More pretty animations!

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

More pretty animations!

Postby quickfur » Wed Aug 20, 2008 4:55 pm

Alright, as I promised, here is a projection of the 600-cell into 3-space, with hidden cells removed:

Image

To show the intricacy of the inner structure more, I've highlighted the 20 tetrahedra surrounding the central vertex (which is actually the closest vertex to the 4D viewpoint). As you can see, they are joined in an icosahedral configuration. Two other layers of tetrahedra are visible, wrapped around this central icosahedron. It's a bit hard to discern individual cells because of the limitations of a 2D screen. Maybe I should make some static images with different cells highlighted.

And now, a dream come true: the 120-cell!

Image

The closest cell to the 4D viewpoint has been highlighted, to make it easier to discern the structure of this projection. As you can see, the central dodecahedron is surrounded by a layer of 12 dodecahedra. You can barely see some cells from a 3rd layer (the outer hexagonal faces are actually dodecahedra seen from a 90° angle, so they appear flat). The rest of the cells are obscured from the 4D viewpoint. Quite interestingly, this particular 4D viewpoint gives us a projection envelope in the shape of a buckyball (truncated icosahedron).

Note that these projections are rotating only in 3D; the 4D viewpoint is fixed. I'm going to work on rotating the 4D viewpoint as well. :)
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: More pretty animations!

Postby Keiji » Fri Aug 29, 2008 9:26 pm

Beautiful :D

Mind if I put these on the HDDB? I'd host the images, of course.

Edit: Octagonny, too
User avatar
Keiji
Administrator
 
Posts: 1984
Joined: Mon Nov 10, 2003 6:33 pm
Location: Torquay, England

Re: More pretty animations!

Postby quickfur » Sat Aug 30, 2008 12:51 am

What is HDDB?

It's OK to put these images up, although they should be accompanied by some description of what they are. Note especially, that the 120-cell animation is deficient, because the 4D viewpoint is too close to the 120-cell, and so perspective projection has made only the first two "layers" of dodecahedra visible. (Think of it as looking at something really close up, so the outer limb is hidden even though they lie on the hemisphere facing you.) There is actually a third layer of dodecahedra on the same hemiglome of the 120-cell, before you reach the 3-quator and get to the far side. So while this animation does represent the 120-cell, it doesn't really convey its full beauty (I guess that means I'll have to post a better animation :P).
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: More pretty animations!

Postby quickfur » Sat Aug 30, 2008 1:20 am

Actually, I already have a better 120-cell animation. Why not post it now rather than later...

Image

This animation shows the 3 layers of dodecahedra on the hemiglome facing the 4D viewpoint. Each layer is rendered differently to help distinguish them better. We start with the central red dodecahedron, which is actually the closest cell to the 4D viewpoint, surrounded by 12 yellow dodecahedra (the 2nd layer). Then we have the dodecahedra that attach to the dodecahedra in the 2nd layer. These lie almost at the limb of the 120-cell, and connect to a 4th layer which is facing away from the 4D viewpoint. The 4th layer is then connected to 12 dodecahedra in a 5th layer, corresponding with the 2nd layer but on the far side of the 120-cell, and finally the antipodal dodecahedron.
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: More pretty animations!

Postby Keiji » Sat Aug 30, 2008 3:52 am

quickfur wrote:What is HDDB?


Higher Dimensions Database - based off of the wiki that used to be here on Tetraspace before getting raped by spambots and then just plain not working for whatever reason.

I added the projections, and a page for the octagonny/koshire/icositetrachoric rectate:

120-cell
600-cell
コシレ

Hopefully I got the descriptions correct!
User avatar
Keiji
Administrator
 
Posts: 1984
Joined: Mon Nov 10, 2003 6:33 pm
Location: Torquay, England

Re: More pretty animations!

Postby quickfur » Sat Aug 30, 2008 3:58 pm

Cool! Maybe I'll work on articles there instead of wikipedia. :P (Wikipedia has a minority of deleters with a vendetta to delete stuff considered as original research, even if there is already a community of people to whom this stuff is common knowledge. There's a lot of stuff on rotachora that I haven't dared to add yet, for fear of getting speedy-deleted.)
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: More pretty animations!

Postby Keiji » Sat Aug 30, 2008 8:15 pm

Awesome! HDDB hasn't had much activity in ages until recently.

I hope you can get used to my conventions on things, though. :)
User avatar
Keiji
Administrator
 
Posts: 1984
Joined: Mon Nov 10, 2003 6:33 pm
Location: Torquay, England


Return to Other Polytopes

Who is online

Users browsing this forum: Google [Bot] and 10 guests