# Hinged Triangle-Square

## Hinged Triangle-Square

Hinged Triangle-Square 710 528 mathgrrl

One of our favorite 3D designs is a model of Dudeney‘s famous hinged dissection of a triangle to a square, also known for some reason as the Haberdasher’s Puzzle. Today we’ll revisit this design and update it for printing on SLS machines.

We designed our original triangle-square model a few years ago with OpenSCAD, using data from an article by Mark Meyerson. You can read more about the original design on our old blog MakerHome, and download a free 3D-printable model on Thingiverse:

The triangle-square is a surprisingly reliable model for printing on a desktop 3D printer, considering that it prints all in one piece with hinges completely assembled. The hinges are the same as the ones on our Fidget Star and Fidget Cube, but on the triangle-square the hinges are more reliable because they all point in the same direction and don’t form overhangs. I’ve had the best success with these at .2mm layer height on a MakerBot Replicator 2. You can make a lot of triangle-squares fairly quickly with a high success rate.

Since we made that model, some mathematical advances were made about hinged dissections! Specifically, in 2007, Eric Demaine and a team of authors proved that any finite collection of equal-area polygons has a common hinged dissection! In other words, as they put it, “for any such collection of polygons there exists a chain or polygons hinged at vertices that can be folded in the plane continuously without self-intersection to form any polygon in the collection.” (!!)

For more information on a wide variety of dissections, check out Frederickson’s book Dissections: Plane & Fancy, as well as his books on Swinging and Twisting Hinged Dissections and Piano-Hinged Dissections, or his book on Ernest Irving Freese’s Geometric Transformations. In the future we hope to adapt our model to other interesting hinged dissections!

In the meantime, we do have a da Vinci Color printer and thought it might be nice to print a color version. We’re still working out some ink issues but here is how the model came out:

We also made an SLS-optimized verison of our triangle square for Shapeways, so if you don’t have a 3D printer you can now order one if you like:

Or, if you REALLY love triangle-squares, then for £100 you can get a huge aluminum version Dudeney’s Dissection at the wonder Grand Illusions shop. Here is that model in action:

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