Petrie polytope (Petrie p-gon analogy for higher dimensions)

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

Petrie polytope (Petrie p-gon analogy for higher dimensions)

Postby hcesarcastro » Fri Apr 06, 2012 9:07 pm

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
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Re: Petrie polytope (Petrie p-gon analogy for higher dimensi

Postby wendy » Sun Apr 08, 2012 7:41 am

A petrie polygon is caused by the cycle of mirrors. One uses it to find the number of mirrors in a polytope: 2m = nh (2×mirrors = dimension × petrie polygon), eg 4B is 6 dimensional, has 36 mirrors, sp h = 36×2/6 = 12. {3,3,5} has n=4, m=60, so 2×60/4 = 30, is its petrie polygon. What is the form of the petrie polytope, save that it is a medial split between opposite poles.
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Re: Petrie polytope (Petrie p-gon analogy for higher dimensi

Postby hcesarcastro » Sun Apr 08, 2012 3:31 pm

wendy wrote:What is the form of the petrie polytope, save that it is a medial split between opposite poles.


I once read in http://en.wikipedia.org/wiki/Petrie_polygon that Petrie polygons are useful in visualizing symmetric structure of the higher dimensional regular polytopes.

I only thought that one could get a better idea of its symmetric structure if he/she could get a higher dimension figure (a polyhedron, a polychoron and so on). I also liked the idea of skew polygons, so I thought how they would look in higher dimensions.


Answering my own question, I visualized what would be the Petrie poly-(n-1)-topes of the 3 regular polytopes of any dimension n >= 5.

n-simplex "Petrie poly-(n-1)-tope" would be a (n-1)-simplexal bipyramid (triangular bipyramid, tetrahedral bipyramid, pentachoric bipyramid and so on);
n-measure polytope "Petrie poly-(n-1)-tope" would be the dual of an expanded (n-1)-simplex. The "petrie polyhedron" would be a special case, as the cantellated (expanded) tetrahedron is the cuboctahedron, and thus the hypercube "Petrie polyhedron" is something similar to the rhombic dodecahedron;
n-orthoplex "Petrie poly-(n-1)-tope" would be a gyroelongated (n-1)-simplexal bipyramid. Something like a (n-1)-orthoplex, in which two opposing facets are removed . One (n-1)-simplex with one of its facets also removed is inserted on each space of the (n-1)-orthoplex that once was occupied by a facet.
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Re: Petrie polytope (Petrie p-gon analogy for higher dimensi

Postby wendy » Mon Apr 09, 2012 9:33 am

You can indeed build maximum sections etc of polytopes, including outlines of projections, but these are hardly 'petrie polytopes'.

If ye look on the 'talk' page of that same wikipedia, ye would see many comments there that i made. Tom Ruen wrote most of the stuff, but he did exchange a number of emails with me. Because i do a lot of original research in the area, i usually don't write on the front pages of many of the pages, although i do write on things like DOS programs and some weights and measures.

Petrie polygons have some rather useful rules and limitations about how big a polytope might be. They're a result of þe operation of symmetries, a polytope having x marked nodes, will have a petrie polygon of xh sides. Because the maximum value of x is n, every polytope constructed from a simple group must be no larger than an {2m}, where m is the number of mirrors.
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Re: Petrie polytope (Petrie p-gon analogy for higher dimensi

Postby hcesarcastro » Mon Apr 09, 2012 2:20 pm

I understand what you mean. I only thought if there could be shapes/solids equivalent to the Petrie polygons on higher dimensions.

And if not calling them "Petrie polytopes" how could I call a skew polyhedron with non-skew faces. I searched for "Petrie polyhedra" on Google and found the Wikipedia page about "Infinite skew polyhedra". That was not exactly what I was looking for, I wanted a close and convex element (not an infinite one) with non-skew faces, but that could not exist in three dimensions. Like the 12-cube solid I described. Its faces are planar, but the polyhedron itself is skew. And how many of them could be built in 3, 4, ... n dimensions. It would only be an extension to the definition of polytopes.


Wendy, about the wikipedia talk page, I read it. All definition you wrote about the Petrie polygons made me understand it a little bit better, besides I still don't understand very well what are these "mirrors". But I don't understand it very weel the same way I don't get how Lie groups and Lie algebra work (and I already studied a little bit of group theory at the university). Anyway, this is not my research field (I do research on Computational Linguistics - I prefered to work with strings... LOL), I can only say I am a polytope aficionado.
I also saw your wikipedia user page. It is good to know that there are more people in this world other than me that like polytopes and linguistics (two very different fields of research). Like "quickfur" says, I am a "fellow conlanger" ;)


I would be glad if you could give me any help with this kind of weird polytopes that I came with.

Best regards
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