# What is the name of the polyhedral shape of the Humanity Star?

Rocket Lab has successfully launched a rocket from New Zealand and placed several objects in orbit, see here for example. One payload is New Zealand's first satellite the Humanity Star, a reflective geodesic that can be seen at night by reflected light. According to this it has 65 sides.

What is the name for this shape?

Open image in new window for full size (source):

• For comparison, Mayak was a tetrahedron, but failed to deploy. – uhoh Jan 25 '18 at 2:20
• Looks to me like it is $80$ sides. $65 = 5\cdot 13$ and $13$ would make for a much more complicated symmetry. en.wikipedia.org/wiki/Pentakis_icosidodecahedron – Doug M Jan 25 '18 at 2:31
• @Doug M: The image in the link that claims 65 sides seems to have too many meet-in-6 vertices close to each other to be a pentakus icosidodecahedron. I think it must be something that doesn't have icosahedral symmetry. – hmakholm left over Monica Jan 25 '18 at 3:10
• Looks like it could be an icosidodecahedron where the pentagonal sides have been stellated. – Cheerful Parsnip Jan 25 '18 at 3:12
• @HenningMakholm It certainly looks like it has icosahedral symmetry. We can clearly see two vertexes with 5 faces meeting, and one vertex in between with 6 faces meeting, and that would fit. I think that 15 faces are non-reflective (or have been removed). It is supposed to shimmer as it rotates. – Doug M Jan 25 '18 at 3:14

Here I've put red dots on corners where six triangles meet and green dots on corners where five triangles meet:

This has too many red dots together to be the pentakis icosidodecahedron suggested by some commenters -- in particular, the two red-red-red triangles sharing an edge slightly below the middle of the image are not to be found in that polyhedron.

What it does look like to me is a pentakis icosidodecahedron that one has cut up along the one of the "equators" and then twisted one half by $36^\circ$ before gluing them together again.

This figure still has 80 sides rather than the claimed 65, though. But on the other hand, I can count 35 sides visible in this picture, and so many triangles should not be visible from one point in a somewhat regularly constructed 65-sided polyhedron anyway.

A systematic name for it would be a pentakis pentagonal orthobirotunda.

• Yes, this is why it looks familliar and yet somehow odd. i.stack.imgur.com/Y3Bxp.png – uhoh Jan 25 '18 at 3:36
• Yes, you are right. The adjacent red-red triangles rule the pentakis icosidodecahedron out. – Cheerful Parsnip Jan 25 '18 at 3:41
• Could it be a pentakis icosidodecahedon where the top and bottom hemispheres have been rotated with respect to each other? – Cheerful Parsnip Jan 25 '18 at 3:43
• @CheerfulParsnip: Yes, that is my suggestion. – hmakholm left over Monica Jan 25 '18 at 3:44
• Ha, now I see that you wrote that. – Cheerful Parsnip Jan 25 '18 at 3:45

The picture on the right, if you look closely at the base, seems to show that the lowest pentagon on the underlying icosidodecahedron (or pentagonal orthobirotunda as some have observed) has not been augmented with five triangles. They may have left those off to provide a mounting surface. In that case there would be 75 triangular sides, so still 10 unaccounted for.

Edit: "picture on the right" is from this link https://space.stackexchange.com/questions/24616/does-humanity-star-have-non-reflective-triangular-panels-if-so-what-are-their

The question is discussed here: http://anewdomain.net/humanity-star-rocket-labs-f-in-math/

• Thank you for the link! I've used it to add an answer here. – uhoh Feb 16 '18 at 14:56