skew polytopes / skeletal polytopes

[Klitzing]

Petrie once brought up into interest the skeletal regular polygons.

Those are defined solely by their vertices and their edges (sides). But the spanned membran (their area or body) simply is neglected. Then regular polygons no longer need to be flat. They can be finite or infinite zig-zags, can be helical (with a flat regular polygonal projection image) or can be simply the alraidy well-known linear apeirogon. In his days he used them to introduce as what then became the Petrie polygon of a regular polyhedron or even of higher dimensional regular polytopes.

It even is possible to define a Petrie dual for regular polytopes by re-using the same edge skeleton, but interchanging the former faces by the Petrie polygons. This procedure then happens to be involutoric, that is, being applied twice, it does return to the starting figure.

Coxeter then used such skeletal polygons for vertex figures of "new" regular polyhedra. Those were {4, 6 | 4}, {6, 4 | 4}, and {6, 6 | 3}. I.e. 6 squares per vertex in the first case. But those are not arranged flat (as for hyperbolic tilings), rather the vertex figure is a skew zig-zag-Hexagon. Thereby induced one gets further recurrencies. This is what the number behind the pipe are for: in the first 2 cases there are square loops (holes). In fact the first one is a substructure of the squares of the cubic honeycomb x4o3o4o = x4o3o4x, the second is the skew polyhedral manifold of the hexagons in the bitruncatedcubic honeycomb o4x3x4o, and the latter one is the skew polyhedral manifold of the hexagons in the quarter cubic honeycomb x3x3o3o3*a.

Recently now researchers around Branko Grünbaum and Egon Schulte are considering those skew skeletal polygons not only for vertex figures, but also for true faces of according regular polyhedra. This then brings back the symmetry of {p, q}. Also potential Wythoffians of all those are considered.


So far on the current state of the art.
Here now my question: could those Wythoffians likewise be described by (somehow extended) Dynkin symbols?

Schulte already kind of distinguishes the polygons, writing e.g. {4_c, 6_s | 4_c}. "c" here means a (flat) convex regular polygon, "s" here means a skew regular polygon. So we probably have to distinguish these by according Indices. But then (provided the "correct" symmetry (undecorated diagram) is being found, a decoration could be applied as for "normal" Wythoffians. Ain't it?

--- rk

[Klitzing]

But those are not arranged flat (as for hyperbolic tilings), rather the vertex figure is a skew zig-zag-hexagon. Thereby induced one gets further recurrencies. This is what the number behind the pipe are for: in the first 2 cases there are square loops (holes).

If I get it myself correctly, then both structures, the hyperbolic flat tiling and the corresponding infinite skew polyhedron, both have the same local incidence structure. OTOH the longer range structure differs however. The latter seem to be some (still infinite) modwrap of the former. As in those mentioned regular skew polyhedra there are holes to run around. When considering the "same" paths on the hyperbolic tilings, one gets to completely separate positions. Accordingly those seem to be additionally identified now.

Question: Could that not only be derived within the explicite examples a posteriori, but in contrary already right before considering the geometric realizations in detail, e.g. just by considering the numbers / symbols? Esp. is there a way to derive the respective genus of the latter ones and to deduce the numbers after the pipes already from the numbers before the pipe? Or, if several such modwraps could be applied, would there be a chance on how to read off those possible identifications e.g. right from the hyperbolic patterns?

--- rk

[Klitzing]

Btw., nice pixies of Coxeter's 3 infinte skew regular polyhedra together with some of Gott's aditional "pseudopolyhedra" (those would not qualify in the former sense, as here always some incident faces become coplanar) can be seen here on Steve Dutch's website. He even depicts here several further infinite skew polyhedra from Wells. Both extensions so are no longer regular ones, as they contain non-equivalent faces. But still all only use flat regular convex polygons for faces...
Also at the webpage of Melinda Green nice pixies and also according interactive VRMLs are provided, cf. here.
--- rk

[Klitzing]

If I get it myself correctly, then both structures, the hyperbolic flat tiling and the corresponding infinite skew polyhedron, both have the same local incidence structure. OTOH the longer range structure differs however. The latter seem to be some (still infinite) modwrap of the former. As in those mentioned regular skew polyhedra there are holes to run around. When considering the "same" paths on the hyperbolic tilings, one gets to completely separate positions. Accordingly those seem to be additionally identified now.

Question: Could that not only be derived within the explicite examples a posteriori, but in contrary already right before considering the geometric realizations in detail, e.g. just by considering the numbers / symbols? Esp. is there a way to derive the respective genus of the latter ones and to deduce the numbers after the pipes already from the numbers before the pipe? Or, if several such modwraps could be applied, would there be a chance on how to read off those possible identifications e.g. right from the hyperbolic patterns?

Seem to have found already myself an according input in an article of Egon Schulte and Jörg M. Wills, "Convex-Faced Combinatorially Regular Polyhedra of Small Genus", Symmetry 2012, 4, 1-14; doi:10.3390/sym4010001 (or at this weblink). Therein in section 2 they state:

...
For a regular map P of type {p, q} with f₀ vertices, f₁ edges, and f₂ faces on a closed surface, the order of its automorphism group Gamma(P) is linked to the Euler characteristic chi of the surface by
the equation

(1)    chi = f₀ - f₁ + f₂ = |Gamma(P)|/2 * (1/p + 1/q - 1/2)

If {p, q} is of hyperbolic type, this immediately leads to the classical Hurwitz inequality,

(2)    |Gamma(P)| <= 84 |chi|

with equality occurring if and only if P is of type {3, 7} or {7, 3}. In particular, this shows that there can only be finitely many regular maps on a given closed surface of non-zero Euler characteristic. For the genus range under consideration, the inequality in Equation (2) also establishes, a priori, an upper bound for the order of the automorphism group of the map, and hence for the geometric symmetry group of any of its polyhedral realizations in E³.
The combinatorial automorphism group Gamma(P) is generated by three involutions rho₀, rho₂, rho₂ satisfying the Coxeter Relations

(3)    rho₀² = rho₁² = rho₂² = (rho₀ rho₁)^p = (rho₁ rho₂)^q = (rho₀ rho₂)² = 1

but in general also some further, independent relations. On the underlying surface, these generators can be viewed as “combinatorial reflections” in the sides of a fundamental triangle of the “barycentric subdivision” of P. For a finite regular map, the relations in Equation (3) suffice for a presentation of Gamma(P) if and only if P is a Platonic solid; that is, if and only if g = 0. Thus at least one, but generally more additional relations are needed if g > 0. Two kinds of extra relations, usually occurring separately, are of particular importance to us and suffice to describe seven of the eight regular maps under consideration, namely the Petrie relation

(4)    (rho₀ rho₁ rho₂)^r = 1

and the hole relation

(5)    (rho₀ rho₁ rho₂ rho₁)^h = 1

These relations are inspired by the notions of a Petrie polygon and of a hole of a regular map, respectively. Recall that a Petrie polygon of a regular map (on any surface) is a zigzag along its edges such that every two, but no three, successive edges are the edges of a common face. A hole of a regular map (on any surface) is a path along the edges which successively takes the second exit on the right (in a local orientation), rather than the first, at each vertex. The automorphism rho₀ rho₁ rho₂ of P occurring in Equation (4) shifts a certain Petrie polygon of P one step along itself, and hence has period r if the Petrie polygon is of length r. Similarly, the automorphism rho₀ rho₁ rho₂ rho₁ of Equation (5) shifts a certain hole of P one step along itself, and hence has period h if the hole is of length h. Thus, if the relation in Equation (4) or Equation (5) holds, with r or h giving the correct period of rho₀ rho₁ rho₂ or rho₀ rho₁ rho₂ rho₁, respectively, then P has Petrie polygons of length r or holes of length h.
...

Thus I think the answer to my question ought be somewhere deducible from these lines. Any help on its application however would be appreciated. Simply because, as usual, any discussion thereon often helps to clear lots of points ...

Can anyone make sense of those line?
Could those more technical lines be "translated" into easier to grasp common sense wordings, clearify on the individual Actions of those rho's, and esp. on how these operations of (4) and (5) would look within the original hyperbolic tiling?
Esp. what then could be answers to my original questions, cf. above?

--- rk

[Klitzing]

Hehe,
once more one step ahead!

Cf. this page.
It provides a graphical device onto the understanding of the Petrie polygon and the hole:

• For a face of a regular tiling the edge sequence always makes the sharpest possible turn, always into the same direction.
• For a Petrie polygon of a regular tiling the edge sequence always the sharpest possible turn, always into alternating directions (zigzag).
• For a hole of a regular tiling the edge sequence always makes the 2^nd-sharpest possible turn, always into the same direction.

The idea then is as follows:

• {p, q} is a spherical, euclidean or hyperbolic regular tiling
• {p, q} _r then is that modwrap of the former, where each r-th vertex on the Petrie polygons is identified - provided such a figure exists
• {p, q | r} then is that modwrap of the first, where each r-th vertex on each hole is identified - provided such a figure exists

--- rk

[wendy]

Although John Petrie discovered both the polygon and skew polyhedra, it's best not to suppose they're identical.

The petrie polygon is given by a march of the vertex through the nodes of a symmetry group in cyclic order, and the PP is thus hx vertices, where x is the number of marked nodes, and nh = 2m n=dimension, h = petrie polygon, m = nr of mirrors. The 2_21 has h=12 for example.

{2p,2q | r) is a group designation corresponding to the skew p-gons in xPxRoQoRz. The dual exchanges x and o. But they correspond to the same group as a subgroup of {2p,2q}

Coxeter and Moser lists the examples of {p,q}_r. These are already designated as xPoQoAR%, with the CM-list at hand. One can rotate te x, o and % to six different positions, meaning that each of these figures, the edge is tri-diactic (two vertices, two faces, two petrie polygons).

[Klitzing]

I'm still struggling with your symbol "xPoQoAR%". Supposedly "xPoQoARo" would translate into "xPoQo *bRo". But then, what means your extra node symbol "%"?

Yes, you are right. {2p, 2q | r} is defined as a modrwrap of {2p, 2q} = x-2P-o-2Q-o along every r-th vertex along a "hole" (= 2nd sharpest bend along edge sequences, always into the same direction) and happens to be a facial substructure of xPxRoQoR*a.
But then it even would be possible in generality to consider

Code: Select all

  2p       
{    | 2r }
  2q       

likewise being defined as the modwrap of o-2P-x-2Q-o along every 2r-th vertex along a hole. This then would be the facial substructure of xPxRxQxR*a.

--- rk

[Klitzing]

Oops, forget about my erronious mentioning of o-2P-x-2Q-o within my previous post. I clearly meant instead: x-2P-o-2Q-x.

Thus, the facial substructure of xPxRxQxR*a rather is by modwrap of x-2P-o-2Q-x by every 2r-th vertex along a hole.
Accordingly one supposedly ought write s.th. like x2Po2Qx|2R here, or in Wendy's decorated Schläfli symbols one might write {;2p,2q;|2r}.
The local vertex figure surely is for both, the skew polyhedron as well as the flat (non-modwrapped) tiling: [2p,4,2q,4].
In fact, within xPxRxQxR*a we simply omit the cells (for sure) as well as all . xRx . and x . . xR*a, i.e. all the 2r-gons.

In other words: t_02( {2p, 2q | r} ) = { t_02( {2p, 2q} ) | 2r}.

--- rk

[wendy]

oPoQoARp (it's actually p, not %. I looked it up :~ ) isn't a real polytope in the sens that the Petrie-Coxeter thing is. So what you are likely to get at most is some sort of map.

This is a list from hsm coxeter, wo moser 'generators and relations for discrete groups' 2nd ed, Ergebnisse der mathematik und ihrer grenzgebiete - neue golge - band 14 (springer-verlag)

It lists all of the finite examples that c&m could find, but John Conway told me the list was incomplete, without any further example quoted.

The lost gives p,q,r, and the order of the group p,q,r. You can put P, Q, R in any order of

xPoQoARp N_0 = g/2q N_1 = g/pq N_2 = g/2p, N_p = g/2r. Where N_p = number of petrie polygons.

Each edge of these figures has an incidence ray to N_0, N_2 and N_p, that is, they belong to two vertices, two faces or hedra, and two petrie polygons.

The effect of this symmetry is that where the nodes run AOBC above, that AA = BB = CC = 1, OO = 1 and

AB = BA, AC = CA BC = CB
AOA.. = OAO.. OBO = BOB.. COC.. = OCO.. (to the number of the branches OA, OB, OC
Petrie polygon A = OBC = BCO = COB, B = AOC = OCA = CAO, C = ABO = BOA = OAB

Code: Select all

  3,  3,  4     24 
  3,  5,  5     60
  3,  5, 10    120   
  4   5   5    160
  4   5   6    240
  3   7   8    336
  3   7   9    504
  5   5   5    660
  3   8   8    672
  4   5   8   1440
  3   7  13   1092
  3   7  12   2184
  3   7  14   2184
  3   9   9   3420
  3   8  10   4320
  4   6   7   4368
  4   5   9   6840
  3   8  11  12144
  3   7  15  12180
  3   7  16  21504
 
  2  2q  2q     8q
  2   p  2p     4p    p odd
  3   6  2p    12p²
  4   4  2q    16p²


At the moment these groups are taken as a kind of isomorphism, in that we can put 'edge, polygon, petrie' in any order, and the same symmetry exists. But since petrie is an isomorphism that has its centre at the centre of the polytope, the effect as far as making models goes is the same as the 'turn-me-inside-out' figures.

The overall symmetry is then a kind of symmetry that exists with when one supposes an isomorph like 7 -> 7/2 -> 7/3 -> 7 is projected on top of the symmetry [7], so we get an overall symmetry of the heptagon is not just [7] of order 14, but [7][3]+ of order 42.

The first three have reflexes in terms of the regular polytopes, those of 1 and 3 particulary match the order of the {3,3} and {3,5} resp.