A Plurality of Polyhedra

A little over a year ago (February, 2015) I was contacted by Los Angles artist Clare Graham about making some models. He had become interested in the illustrations of polyhedra in the 1509 book De Divina Proportione, by Luca Pacioli. What’s significant about these  illustrations is that they were done from woodcuts by Leonardo da Vinci. Many of them represented the first depictions of polyhedra which allowed one to easily see their internal structure.

Screen Shot 2016-03-11 at 1.25.18 PM

Da Vinci made 61 illustrations for the book. About half of these are solid polyhedra, and half are the more open, skeletal designs like the one in the image above. He also included three solids that are not polyhedra at all: a sphere, cone, and cylinder.

After a few months of going back and forth with Clare about decisions on size, material, finish, etc, I set to work reproducing these. Each one took anywhere between 30 seconds and 3 or 4 hours to design.


Here’s how I did it. First, I installed the “Polyhedra” plug-in for Rhino3D. Most of the polyhedra I needed had already been programmed by the author of this plug-in, so all I had to do was type in the name of the shape I wanted. There was a little work to do to translate the latin names to the modern nomenclature, but most of that was fairly easy to guess. To create the open designs, I simply used the “extract wireframe” command to get the edges of each polyhedron, and then the “pipe” command to thicken them.

The solids were trickier. Each model was printed at Shapeways.com. Since they charge by volume, a large solid piece would be extremely expensive. So I made each one hollow with a removable tip that could be glued in later.


To get a tight fit for gluing, I chamfered the edges of each glue-in piece. It took a few hours of playing to figure out an efficient way to do this, but eventually I came up with a method that took under 5 minutes per model. The tricky part was getting the exact same chamfer on both parts to be glued.

A few of the da Vinci designs were not in the “Polyhedra” plug-in. Most of those were stellated versions of some of the polyhedra I did have access to, so I wrote a custom Grasshopper script to stellate any input polyhedron. If anyone out there is interested in that, I’d be happy to share it.

After the models arrived in a big box, I glued all of the solids together with super-glue. The Shapeways “White, Strong, and Flexible” plastic is notorious for picking up smudge
IMG_5107marks, so I thought they should also be sealed. After hours of internet searching I discovered “Pledge Floor Care Finish”, which is basically a clear acrylic sealer. This worked OK, but the next time I do something like this I’m going to try a penetrating stone tile sealer. (I just used one brand on some Saltillo tile I installed in my home, and was really impressed!) Here they are arranged on some plastic in my office, ready to be sprayed with sealer.

Finally, I handed the finished models over to Clare, who has them arranged in a  beautiful glass case in his gallery in Eagle Rock. (Apologies for the crappy quality iPhone 4S picture!)




All of these models can be purchased here at Shapeways. There’s a modest $10 mark-up fee that would go to me if you buy one there, but if you have just a little knowledge of Rhino (or any CAD package) you could also make them yourself in very little time and save that.

A Plurality of Polyhedra

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