
 0.18 CrossSection of Great Icosahedral 120Cell 
The great icosahedral 120cell is number 14 in the list of uniform polychora
in Stella4D,
with the abbreviated name "gofix". It is a regular polychoron, with 120
great icosahedra as its cells. 12 such cells meet
at each vertex, forming a small stellated
dodecahedral vertex figure.
The model you see here is a 3D crosssection through that 4D polytope, taken
18% of the way through the model, along one of the 60 icosahedral symmetry
axes.
The model consists of 13 small stellated dodecahedra: a regular one at the
centre (representing the first vertex sliced by the sectioning plane), and 12
distorted ones around the outside, each with one spike that intersects one of
the spikes of the central polyhedron. The distorted ones represent
crosssections through vertices at an angle, causing the distortion.
It makes a striking model, looking like some sort of alien weapon or space
craft!
See this tutorial for some comments on
obtaining nets for building paper models of 4D polytopes.

The faces of this 3D crosssection are themselves 2D crosssections of
the great icosahedron. First, there are
small, medium and large regular pentagrams, representing crosssections
orthogonal to the 5fold symmetry axes of the great icosahedron
(largest pentagram outlined here).


Secondly, there are small distorted pentagrams, near the outside of the
model, occurring twoperplane. These represent crosssections
orthogonal to the 2fold axes. Such sectioning planes will slice two
vertices of the great icosahedron at once, generating two faces per
plane.


Finally, there are some larger distorted pentagrams poking into the
central small stellated dodecahedron. These occur 3perplane and
represent crosssections orthogonal to the 3fold symmetry axes. Such
planes will cut through 3 vertices at once.


I started down the wrong path when making this model. I originally
planned to attach the spikes together as shown, with each spike
truncated to leave a pentagon, and the two parts glued together at this
flat face. I decided this would be a bad idea, not giving enough
strength at the narrow join. So I started again! I also didn't like
my colour choice so took the opportunity to give the whole thing a more
menacing metallic look!


Instead of a flat join, I allowed full length spikes on the outer
parts, and built inverted spikes for them to sit in on the inner part.


When the outer parts have their spikes glued into these cups, it should
form a strong join. Handy if I want to pick this thing up by just one
of the outer parts!


I added a pentagon across the base of each section for strength, but
had to leave a hole for the outer spikes to stick through.


Put these sections together like a
dodecahedron to complete the central
part.


Here are the different types of nets for the outer parts (distorted
small stellated dodecahedra).


Again, I strengthened them with pentagonal parts at the base of each
spike.


Make 12 such parts and you're ready to glue them into the central part.


Ready for battle!


Each part fits snuggly into the central cups. Just put glue around the
inside of the cups and hold the outer parts in.


Another photo of the finished model.


Closeup.


Another closeup.


Here's the model on display at the Bridges 2016 conference in
Jyväskylä, Finland.

Makes a great star for a Christmas tree! You can buy gift cards with this
design here:
http://www.cafepress.com/stella4d.
