Kawasaki Cluster

After this little foray, if you are brave enough, please consider this cluster as just one cell of a much bigger tessellation.  Also, you should be able to "Kawasaki" any standard origami tessellation.  The best candidates for twisting are tessellations that look kind of boring though: think of the underlying tessellation as a base.  A field of twists, preferably closed backed and spaced well apart make good tessellations, here are examples of the kinds of things you can do:
Other Kawasaki Rose Tessellations

Anyway, on to the folding:
1. Start with a square, and a pair of scissors, and this method of creasing a maximal hexagon into the square:

2. Use the scissors ... unless bleeding paper makes you queasy.  Crease mountain creases along the 3 diagonals of the hexagon.  We're going to make a regular closed-back hexagonal twist right in the center.

3. You can, of course, make the next fold precisely, but I just make it by judgment.  Make valley folds on each side of the mountain folds.  Their exact width isn't important, but they should be at least a finger span wide or you will have trouble opening up the center twist at the end of the model. Here, I've penned in the valley folds (there are six of them).

4. Fold those valleys, note how there is now a star of david folded into the center of the hexagon.

5. We will make the hexagon twist now, in a straightforward generalization of this method for square twists: Make the indicated hexagon all mountain folds.

6. Fold in half along one of the main diagonals, but don't completely compress the model.  Keep the depressed hexagon in the center!

7. Fold the upper third and the lower third in opposite directions ... on the way to folding completely in half.

8. And now flat.

9. Fold the mountain edge of the lower third as below ... this should expose the central hexagon

10. Now, unfold the model keeping the center hexagon flat, and you should have the center hexagon twist done.  There are lots of ways to make the twist ... let me know if these directions didn't work out.  I could provide a crease pattern ... but what's the fun in that.

Here is what the twist looks like from the other side of the paper ... where I would call it a hexagon sink:

11. We're going to make triangle twists OF THE OPPOSITE HANDEDNESS along each of the six arms of the twist.  Again, I am going to do this by judgment folding, you could make this precise by figuring out how to make adjoining twists so that they are equally spaced along the edge of the model. First, open up one of the pleats from the white side of the paper, and inside reverse fold these triangular sections with two mountain folds:

Can you see two equilateral triangles here ... after I've folded these mountains, the part of the main mountain fold going to the edge is going to change to a valley fold:

Do this the same way to each of the remaining five arms.

Now ... Here is how to form the triangular twists at each of these "triangular" water bomb summits.  I will let a short animation speak for me, although you could make the appropriate precreases, just as for the hexagon twist above.  The method I use here is written down in "Origami for the Connoisseur" by Kasahara as the method for making a square twist.  I call it the "squash the point":
How to make a triangle twist:
Notice that in doing this, you will be making judgment VALLEY folds along each of the mountain folds going up to the summit.  You will only have one choice for which way to fold the spar leading to the hexagon twist at the center.  The other two pleats can only be formed one way so that everything folds flat up to some critical point near the summit.  As in the animation, keep smoothing out the paper closer and closet to the summit from each of the three directions, and eventually the point of the summit will squash itself flat, at which point you can go and clean up the creases with your finger.

12. The base is done, a not very interesting tessellation that folds flat.  Here is the back:

And, here is what it looks like from the colored side:

Now ... you've probably been at a similar point with lots of folds.  Now is the time to do something new with it. When we are done, a twist rose of Kawasaki, one with six petals in the center, and six with three petals each around the edge will be formed.

13. "Kawasaki" the hexagon sink.  This is best described in the following animation, although the point is just to fold the correct valley folds from the white side of the paper so that the center rose can twist open.  This involves a construction just like one you may be familiar with from folding the good old Kawasaki rose.  Basically, you are folding each pleat of the model so that it will lay against the next one.
kawasakize.mpg
When you have done all six petals, your model will look like this ... and I wish I could execute this part w/o wrinkling the paper so much.  Do you ever wish that you could fold your own models ... better?

14. Now, we Kawasaki-ize each of the six outer triangle sinks.  Again, a movie is the best way to show how to do this.  Once you have this move down pat, you should be able to fold the Symmetric Kawasaki Rose Crystallization, the asymmetric version, and each of the versions I've put on this page: Kawasaki tessellations.
Kawasaki the tri-sinks
When you have done, here is what the model looks like from the front

15. Now, we are ready for the collapse.  Squish the model from the edges, and the center hexagon twist should wrap up tightly.
Collapse the model.
As you can see, there is a substantial amount of "shaping" to do at this point.  The collapse will be helped out by creasing sharply from the white side all of those Kawasaki fold you put in on the previous two steps.  These creases do not always meet at 120 degree angles!  The triangle folds and the hexagon folds are perpendicular:

16. Now, open up all the sinks from the colored side so you can see the "rose" effect:

I've worked on the hex in the center, as well as the triangles.  To get the hex to open up, use a click pen (w/ the point retracted) or chopstick:

The best tool, once you've got a large enough opening, is your finger:

Stick your finger (or thumb) into the hole, and twist with the same sense as the pleats rotate ... the center will tighten around your digit, and the twist will become very well creased near the center.

17. Finishing:
There will be triangular flaps at the edge of the model when you are done.  You can hold on to them and use the model as a flasher:
Cluster Flasher
Or, you can lock them into the model this way:
Lock the sides
And here is the finished product.  Make a hexagonal masu-like box to hold it, and it will look really sharp:

And from the back:

Notice that the back of the model consists of a six fold tube holding the hexagon twist, and a bunch of three-fold tubes containing the satellite triangle twists.

My square started off as 12 inches:  and the final model fits well into a coffee cup:

Sprinkle the model with water in the cup, and then microwave until it is dry ... 30 sec or so ought to do it.  Instant wet-folding to hold the locks set.  Maybe "microfolding"?

If anyone possibly manages to follow these instructions ... reward me with a picture of your cluster, or Kawasaki-ed tessellation!