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Penta-hexagonal dome inspiration

In these latest nights of geodesic meditations, I have been exploring an apparently simplified dome model that is based on hexagons and pentagons, rather than triangles but basically just grouping them up.
Because of this, it turns out to be a little bit more sacred geometry alike, so it fits me way better. It's all just empirical reasoning and few hand drawn sketches but I feel this is a good concept.

The first model is actually made of pentagons and triangles, equivalent to the 2v frequency. The section in red is the fifth section of the ideal cover.
The next level up frequency that works with this concept is the 3v that gives a classic football-ball feeling. An interesting property of this design is that every joint connects always three corners, which is a fact that cannot be underestimated.

As the design grows in complexity it jumps up to 6v frequency, which is actually simplified down to only 4 patterns with 4 struts-length: one regular pentagon, one regular hexagon and two irregular hexagons. The lengths of the four struts (a,b,c,d) are calculated in a percentage of the longest strut (side of the regular hexagon). The number of each kind of pattern can be easily calculated: as we keep going up in complexity, the number of pentagons does not change (they're always six) as well as the number of regular hexagons (always ten, but just smaller and smaller compared to the radius of the dome). We just keep adding more and more kinds of irregular hexagons. The next levels up should be frequency 9v, 12v... always adding three extra: I didn't go further than this for now and probably it should be enough. 
The tricky point is calculating the correct angles of each irregular hexagon, planning to make each pattern flat. 
This is going to be the main subject of my next meditations.

1 comment:

  1. Hi! I`m building dome with 5v frequency and trying to calculate which method is beter from view of material waste. Popular method is make triangles but there is a lot of waste material and lot wokr. Another way is slices but it is not easy for dome with diameter 10m. So use hexagon is better choise but I didn`t find anywhere formula to calculate heaxagon count and sizes. Could you help with that?