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Pseudogeodesic sphere to scale 2
When you try building a pseudogeodesic sphere to scale 2, it's quite easy to progress to this point (a little more than 62% of the total):
 Pseudogeodesic sphere to scale 2 (incomplete)
But when you begin to close the top part of the object, the base breaks very easily. Usually, the rods which go up from the vertices of the base pentagon, forming the boundaries between adjacent hexagons, detach from those vertices simultaneously, and the whole construction collapses inwards.
Peter Jepsen (a very experienced constructor), said in a post at the Geomag forum that he was able to complete the object after many failed attempts, an so were some company employees. However, he didn't give any additional details about techniques or tricks used.
I decided to try the construction using five supports, in the same way as I have explained for the sphere to scale 1. I thought that, as the problem seemed to be in the base, distributing the load among five different points should work. And it did:
 Pseudogeodesic sphere to scale 2 on five supports: horizontal view 2897 pieces: 707 balls, 2150 rods, 40 pentagons (16.31 kg) The sphere with no supports would have 1922 pieces: 482 balls, 1440 rods (10.91 kg)
The columns are made of pentagonal antiprisms. The «capitals» are made of two double-scale pentagonal pyramids joined at their bases, with those pointing up of each pair being the ones which actually make contact with the sphere, and matching the ones pictured here. Notice that the external portion of these pyramids could be (and, in this case, have been) truncated, so that they don't overhang the columns.
 Pseudogeodesic sphere to scale 2 on five supports: columns and bottom part
Two more views of the finished object:
 Pseudogeodesic sphere to scale 2 on five supports: top oblique view Pseudogeodesic sphere to scale 2 on five supports: bottom oblique view