
illustrate a fifth-degree equation with particular coefficients by
displaying an icosahedron which has been distorted or transformed or
mapped to another one in some way? Could the distortion or
transformation then give us insight into why that particular
higher-degree equation is or is not solvable in explicit form via root
extractions, etc.?

It would be interesting to know if all the midsphere-inscribed
polyhedrons can be found via "taxicab" distance.

A little clearer if we gray out everything outisde of the green contour:

Or make everything outside black and use a black background:

By parametrically plotting each quadrant of the inscribed rectangles,
we could get the corners out of the way comletely.

These versions might work especially well for 3D pairs.

More amusing (but a corollary) is to inscribe the icosahedron in the
octahedron: eight of the faces of the icosahedron lie on the same
planes as a suitably sized octahedron. (See attached photo: the icosa
is in yellow, embedded in the model)

There are five choices for this set of eight planes, corresponding to
the five ways you can pick three mutually perp pairs of edges in the
icosa (or to put it another way, the five ways you can pick some
'axes')

So in turn this gives a nice way of circumscribing the icosa with 5
octas, the famous compound of 5 octas. Since their faces all lie on the
facial planes of the icosa (and some other technical stuff) this is
known as a "stellation" of the icosahedron. (Careful: there are two
distinct, and non-interchangeable notions of stellation; this is the
one preferred by polyhedralists)

If we take just every other face of the octahedron, these lie on the
facial planes of a tetrahedron. Correspondingly, an icosahedron can be
inscribed in a tetrahedron (see attached photo), and a compound of 5
tetrahedra is another interesting stellation of the icosahedron. In
fact, this choice of inscription comes in two flavors: right and left
handed, and so there are right vs left handed compounds of 5 tets. Or
we can combine them and produce yet another stellation, a compound of
ten tetrahedra, with facial planes exactly those of the icosahedron.

One can "dualize" all of this, and produce, for example, a compound of
five cubes inscribed in the dodecahedron.

What drives all of this? One can produce deep explanations I'm sure,
but most simply, its only because of the nice coordinatization of the
icosa, that allows it to be so wonderfully lined up with the x, y, and
z axes.

Some nice books to start with are "Polyhedra" by Cromwell, "The 59
Icosahedra" by Coxeter et al., "Polyhedron Models" by Magnus Wenninger,
"Shape Space and Symmetry" by Holden.

Currently, it is quite tedious to make these pix by hand in GC. One
approach I'd like to experiment with is to have an external
program/script generate the GC files.

Chaim

> It turns out that if you connect the midpoints of all the short ends of
>  the rectangles, you get an octahedron nested inside the icosahedron. I
>  got this from page 54 of the excellent book "Platonic & Archimedean
>  Solids: The Geometry of Space" by Daud Sutton (Wooden Books, 2002),
>  which has pictures (a little hard to interpret) showing "Each [regular
>  polyhedron] Embracing Every Other".
>
>  What it all means, I'm not sure. Given the connections between the
>  symmetries of the icosahedron and the solvability of the 5th degree
> and
>  higher equations, it might have some real significance. I understand
>  virtually nothing about this, despite having attempted to fathom Gabor
>  Toth's explanation in "Glimpses of Algebra and Geometry" and similar
>  texts.
>
>  For any algebraic geometry experts out there: Is there some way we can
>  illustrate a fifth-degree equation with particular coefficients by
>  displaying an icosahedron which has been distorted or transformed or
>  mapped to another one in some way? Could the distortion or
>  transformation then give us insight into why that particular
>  higher-degree equation is or is not solvable in explicit form via root
>  extractions, etc.?
>
>  It would be interesting to know if all the midsphere-inscribed
>  polyhedrons can be found via "taxicab" distance.
>    <J1e. Making inscribed octahedron clearer.gcf>
> <pastedGraphic12.tiff>
>
> A little clearer if we gray out everything outisde of the green
> contour:
>
> <pastedGraphic15.tiff>
> <J1f. Making inscribed octahedron clearer.gcf>
> <pastedGraphic14.tiff>
>
> Or make everything outside black and use a black background:
> <J1g. Making inscribed octahedron clearer.gcf>
> <pastedGraphic17.tiff>
> <pastedGraphic16.tiff>
>
> <pastedGraphic18.tiff>
>
> By parametrically plotting each quadrant of the inscribed rectangles,
> we could get the corners out of the way comletely.
>

On Aug 8, 2006, at 9:19 AM, C Goodman-Strauss wrote:

> So in turn this gives a nice way of circumscribing the icosa with 5
> octas, the famous compound of 5 octas. Since their faces all lie on
> the facial planes of the icosa (and some other technical stuff) this
> is known as a "stellation" of the icosahedron. (Careful: there are two
> distinct, and non-interchangeable notions of stellation; this is the
> one preferred by polyhedralists)
>
> If we take just every other face of the octahedron, these lie on the
> facial planes of a tetrahedron. Correspondingly, an icosahedron can be
> inscribed in a tetrahedron (see attached photo), and a compound of 5
> tetrahedra is another interesting stellation of the icosahedron. In
> fact, this choice of inscription comes in two flavors: right and left
> handed, and so there are right vs left handed compounds of 5 tets. Or
> we can combine them and produce yet another stellation, a compound of
> ten tetrahedra, with facial planes exactly those of the icosahedron.

Very interesting, thanks. I'm going to keep this on file for future
efforts.

> One can "dualize" all of this, and produce, for example, a compound of
> five cubes inscribed in the dodecahedron.

This sounds like a super-cool shape to make. Will see what I can do.

> What drives all of this? One can produce deep explanations I'm sure,
> but most simply, its only because of the nice coordinatization of the
> icosa, that allows it to be so wonderfully lined up with the x, y, and
> z axes.
>
> Some nice books to start with are "Polyhedra" by Cromwell, "The 59
> Icosahedra" by Coxeter et al., "Polyhedron Models" by Magnus
> Wenninger, "Shape Space and Symmetry" by Holden.
>
"Polyhedra" looked like a fine book, graphically and matematically. The
old Coxeter classic, "Regular Polytopes", has an amazing amount of
information crammed into it, but "unpacking" it is a huge job ---
hardly any diagrams and densely written.  A job for GC.

> Currently, it is quite tedious to make these pix by hand in GC. One
> approach I'd like to experiment with is to have an external
> program/script generate the GC files.

That sounds like a fine idea. But even easier to use, I think, would be
to have multiple parameters lists and to have the bugs taken out of the
"indexing", where you have functions defined using integer arguments.
As I understand it, Ron is working on these things for the next
release.