Geometric construction of Truncated Cuboctahedron Polytope

[asherwin]

Hi,

I recently got really into geometrically constructing polytopes in GeoGebra (probably not the best software to use) such as the 120-cell (or part of it) and was curious as to what the geometric construction for the Truncated Cuboctahedron was. I've seen at least one YouTube video that tries to explain it and wasn't really helpful. Unfortunately, Stella4D isn't helpful with this specifically either. I've also tried to do this on GeoGebra and seems to be off somehow when I compare a Stella4D rendered image (I think) of the polytope found in perhaps the Wikipedia Commons. I have tried turning to Wikipedia for the Cartesian coordinates for the vertices of other polytopes and I definitely don't get what I was looking for. I hope someone can answer my question.

[robertw]

The topology is that of a truncated cuboctahedron, but it doesn't really work that way geometrically. If you truncate the cuboctahedron you'll get rectangles rather than squares. The result has to be adjusted to make all the faces regular. I'm not sure how you could construct it other than by building it face by face.

[asherwin]

Ok, well I do know that part about the truncated characteristic and how it's squares and not rectangles. However, related to the Stella4D 4D library, I'm referring to the "331-Gidpith proj" file. When I geometrically constructed the 120-cell using GeoGebra, I started with a regular dodecahedron. I then made a plane attached to one face and reflected the point (0,0,0) across the plane. I used a segment to to connect the reflected point to three points on the dodecahedron's "side" and found their points of intersection with the plane. So simply put, I guess this is how I used the Schlegel diagram of a dodecahedron to construct at least a few cells of this polytope. I'm essentially asking how or if I can use the same process with other uniform solids.

If you watch the YouTube video from this link, https://www.youtube.com/watch?v=pWLji1QJ8mk&t=307s, at around 14:15, it happens to show this process with a truncated Cuboctahedron where the entire solid is projected onto one of the octagonal faces. My next question from here is about where the center point of projection is and how this is used to make one of the outermost cells corresponding to the octahedral face. I hope this clears up my question.

[robertw]

Oh, sorry, thought you were asking about the construction of the truncated cuboctahedron, which seemed odd if you'd already managed the 120-cell!

Stella4D can construct it, but it doesn't do it based on reflection, so I don't know. I'll leave this for other to answer if they have any insights.

[asherwin]

Ok. Were you the one who uploaded the different models on Stella4D? Actually, I am trying to figure out how to geometrically construct the snub cube. Sorry that I didn't make a new topic to post this on.

[robertw]

Yep I added all the models for Stella4D. The 4D library consists of the 3D vertex figures, from which Stella4D can generate the full 4D polytope.

[asherwin]

Huh. Did you have to program them in or were they like already made digital files that you just added?

[robertw]

The 3D vertex figures you mean? Some are simple and already available. Some I had to program in. And some I created with help/advice from others about what was required.

[asherwin]

I see. So what about the 3D rotational view of the truncated cuboctahedron polytope? Did you have to write the code for that file?

[robertw]

You can display the vertex figure in Stella4D. In this case it's an irregular tetrahedron. I wrote my own code to generate tetrahedra with any edge lengths which I used to create vertex figures like this. I'm not going to go into details about how I did anything specific in the code though.

[asherwin]

Wow, ok, I'm not sure I really understand all of that. Who might I ask if I'm looking for a detail-oriented explanation for building that polytope from scratch?

[asherwin]

Hi Robert,

I know it's been a while since we last communicated. I was just really hoping you could let me know who to reach out to if I'm looking for more information on 4-polytopes and possibly where you got the information you used. Thanks!

[robertw]

Hi, sorry for not getting back to you. I probably just put it in the too hard basket and never got back to it. I probably figured it out bit by bit from websites and people here and there, but I don't know who or what to suggest now. It was years ago now. And I would have worked out a fair chunk on my own. If you google polychoron or 4D polytope or similar, you should find lots of info out there. And you can keep asking on forums like this, if you have any specific questions.