Thursday, October 28, 2010

C60 in paper - partial success

kw: hobbies, photographs, mobiles

More than two years ago I posted instructions for making an origami module for a "spiky edge polyhedron". This followup has been a long time coming.

I decided to make a shape based on the geometry of a buckyball, or buckminsterfullerene, or C60 (formally, C60, but I'll forgo the subscript codes, as most writers do). The 60 carbon atoms in C60 are held together by 90 bonds, and the origami modules I am using correspond to chemical bonds, so I needed 90 of them. I decided to make it colorful by making the 60 bonds that surround pentagons to be of five colors, and using a sixth color (gray) for the rest of the bonds. I spent months doing one or two of these per weekday, because it is really tedious making more than a couple at a time.

The dihedral bond angle needed for C60 is about 20°, while the smallest dihedral that is practicable with these modules is 36°. Thus I know there will be some steric strain; the ball will "want" to be a lot smaller than I am making it, and will strive to close early. Fortunately, paper ought to be flexible enough for this to work.

I began by putting together the twelve pentagons. Here are two of them. That used up all the colored modules. By inserting a tab from each into a pocket of the next in round-robin fashion, I have a tab left at each corner for connecting to the gray modules that will hold the pentagons together while outlining the hexagons.

This shows the twelve pentagons laid out in a planning pattern, ready for me to connect them with gray modules. There are several ways to color a buckyball, and I can see that this is not really the most symmetrical, but it will serve.

To make the twelve pentagons, I just had to insert one tab into one pocket at each step. From this point, I have to insert two tabs into two pockets simultaneously, and do it properly sixty times (once for each end of the linking module). Sorry, but I neglected to photograph one of these double-tab insertions.

Here is a top view once the first five linking modules and five pentagons have been assembled around the one chosen to be central to the building process.

This is an oblique view at the same stage. Just below center you can see where a linking module will be put onto a pink and a purple tab to complete the facing hexagon. The hexagon is trying to be a pentagon-and-a-half, showing the steric strain I mentioned. The dihedral bond angle these modules are built for is actually better suited to attaching the twelve pentagons together without linking modules at all.

Look closely at where the pink tab bends away from the pink module it is part of. There is a blue tab going into the pink module's pocket. The blue pocket is open. When I attach one end of a linking gray module, its tab will go in the blue pocket, and the pink tab will go in the gray pocket. I insert them halfway, apply Elmer's Glue-all, then push all the way in and press tight for a bond.

Shapes without steric strain can be assembled without glue, though they slide apart at a rather light touch. I glue all shapes since I want them to last for years.

Here is the completed ball. It was a bit of a tussle getting the last five linking modules and the final pentagon in place. The hexagons and pentagons are showing a bit of distortion, another symptom of the steric strain, but the piece is quite pretty nonetheless. Over the coming months (years?), I'll produce some less elaborate shapes to go together with this one, the centerpiece for a new mobile.

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