It's highly likely you've seen a geodesic dome before.
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgZr_iNf5qphcs-lu8sZlU3gdNi_KeGKXt_TS93r5Mjr-LPj3zOylGWXdAMil1t42qTMre_kqY3yA_L390i-Vv1A2vY5VL-D1owUI1SefC6ccjOnZaMnvucG0KWC-ypbO9wrJRRDC1xrUw/s400/Epcot07.jpg)
After briefly starting to optimize triangle counts for textures in MagicTile, I had a fun realization. The triangle patterns sparked the idea that there could be a precise hyperbolic analogue to a geodesic dome. I was compelled into the diversion, and with minor code changes made some pretty pictures of "geodesic saddles". (That seems like a nice name for these objects anyway.) Alas, my intended optimizations are yet to be done, but at least I can present this geodesic saddle based on the {3,7} tiling :)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi1V3LxjAYB5QhC06up4cvxjtvkPsdKUgqQOrWAg3sJPFKXW0n0PxceOotwgtkGfqZ87xiEigMoKo8PriNTTo2ybLnwnv556mdcFoePdj3truVFNApII9jJR7dT-gnMjT9s1EHtFNnKGQo/s400/%257B7%252C3%257D-geodesic-saddle.png)
Can you find some of the "knots"? That may not be the proper term, but I mean those rare points in the saddle where seven triangles meet at a vertex instead of six. On a geodesic dome, which is usually based on the spherical {3,5} tiling (aka icosahedron), the analogous points are the rare vertices where five triangles meet instead of six. I find knots easier to spot on a geodesic saddle derived from a {3,9} tiling.
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEglhvqr5ApizL-MaKbgxzabDgHDlqUsTpiyIvjbtYegBOcVFLMhgCm1T2C_hdqvL9f4imspqwh8VPKP2vGX85_bqZMcdVfduQPgT8i7CLsvxznhAQcWgAHaj5IYR1fGrSYodOuKmrWkkTY/s400/%257B9%252C3%257D-geodesic-saddle.png)
Geodesic domes and saddles are generated by taking the tiles in a triangular tiling and subdividing each of them into smaller triangles. Hence, triangular numbers make a cameo in the calculations. For these pictures, I chose to subdivide the original triangles with eight new triangles per side.
But there was one thing that tripped me up quite a bit. I began by mistakenly thinking I could subdivide the triangle edges of the original tiling equally, and then interpolate interior points thereafter. As much as I tried, things just wouldn't line up quite right, and I wasn't seeing the geodesics that I expected. It turns out that all the small triangle edges have varying lengths, something that is also true for a geodesic dome. Compare the proper {3,9} geodesic saddle above with the waviness of an incorrect effort.
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhP4t4fq614a1YOwxxJinFQwrNZ4c3XL2cvVwqyY6zs1YysLFP3xdzI_6fc-iyqaXa6PPlbJr9c9iNsRM6VQbYhn30CGkMGheEFj-jOE9dOejaZfslXUHi813bym01ChX8F-xlCvEovC3Q/s400/%257B9%252C3%257D-not-quite-a-geodesic-.png)
I've been showing these pictures in the Poincare Disk, and I don't have unprojected renderings at the moment. But despite this, one thing is certain - a portion of geodesic saddle would make for a unique and fantastic jungle gym!
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiY04qhRgLi14UuS6Bzri_u0hnLC6Ju85p1P33Mp2W7K_0BiFTH3kpMuszHze36wXiYuKsxx8DSt3XfmYhFxNBz1xkzdX9YEtgtiaWYnrMuSIl8cH4ICONZN2rqDAUzylcqeTMWasjtQg0/s400/%257B7%252C3%257D-geodesic-saddle-close.png)
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