### The Laser Cut Truncated Icosidodecahedron, Part II

July, 2010. My nephew was in town, and he wanted to create something cool. We decided to make another laser cut polyhedron, using the same design as the original below, only much smaller. For reference here is the original large one and the new smaller one.

### Making the small one

Here are some pics of the making of the small polyhedron. First note all the pieces fit on one sheet of plastic nicely.

Here are all the cut and peeled pieces with a watch for scale; each had paper on both sides that needed peeled off.

Here is the one Josh assembled.

Here are both of them. Perhaps I'll use glue to hold them together and remove the rubber bands.

### The Laser Cut Truncated Icosidodecahedron

Mar 2, 2010. My wife is out of town for two days, and I wanted to play with some plastic I bought on the laser cutter, so I decided to build some 3D polyhedra shape. To kill the suspense, here is the final result. Click on each image for a higher resolution version.

### The Design

I started out in Mathematica, looking through polyhedra until I found one I wanted. I selected the Truncated icosidodecahedron, also known as a rhombitruncated icosidodecahedron, a great rhombicosidodecahedron, or a omnitruncated dodecahedron. This shape has 62 total faces made of 30 squares, 20 hexagons, and 12 decagons. It has 120 vertices.

The final Mathematica file is around 120MB, and is zipped here (4MB).

I decided to only make the squares and hexagons, leaving the decagons out of the final assembly, to leave the shape somewhat open. I wanted to cut the shapes on the laser cutter, but needed a way to stick them together. I didn't want to use glue, since the transparent pieces would get slightly messy, so I designed a system of wedges to hold the polygons together. I would need 30 squares + 20 hexagons + 90 (three per hex) connectors for a total of 140 pieces.

The wedges look like this:

Next, I played with different outer geometry curves to find one I liked. Here is an example sawtooth one, printed on paper:

In the end I chose smooth outer radius pieces, constructed so that the inner and outer radii track spheres. I designed a system of notches to (hopefully) provide enough strength to keep the final shape sturdy. The result of these decisions is that the connectors are non-symmetric, but carefully designed to create a pleasing final shape.

Here is a final check in Mathematica on the spatial non-intersection of the design:

The final step in Mathematica was to orient enough of each piece into tiles to minimze plastic usage, resulting in three templates, one for each piece type.

These were exported into lists of lines from Mathematica (the DXF exporter in Mathematica sucks and is unusable for this).

After that, to minimize laser cutting and create nice DXFs for the cutter, I wrote a C# program that took the lists of lines in, removed duplicates, merged segments into longer segments, and output nice DXF files for the cutter. Here (37K) is a zip of the program. It is a command line tool that cleans the Mathematica output, used like "reduceLines squareData.txt squares.dxf". Here is a screenshot of some code:

The result was three DXF files, one for each piece type. Here they are: Squares, Hexagons, Connectors.

### To the Laser Cutter!

Before doing a complete run, I cut enough pieces to check the fit. The resulting pieces aresomewhat loose, and can be tightened up a bit on a later pass. However, it is nearly impossible to cut acrylic to fit snugly due to brittleness and variation in the material, so I wanted to err on the loos side and avoid having to hand sand 140 pieces! If these were cut from wood, then they could be made snug, but would (I estimate) be much weaker.

I took the DXF files, loaded them into the cutter program, inserted plastic, and started cutting hexes. Here is an in process, final cutting shot, final result, and two other hex pics.

The pieces are cut with protective wrapping, so splatter from the laser cutter does not mar the finished pieces.

Next I did the squares:

And finally, the connectors:

The final pieces. Very pretty. I modeled the colors of the various stock I had in Mathematica to find a combo that looked pleasing.

### Assembly

Here are all the pieces, and some rubber bands I am using to hold it together. For the next version (if I do another) I'll work out a better method. I didn't use glue so I can disassemble this if needed. I have several ideas for a more polished next version, but for now this method works well.

Now to assemble the lowest layer, which is done by just placing the pieces together, and then using a rubber band to keep them tight.

Using the rubber bands for tension, next add more and more faces until done: