Talk:Petrie polygon

I created this stub article using info from Mathworld, then looked at Coxeter's book Regular polytopes.

For the generalized Petrie polygons, I left out any math details, just gave the results for h as Coxeter listed, and all the graphs I could find that represent the regular polytopes. Fairly complete! I have diagrams from other books for the regular polygons for the 24-cell and 600-cell (maybe I can get these digitized and traced?), but nothing for 120-cell. Mathworld has a sampling of diagrams for the lower hypercubes. The construction still not within my grasp, but I'm hopeful I can compute the correct orthogonal projections now that I know they exist. For now my "column graphs" will have to do, at as least 2n-gons, even if not regular. Tom Ruen (talk) 03:04, 25 July 2008 (UTC)

but no three

 * A Petrie polygon of a regular polygon {p} trivially has p sides as itself.

This violates the "but no three" provision in the first sentence. —Tamfang (talk) 14:37, 6 August 2009 (UTC)

Looks like you're right, as defined, a petrie polygon doesn't exist below polyhedra. I removed it. Tom Ruen (talk) 22:12, 6 August 2009 (UTC)

p.s. I wish we could get the 1_32, 2_41, 1_42 polytope graphs! Someday maybe I'll see if I can figure out how to construct them myself. Tom Ruen (talk) 22:19, 6 August 2009 (UTC)

regular star polytopes
The graphs should exists also for the nonconvex Kepler-Poinsot polyhedrons, and Schläfli-Hess polychorons.

For Kepler-Poinsot polyhedra, by the polyhedron formula, the {3,5/2} and {5/2,3} polyhedra have h= 10/3 (decagram), and {5,5/5} and {5/2,5} have h=6 (hexagon). Tom Ruen (talk) 22:34, 6 August 2009 (UTC)

Here's the Kepler solids, manually rotated to regular polygon boundaries, might be right or wrong! Tom Ruen (talk) 22:58, 6 August 2009 (UTC)

Orthogonal projections with vertices

Graphics marked up with petrie polygons(?)


 * In the cases where the outer edge is two polygons, these correspond to "face-first projections". The petrie polygons are in fact the red hexagon and decagram (10/3), that are not highlighted.--Wendy.krieger (talk) 07:41, 9 January 2010 (UTC)

Coxeter lists them in the tables at the back of Regular Polytopes (3rd edition). h = 6 for {5/2,5} and {5,5/2}, and 10/3 for{5/2,3} and {3,5/2}.

In 4D there is a slight wrinkle; the 5-cell's Petrie polygon can be {5}, but also {5/2} in an orthogonal plane. I will denote this as {5}, {5/2}.


 * {3,3,3}: {5}, {5/2}
 * {3,3,4}, {4,3,3}: {8}, {8/3}
 * {3,4,3}: {12}, {12/5}
 * {3,3,5}, {5,3,3}: {30}, {30/11}
 * {5/2,5,3}, {3,5,5/2}: {20}, {20/9}
 * {5,5/2,5}: {15}, {15/4}
 * {5/2,3,5}, {5,3,5/2}: {12}, {12/5}
 * {5/2,5,5/2}: {15/2}, {15/7}
 * {3,5/2,5}, {5,5/2,3}: {20/3}, {20/7}
 * {5/2,3,3}, {3,3,5/2}: {30/7}, {30/13}

It appears then that the only non-trivially occurring Petrie polygons (i.e. not from α,β,γ,hγn) are hexagons, decagons, dodecagons, pentadecagons, icosagons, icositetragons (appear in E8), and triacontagons. Double sharp (talk) 15:19, 27 February 2015 (UTC)

Hypercube projections
I successfully computed the orthogonal projection basis vector for the hypercube family. Since they already exist as graphs, I collected them as a summary in one chart below. The only improvement of my graphs is that I color the vertices by a spectrum sequence based on visual overlapping positions. I can do the same for the demicube and other families given vertex, edge data and the petrie polygon path. But a good first test for now! Tom Ruen (talk) 05:37, 23 August 2009 (UTC)




 * Hm, is there a pattern to which dimensions have a central vertex in this graph? —Tamfang (talk) 20:22, 23 August 2009 (UTC)


 * I checked up to 16-cube, and consistent pattern of only dimensions that are a power of 2 have no central vertices. Tom Ruen (talk) 20:57, 23 August 2009 (UTC)


 * One can demonstrate, that if a polygon contains 2o, where o is an odd number, then the centre will be occupied, since the radius of a {2o} is the same as the longest chord of an {o}.--Wendy.krieger (talk) 07:38, 9 January 2010 (UTC)

Petrie Polygon, defined and discussed.
Coxeter gives two different definitions for the petrie polygon, which happen to equal in the case of the regular figures.

One corresponds to N, but not N+1 edges in consecutive surtopes.

In the notation of Conway's orbifold notation, the replacement of the and [] brackets in the expansion of uniform polyhedra orbifold-notation, produces a similar symmetry, but the faces and petrie polygons are exchanged. A similar kind of exchange on this line occurs in finite regular polyhedrak maps limited by petrie polygon. See, zb Coxeter and Moser "Generators and Relations for Discrete Groups"

The second corresponds to the transformation of mirrors in cyclic order (eg R0 R1 R2), or R0 R4 R2 R5 R3 R1 ... [the individual order of the mirrors in a cycle does not matter. Because all of the mirrors are used twice, the total operation uses 2m=nh.

For the gosset figures 2_21, 3_21 and 4_21, h is 12, 18 and 30 respectively. This gives 6*12/2 = 36, 7*18/2 = 63, and 8*30/2=120 mirrors for these groups, giving eutactic stars of 72, 126 and 240 vertices (ie 1_22, 2_31, and 4_21 respectively.)

Given the second definition, the size of the petrie polygon of a polytope with k marked nodes, is hk, since an edge is generated for each node in each cycle. Since the petrie polygon girths a real polytope, this puts a upper limit on the size of these things.

--Wendy.krieger (talk) 07:35, 9 January 2010 (UTC)


 * You can generalise the concept of "Petrie polygon" to any graph, however irregular, that is embedded in a 2-manifold: start along some edge, and take the first left and the first right at alternate vertices, until you end up where you started going in the same direction and in the same phase. Maproom (talk) 17:58, 5 April 2012 (UTC)
 * However such polygons are in general not congruent. The important properties that Petrie discovered are relevant only certain highly symmetrical figures, such that all Petrie polygons lie in a common symmetry orbit. &mdash; Cheers, Steelpillow (Talk) 20:41, 5 April 2012 (UTC)

I'm not sure I agree with this sentence:
From the lead, "The Petrie polygon of a regular polygon is the regular polygon itself". I agree that the Petrie polygon of an even polygon (a cyclic graph embedded in the plane or sphere) is the same cyclic graph, similarly embedded. But I claim that the Petrie polygon of an odd polygon (embedded in the plane or sphere) is the same cyclic graph, but embedded in the projective plane. I am using the definition of "Petrie polygon" used by those who study regular maps, rather than the one supplied in the lead and based on polytopes in Euclidean space. Maproom (talk) 17:06, 12 August 2015 (UTC)
 * John Flinders Petrie first described these polygons for the regular polyhedra and other writers have since generalised the idea. This article defaults at lest to the spirit of his original conception. The article on regular maps which you link to is a disambig page, and neither of the two pages which it disambiguates mentions Petrie polygons by name. I am therefore at present unable to understand your problem. There are so many variations on any theme in polytope theory that to make headway I'd suggest that you give clear definitions of terms such as a polygon, a graph and a map, and explain why the type of space - Euclidean, spherical, projective, whatever, is a significant concern. &mdash; Cheers, Steelpillow (Talk) 19:42, 12 August 2015 (UTC)

What is regular map?
In sentence A Petrie dual or Petrial of a regular polyhedron is a regular map... what is regular map? Is it Regular map (graph theory) or something else? Jumpow (talk) 21:41, 12 February 2016 (UTC)
 * I believe that link is correct. Tom Ruen (talk) 21:50, 12 February 2016 (UTC)
 * Thank you. I found good explanation of regular map in book Abstract regual plytops of Peter McMullen and Egon Schulte (Chapter 1D Regular Maps, p.17). I think, it will be good to insert reference to this book after words regular map. Jumpow (talk) 19:28, 13 February 2016 (UTC)