## Sunday, January 4, 2015

### Hollow QRSTUVWXYZ Stars

The Hollow QRSTUVWXYZ Stars, based on a design by Meenakshi Mukerji. Click for a bigger version.

Today I present the biggest model I've created so far.  It's called a QRSTUVWXYZ model because it consists of ten intersecting planes.  The naming convention is similar to other planar models like the WXYZ and the TUVWXYZ models.  As far as I know, this is the largest of the planar models, invented by Meenakshi Mukerji.

My model is a variant on the original model.  This variant is hollow at the center, which has a great visual effect not captured in the photo.  But believe it or not, I actually created this variant as a defensive measure.  I thought it would be too hard to have 90 pieces of paper, each folded into 16 layers, all coming together to a point in the center.

For some reason I decided to diagram the steps out.  To make the variant, I added step seven.

Click for a bigger version.  Sorry, I'm not ready to diagram the assembly.  I also found these diagrams of the original model, if that helps.

90 pieces of paper!  As you can imagine, the way they connect is very complicated.  It also seems like 10 is the magic number of planes, creating a figure that is far more symmetrical than a 9-plane or 8-plane model.

I didn't know why that was, until I realized that an icosahedron has 20 faces.  And each pair of antipodal faces on an icosahedron are on parallel planes.  So if you take the 20 planes of the faces of an icosahedron, and move them all to the center without rotation, then you'll end up with 10 unique planes.

With that in mind, I drew an assembly guide, embedded on the surface of an unfolded icosahedron.

Each color represents one of the ten planes--except for gray, which just outlines the unfolded icosahedron.

The ten planes also form an abstract polyhedron, called the hemi-icosahedron.  It's like a regular icosahedron, but with opposite faces and vertices identified with each other.  I don't know that much about abstract polyhedra, but I note that each plane corresponds to a vertex in the dual polyhedron, the hemi-dodecahedron.

Image from Wikipedia.  Each gray circle is a vertex in the hemi-dodecahedron, and corresponds with a plane in the QRSTUVWXYZ model.

If you look at the photo at the top, I have a drawing of an alternate version of this graph.  I was using it to decide on the coloring.

Because of the heavy math involved, this is one of my favorite models of all time.