```Here's a brief explanation of how to put the penultimate modules
together, plus some FAQ's.  I'm also posting longer commentary by
Kenneth Kawamura (originally posted to origami-l).

----------------------------------------------------------------------
Make three "pentagon" modules -- call them
1, 2, and 3.  Each will look roughly like this:

A                                    B      C
--------------------------------------------
|\                             ___--- \    |
| \                      ___---        \   |
|  \               ___---               \  |
|   \        ___---                      \ |
|    \ ___---                             \|
--------------------------------------------
D     E                                    F

All the above are mountain folds.

If you started with 4x3 paper, each of the long edges of the rectangle
(AC and DF) will have two layers.  What you are going to do it slip the
D corner of module 1 in between the two layers of module 2 right at
point E.  What you'll want is for the AE fold of module 1 to line
up exactly on the EB line, in the inside of module 2.  Point E of
both modules should be in the same place.  You'll end up with
something that looks like:

A                                    B      C
--------------------------------------------
|\           module 2          ___--- \    |
| \                      ___---        \   |
|  \               ___---               \  |
|   \        ___---                      \ |
|    \ ___---                             \|
--------------------------------------------
D     E\           \                       F
\           \
\           \
\           \
\           \
\           \B
\ module 1  \
\           \
\          _\C
\     __--
\__--
F

At this point, nothing will be solid.  Now, take the D corner of
module 3 and put in between the two layers of module 1 right at point
E, and line up AE of module 3 with EB of module12.  Finally, take the
D corner of module 2 and put in between the two layers of module 3
right at point E, and line up AE of module 3 with EB of module 3.
This will be a little harder than before, but what you'll end up with
is a rigid vertex of a dodecahedron.  Do the rest in this manner --
it will take 30 modules.  Modules meet at the 20 corners in groups of
three, and make five-sided faces.  Looking at my pictures of
dodecahedrons while you're doing this may help.

Once you've made a dodecahedron, making the others should be
straightforward.

Jim Plank
plank@cs.utk.edu
http://web.eecs.utk.edu/~jplank

----------------------------------------------------------------------
>Hi james!  Could you please explain to me what you mean by saying,
>for example, "pentagon/hexagon" hybrid.  I'm assuming its a unit that
>has combined features, but how do I go about creating one?  The
>directions on your website has directions for only non-hybrid kinds.
>could you please shed some light?

Hi - in the hybrid modules, the angles about points B and E.  For
example in a pe-he module, after making the diagonal fold
(EB), you make the left half by folding the whole module along
AE.  Then you make the right half by making the 120 degree
angle using the instructions for the hexagon module.

A                                    B      C
--------------------------------------------
|\                             ___--- \--  |
| \                      ___---        \ --|
|  \               ___---               \  |G
|   \        ___---                      \ |
|    \ ___---                             \|
--------------------------------------------
D     E                                    F

When you're done, you'll have a 108 degree angle (BEA), and
a 120 degree angle (EBG).  The 108's make pentagons and
the 120's make hexagons.

See web.eecs.utk.edu/~jplank/plank/pics/origami/penultimate/help.html

Good luck,

Jim Plank

----------------------------------------------------------------------
From origami-l@nstn.ca Mon Jan 27 03:21:29 1997
To: origami-l@nstn.ca
Subject: Re: Origami-L: Pentas AND Modules AND School

In a message dated 97-01-26 12:02:47 EST, you write:

<< My problem is . . . I got lost after "accordian pleat a square into
fourths"!  >>

If that's the Robert Neale module for the dodecahedron (from 30 modules),
that I remember, the accordian pleat can be done by: (see the disclaimer
further down)

(1) position the square in front of you, bottom edge going left to right, and
colored-side up;

(2) (fold) raise the lower edge to the upper edge, for a valley fold, to
bookfold the square in half, horizontally, colored-side in (this is an
orthogonal fold, edge-to-edge, as opposed to a diagonal fold);

(3) (position), rotate 180 degrees (half-circle) clockwise, so the folded
edge is the upper right-to-left edge;

(4) (fold) then raise the raw long bottom edge of the upper layer half-square
flap, to the folded edge, and valley fold another lengthwise bookfold, so the
long raw edge lies on the folded edge from step (1), colored-side out;

(5) (position) and turn the paper over, left-to-right, to get at the other
half square flap, and repeat step 4 on this second flap.

To finish the module:

(6) (orientation & strategy) You should now be looking at the "accordion
pleat" in fourths, colored side out. It is a 1x4 rectangle. Think of it as
being divided into four 1x1 squares. You want to fold three diagonal creases,
in a zigzag pattern. The left and right will be diagonals of the left and
right squares; and will be parallel to each other. The middle crease will be
a diagonal of the 1x2 rectangle formed by the two middle squares. And these 3
diagonal creases connect end-to-end.

(7) (arbitrary choice) One possibility is to valley fold the three diagonals
as follows, from the square on the left, upper left corner to lower right
corner, then across the middle two squares lower left corner to upper right
corner, then across the remaining square upper left corner to leower right
corner.

It doesn't matter whether you start upper left corner to lower right corner
in the leftmost small square, or lower left corner to upper right corner, as
long as you are consistent with all 30 modules. (This means you have a
choice, between all "left-handed" modules, or all "right-handed" modules, but
you can't mix them. At least I think you can't. If you can, I'd be interested
in the result.)

(Note, I said valley fold, only because, my experience is that a valley fold
is easier to line up, than a mountain fold. Also, the valley-folded side of
the module will face the interior of the dodecahedral ball. If you have a
module that looks better on one side, you can choose which side goes out.
This helps if you're using wrapping paper, or have an accident.)

8) (additional orientation) The two little end triangles are tabs that  go
into the "pockets" between the layers of other modules, and "hook" over the
middle diagonal crease.

(There is actually one "pocket" too many, since the accordian pleat left you
with two on one long side, and one on the other long side. This model could
be made from 3x4 rectangles, pleated in thirds lengthwise, giving up the
extra flap, but it is stronger with the extra layer.)

Assembly:

9) (strategy) The modules go along the edges of the dodecahedron. That middle
diagonal is the edge. (Note that the angle between the diagonals of one
module where they meet, is almost exactly the angle between the edges of a
pentagon. Real close, but not mathematically exact.) They will group into 12
pentagonal rings of 5 modules, with neighboring rings sharing an edge, and
exactly 3 modules/edges will meet at each of the 20 corners of the
dodecahedron.

19) (assembly) The tabs are slipped into the pockets, between the layers, of
the neighboring module, far enough to "hook" over that module's middle
diagonal. Then slide the first module down, so it's hooked on at one end of
the second module's middle diagonal. When this is done right, the middle
creases of the two modules will be end-to-end, at almost the angle of a
pentagon.

I know this isn't as clear as it could be, but hopefully this helps?

Jim Plank's pages show his extension of this base module, by varying the
angle at the end of the edge, so the ring on that side of the edge is a
different shape, and can be used for the other polyhedra. One of those neat
ideas I wish I'd thought of.

I see from my notes, that he's simplified the module to the 3x4 rectangle, to
get rid of the confusing extra pocket, and have a different inside and
outside color.

Disclaimer:

This is not the only way of putting in the creases. Feel free to fold in
whatever way works for you. Find the direction and folding method that is
most comfortable and accurate for you.

These instructions are for the way I fold, since, for me, folding an edge up
to and edge, feels easier, than folding an edge, left-to-right to an edge;
and valley folding usually feels easier than mountain folding.

Also, I am left-handed, so don't take my left-to-right directions as gospel.

Sometimes it's easier to move a corner to an edge, and then adjust the
position until points, corners, edges, etc. line up, and then squash the
paper.

Sometimes, I move the paper to align landmarks, and lightly pinch the paper
to mark the new in-between landmarks, that I can then use as starting and
stopping places for edges or folds.

I've tried closing my eyes, and letting my fingertips feel for the edges
lining up, especially when the lighting is low or my eyes are tired.

Mountain folding does feel easier, when I'm folding a crease from one point
to another, where I can pinch at the two points, and stretch the paper
between the two points and squash the mountain fold into place.

I've also partly learned the trick of putting two landmarks together, holding
them tight, and using the other hand to pull and squash the paper away from
the holding point. That lets you put two corners together, and bookfold the
square in half in one or two sweeps, and have the corners and edges line up,
provided you held on tight, and kept an even tension during the "sweep".

But sometimes, I just pinch-pinch-pinch along the line, to locate it and pin
it dowen accurately, then go over it with one or two squashing sweeps, to get
a nice sharp final crease.

When I learn a fold, I may adjust the folding sequence to:
a) make it easier and more comfortable;
b) locate the folds, neatly, and accurately;
c) add a little slack, where the model wants it, for instance, inside the
head and tail of the crane;
d) have the model proportions look "right";
e) Adjust the folding to minimize the wrong  side of the paper showing;
f) try to eliminate extra creases that show on the outside of the model;
g) take advantage of the pattern of the paper.

Aloha,

Kenny1414@aol.com

Kenneth M. Kawamura
PO Box 6039
E Lansing  MI  48826-6039

```