## Hexayurt dome details and models

**Edit 4/8/12: **Andrew Maxwell, Tracy Suskin, Ying Yang, students at SAIT polytechnic in Canada, have put together the engineering details for the tri-dome.

People are now starting to build my tri-dome and quad-dome versions of the hexayurt, so it is time to give some of the technical details. To start, however, here is an application of the intermediate value theorem!

When I started working on the details for the tri-dome I realised I had made a bad assumption (thinking that the form was geometrically pure). This means that some of the details in my original write up were wrong. All a little embarrassing. Ironically, I might have missed a form that does actually work, had I not made the bad assumption. The shape, like the hexayurt, starts with a hexagonal based pyramid. In a traditional hexayurt this lies on top of a hexagon of vertical walls. Instead of this we attach a square to three of the edges and the classic hexayurt triangle (isocoles triangle with base and height the same length) to the other three. We can look at what happens as the pyramid is moved away from the ground, while the edges of the shapes rest on it:

This does not give a great building; there are holes. The holes are triangles and two of the sides have a fixed length. The final edge changes length, starting long, and ending short. We know we can fill the holes with classic hexayurt triangles. Two of the edges are the right length we just need the third. The length changes smoothly as we raise the roof, and starts shorter and ends longer than we want. Here we can apply the intermediate value theorem, so the correct length must be passed. As a mathematician I would stop there, the system works; however people are building the things…

So to calculate the correct angle for the square sides of the model we can look vertically down, calling the angle of the square face θ, (and assuming that the boards we are using are 8′ by 4′) needing as the classic maths problem asks to “find x”.In this case

,

we want so:

Solving the quadratic:

Which gives an angle of about 49°, and the height of the roof (assuming 4’x8′ panels) is , just over 6′ at the edge and 10′ in the centre. We can use these, and useful facts about general tetrahedra to calculate all the angles between faces by using the lengths of their edges. If you want to follow the details yourself, you need to add vectors to get some of the edge lengths, then use the Cayley-Menger determinant to find the volume of the tetrahedron, and then the generalised Sine rule to (halfway down this page) to give the angle.

Finally here are the hexayurt models (rhino 3dm and vrml formats) of the hexayurt, H13, TriDome, QuadDome, plus a couple of others, including a very large one.

Dang, wish we had this last month!!. We built our Quad Dome with non-beveled edges (and lots of tape). When we get back from Burning Man we’ll probably be doing some cutting to make our seams tighter. Or maybe we’ll just crush the foam edges into place under the tape. Hm.

Any suggestions about how to keep one tied down on a flat surface? Can’t figure out the best way without damaging the edges, since we didn’t bevel.

But thank you, thank you for posting these specs!!

Cassidy

My apologies, I was trying to get the data out sooner, but other things got in the way! I look forward to seeing pictures. Please let me know if there are any specs missing.

Just wanted to let you know that I also built a quad dome this year at burning man. I’ve documented my experiences here: http://www.morganengel.com/category/burning-man/ Thanks for the information, it was a huge success!

Hey, thanks so much for this! I’m working on a large-ish shelter for burningman this year and I’m very interested in the quad dome geometry. Vinay sent me over to your blog.

I’ve been doing some detailed design cad models, (I’m working on a new hinging system) and I tried to import your models… my software doesn’t really like the format… I wonder if you would be willing/able to provide step or iges files? It would be quite helpful to me, and maybe others who are working on similar projects.

Thanks again for the great work and info!

Brian, that should not be a problem, I have the models in Rhino which can export those formats. Will try to get them done soon. Send me an email poke in a couple of days.

Brian, I built 2 scale models before I did the real thing, and they helped immensely. I highly recommend using cardboard and then foamboard to simulate your needs. The price and time is well worth it.

Thanks for the tip. I was planning to make some more physical models too, and I already have made some. But I’m trying to mess with the geometry a bit if I can, so there are too many iterations to make them all even tiny… I like to go back and forth between physical and virtual models, because they both have different strengths and weaknesses.

Like when you’re messing with the geometry, it’s easy to think a small scale fits and convince yourself that it just seems wonky because you cut the parts wrong, whereas the computer models are pretty darn precise… I had this issue with a design I thought was solid until I got into the computer and started trying to recreate it, where I realized it was the geometry that was wonky, not my (just) my cutting.

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