How to Fold a Pentakis Dodecahedron

Math Craft admin Cory Poole provided quite a few recipes for sonobe models in his blog, and I followed one to make the pentakis dodecahedron here.

I wasn't that happy with my first result because the colours kept on coming out wrong—i.e. two of the same colour next to each other. I am still not sure whether it is possible to make one with three colours without getting this happen. You can definitely make it with four colours, but three, not so sure.

To work out a colour scheme to plan any new model, I draw a graph of it. A graph can be thought of as a two dimensional representation of a three dimensional surface. The best way to think of it is to think of a model made with a sheet of super stretchy rubber, with a hole in one face, and the edges of that hole stretched it out until it was completely flat. This is a graph of a dodecahedron—the wiggly line around the pattern is the hole in the white face that has been stretched out. The dots are vertices and the lines are edges. You can see that the faces are distorted, but they still all have 5 edges, and the relationships between the faces stay correct.

Make Your Sonobe Units   

You will need 60 sonobe units—5 each of twelve different colours. Go here to learn how to fold a sonobe unit.

These are made from 50mm squares (i.e. nine pieces from each sheet of 150mm origami paper.)

Turn Your Sonobe Units into Compound Sonobe Units   

You now need to make a two module compound. **IGNORE THE COLOURS** on my photos of this step, as they are made from a different model. :-)

First, make the creases. The sonobe has a face with the pockets where the edges cross and one which is plain; with the pocket side facing you, fold the RHS triangle away from you and the LHS triangle towards you.

Next, fold across the sonobe from the point of the triangle on one side to the point on the other.

Tuck the left hand triangle of sonobe A into the left hand pocket of sonobe B.

Then fold the right hand triangle of sonobe B in the right hand pocket of sonobe A.

Complete and tighten (whoops, bad picture).

Make a Face

Again, ignore the colours—these photos come from an aborted first attempt where I started with the white face.

1. Assemble the sonobe for a face. As this is first face, you get 5 of white and one each from the faces that surround it. Look at the graph at the top of the page; the white face is surrounded by pink, peach, lime, red and yellow. And make sure they are all folded the same way. Then, assemble them to make compounds per the above instructions—each compound being white and a colour.
2. The white flap on the red/white goes into the white pocket of the pink/white.
3. The white flap of the peach/white goes into the white pocket of the red/white.
4. The white flap of green/white goes into the white pocket of the peach/white.
5. The white flap of yellow/white goes into the white pocket of the green/white. To complete the ring, the white flap of the first unit the (i.e. pink/white) goes into white pocket of the last unit (i.e. the yellow/white).

Build the Model   

In my final version, I started with the orange face because the white was getting grubby with too much handling.

If you look at the graph way up at the top you will see that the orange face is surrounded by purple grey green black and blue. 

Purple Face

I have assembled the compounds for the purple face (look at the graph—purple is surrounded by orange (done), blue, lime, red and grey). There are 4 loose units because the orange/purple is already in the construction. Again, you make a ring. The purple flap of orange/purple goes into the purple pocket of blue/purple. Around to the end, the purple flap of grey/purple goes into purple pocket of orange/purple. Additionally, the blue flap of purple/blue goes into the blue pocket of orange/blue and the grey flap of grey/purple goes into the grey flap of orange/grey. This is complicated to explain, but it's obvious when you are doing it.

Grey Face

This time, there are only three loose compounds: grey/orange is in the orange face already, and grey/purple is in the purple face (look at the graph to check the other colours). Again, insert the grey flaps into the grey pockets to make a ring, and the green flap into the green pocket of orange/green.

Green, Black, and Blue Faces

Green and black faces are pretty much the same as grey. The last face blue is different because it connects up with purple. There are only two blues loose, as the other three are already in in the orange, purple and black faces respectively.  Same old, same old—blue flaps in blue pockets; spare peach flap into peach pocket on black/peach, lime pocket filled with lime flap on purple lime.

Pink Face

If you look at the graph, we have now completed the central face and the five around it. If you look at the pink face, you will see it is surrounded by black, green, yellow, white and peach. The black and green are already in place.

Proceed as before: pink flaps into pink pockets. Spare yellow flap into yellow pocket on the green/yellow; spare peach pocket gets flap from black/peach.

Peach Face

From the graph, peach is surrounded by pink, white, lime, blue, black, and we have already made pink, blue, and black, so only white and lime remain.

Lime, Red and Yellow Faces   

These proceed in exactly the same way as the peach face.

White Face

You will be left with the white face. The compounds of which are arleady in the model—you just need to put the flaps in the pockets. This can be really fiddley, I used tweezers in the end.

Everything is a bit loose at this point, and the gaps are obvious.

Tightening Up

Slowly work around the model, gently squeezing whilst supporting to close the gaps. I might also go over the model with a felt tip to color in the exposed edges, and where the colour has started to come off. The black face looks a bit shoddy at present.

But all in all, it's a really nice model and whilst it is a bit of, it is also pretty cool as well.

From this:

To this: