Hi all,
I was reading about Petrie polygons when I suddenly realized how would look their analogues in higher dimensions. E.g. Which polyhedron would be created if one join together the faces around the equator of a polychoron?
I thought I would have something near to one of their envelopes, but when I read this page: http://teamikaria.com/hddb/forum/viewtopic.php?f=3&t=1292, I realized that the envelope is not exactly what I was looking for. A "Petrie polyhedron" of a 120-cell would only have pentagons, but its envelopes have hexagons, some have triangles, square and other kinds of face.
Like Petrie polygons use the edges of the original polyhedron, but they lie in different planes. The analog would happen to "Petrie polyhedra", it would use the faces of the original polychoron, but they would lie in different realms.
Would I visualized:
5-cell "Petrie polyhedron" would look like a triangular bipyramid (all triangles are equilateral - 5-cell ones are, so they must still be equilateral) - The 3 triangles from one pyramid connect the vertex in one realm to 3 edges in other realm, the 3 triangles of the other pyramid lie all in the latter realm;
8-cell "Petrie polyhedron" would look like a rhombic dodecahedron (all faces are squares, no they are not rhombuses. Each face lie in one realm, so they can be regular) - 3 squares lie on the its north pole. 3 on south and 6 in a realm that connect the poles;
16-cell "Petrie polyhedron" would look like a gyroelongated bipyramid (all triangles are equilateral) - Like 5-cell "Petrie polyhedron" the 3 triangles from each pyramid connect each vertex from the outer realms to distinct sets of 3 edges each in the inner realm. The 6 remaining triangles lie in the inner realm, they are 6 of the 8 triangles of an octahedra;
The other 3 polychora are a little bit more complex to explain.
24-cell "Petrie polyhedron" would look like a tetrakis square. All its triangles are equilateral, and they are not coplanar, because they lie in distinct realms;
600-cell "Petrie polygon" would look like a pentakis icosidodecahedron. The triangles from the original icosidodecahedron lie on the equator, the remaining triangles (the ones derived from the kis operation) connect edges on the equator (that create hollow pentagons) with a vertex that lie in another realm. So, all triangles are equilateral.
120-cell has a separate part for it because it is the most complex of all to be explained. On its equator lie 12 pentagons, from their edges there are 5 pentagons in a distinct realm from the 12 equatorial ones, thus 60 more pentagons. 3 more pentagons join these pentagon groups together in yet another realm (there are 20 groups of these 3 pentagons). In order to simplify, visualize a truncated icosahedron and from each pentagonal face appear 6 pentagonal faces, a central one and 5 surrounding it, and from each hexagonal face appear 3 pentagonal faces.
I don't even know how to name its equivalent in three dimensions. I only know that it would have 132 pentagons (I think that there would be 120 isosceles and 12 regular ones). It would be the dual of a pentakis omniaugmented truncated dodecahedron. (Omniaugmented is a neologism based on Johnson augmentation function, for more details see http://en.wikipedia.org/wiki/Triaugmented_truncated_dodecahedron and imagine the augmentation function being applied to every decagonal face).
In order to visualize it, use this site: http://levskaya.github.com/polyhedronisme/, insert dk10tD in the RECIPE box (it is a Conway polyhedron notation) and visualize how it would look like if every decagonal face where transformed into a pentagonal face with other 5 pentagonal faces surrounding it.
120-cell "Petrie polyhedron", like the other ones, only have regular pentagonal faces, because they lie in different realms.
There are some things I would like to know.
1. Can someone tell me if my visualizations were right?
2. Can someone tell me the name of 120-cell "Petrie polyhedron"?
3. Can someone see an easier way to construct these "Petrie polytopes"?
4. Would it be simple to generalize it to higher dimensions?
5. Would it be very difficult to create a "Petrie polyhedron" for the Grand Antiprism or any other uniform polychoron?
Thanks in advance for the help,
Hugo Cesar