Pseudosphere From Geomag Box

Introduction[]

Pseudosphere From Geomag Box
A "near miss" spherical polyhedron featuring Geomag Rhombic Panels

Type	Near Miss
Panels	20 × 60 × 12 × 30 ×
Rods	240 ×
Spheres	120 ×
Author	PolyClare 02:54, 22 January 2009 (UTC)

This page will discuss the idea of building a spherelike polyhedron ("pseudosphere") based on completing the spherical shell shown on the cover of the 350 Geomag Case shown below. I was particularly intrigued since it seems that few interesting polyhedral models are buildable with Geomag that use the 60° Rhombic Panels.

Polyhedral Shell on a Geomag Case Cover (mine)

350 Geomag Case

Polyhedral Shell on a Geomag Case Cover (mine) CLOSEUP

Closeup of Geomag Spherical Shell

Building Attempts[]

I tried to build that object over a year ago, and failed; my panels kept popping out. I was able to complete it by replacing the 60° rhombic panels with pairs of adjacent triangular panels. Upon completion I could see that the model I had created was actually an Augmented Truncated Dodecahedron CT2T. The completed model was clearly nonconvex, with slightly bulging pentagonal cupolae. The modified model was relatively strong, and I could pick it up with some care. Since the Augmented Truncated Dodecahedron CT2T is nonconvex, particularly between the pairs of triangles that replaced the rhombi, I concluded that the original idea to build a sphere based on the Geomag box picture was impossible.

Recently, long after giving up on building this model, I saw this Flickr page by musikart001 entitled "sphere" and realized that the model actually could be built, though it was "unstable". This photo inspired me to try again to build my own model, being extra careful to not even attempt to pick it up. Interestingly, musikart001 was inspired similarly to me, by seeing a photo of a partially-completed shell.

I now belive that this Pseudosphere is really a "near miss" that can barely be built if one is extremely careful and doesn't try to pick it up. The fragility is a symptom of the fact that the fact that the Pseudosphere is only nearly convex, differing from convexity just barely within the limits of Geomag tolerance.

Building Method[]

After being confronted with the bald-faced fact that this model could, in fact, be successfully built, I set out with extra determination to make it work myself. First I bulit the Augmented Truncated Dodecahedron CT2T, shown here (using wall-eyed stereo) for reference:

Truncated Dodecahedron Augmented by Pentagonal Cupolae, CT2T
Left view	Right View

First, I popped out and replaced five neighboring pairs of triangles with yellow rhombi. I also replaced the central pentagon with a yellow pentagon, just for fun. I decided to take pictures, in case my model collapsed before I finished. You can see some the remaining unmodified pairs of triangles around the edges of this model.

Start Building the Pseudosphere: Five Rhombic Panels Only
Left view	Right View

While the model was still relatively strong, I inverted the model, so that the yellow panels were on the bottom. I then proceded to gingerly replace the rest of the triangle pairs with rhombi, also replacing clear pentagons with yellow pentagons as I went along. I would never lift the model, just twist it as I worked to complete all the replacements. This worked! Results below: