## Face Numbers of 4-Polytopes and 3-Spheres (2002)

Venue: | Proceedings of the international congress of mathematicians, ICM 2002 |

Citations: | 10 - 2 self |

### BibTeX

@INPROCEEDINGS{Ziegler02facenumbers,

author = {Günter M. Ziegler},

title = {Face Numbers of 4-Polytopes and 3-Spheres},

booktitle = {Proceedings of the international congress of mathematicians, ICM 2002},

year = {2002},

pages = {625--634},

publisher = {Higher Education Press}

}

### OpenURL

### Abstract

Steinitz (1906) gave a remarkably simple and explicit description of the set of all f-vectors f(P ) = (f0 , f1 , f2) of all 3-dimensional convex polytopes. His result also identifies the simple and the simplicial 3-dimensional polytopes as the only extreme cases. Moreover, it can be extended to strongly regular CW 2-spheres (topological objects), and further to Eulerian lattices of length 4 (combinatorial objects).

### Citations

199 |
The Geometry and Topology of 3-manifolds
- Thurston
- 1978
(Show Context)
Citation Context ...eorem may be phrased as follows: Every connected finite Eulerian lattice of length 4 is the face lattice of a rational convex 3-polytope. 3. The famous Koebe--Andreev--Thurston circle-packing theorem =-=[26]-=- implies that every combinatorial type of 3-polytope has a realization with all edges tangent to the unit sphere S 2 . Furthermore, the representation is unique up to Mobius transformations; thus, in ... |

181 | Lectures on polytopes. Graduate texts in mathematics - Ziegler - 2006 |

80 |
Generalized h-vectors, intersection cohomology of toric varieties and related results
- Stanley
- 1987
(Show Context)
Citation Context ... below. Another important, self-dual inequality reads f 03 - 3(f 0 + f 3 ) # -10, that is, C(P ) # 3. In fact, this is the 4-dimensional case of the condition "g tor 2 (P ) # 0" on the toric=-= h-vector [21]-=- of a polytope; it was proved by Stanley for rational polytopes only, and verified for all convex polytopes by Kalai [13] with a simpler argument based on a rigidity result of Whiteley. It is still no... |

77 |
Generalized Dehn–Sommerville relations for polytopes, spheres and Eulerian partially ordered sets, Invent
- Bayer, Billera
- 1985
(Show Context)
Citation Context ...0 - f 1 + (-1) d-1 f d-1 = 1+ (-1) d-1 . The flag-vector (with 2 d components, including the f-vector) is highly redundant, due to the linear "generalized Dehn-Sommerville relations" (Bayer =-=& Billera [3]-=-) that allow one to reduce the number of independent components to F d - 1, one less than a Fibonacci number. In particular, for d = 3 there is no additional information in the flag-vector, by f 01 = ... |

56 | Variational principles for circle patterns and Koebe’s theorem
- Bobenko, Springborn
(Show Context)
Citation Context ... to the unit sphere S 2 . Furthermore, the representation is unique up to Mobius transformations; thus, in particular, symmetric graphs/lattices have symmetric realizations. (See Bobenko & Springborn =-=[6]-=- for a powerful treatment of this result, and for references to its involved history.) This is the situation in dimension 3. The picture in dimension 4 is not only quite incomplete; it is also clear b... |

49 |
Rigidity and the lower bound theorem
- Kalai
- 1987
(Show Context)
Citation Context ...e 4-dimensional case of the condition "g tor 2 (P ) # 0" on the toric h-vector [21] of a polytope; it was proved by Stanley for rational polytopes only, and verified for all convex polytopes=-= by Kalai [13]-=- with a simpler argument based on a rigidity result of Whiteley. It is still not established for the case of strongly regular 3-spheres, or for Eulerian lattices of length 5.sGunter M. Ziegler 5. The ... |

47 |
Theorie der vielfachen Kontinuitat (1852), in
- Schläfli
- 1950
(Show Context)
Citation Context ...intensive study since antiquity, . properties of convex polytopes are essential to the geometry of Euclidean spaces, . the regular polytopes (in all dimensions) were classified by Schlafli in 1850-52 =-=[20]-=-, exactly 150 years ago, and . modern polytope theory has achieved truly impressive results, in particular since the publication of Grunbaum's volume [9] in 1967, thirty-five years ago. Moreover, we h... |

42 |
The numbers of faces of simplicial polytopes
- McMullen
- 1971
(Show Context)
Citation Context ...duce facets that meet only at the apex (the simplex). The f - and flag-vectors of simple/simplicial 4-polytopes and 3-spheres are well-known --- a complete picture is given by the g-Theorem (McMullen =-=[15]-=-), which for 4-polytopes was first established by Barnette [1] and for 3-spheres by Walkup [27]). Face Numbers of 4-Polytopes and 3-Spheres 7 6. Constructions In order to prove completeness for linear... |

39 | Realization Spaces of Polytopes
- Richter-Gebert
(Show Context)
Citation Context ...ve a realization with rational (vertex) coordinates. We have natural inclusions P Q 4 # P 4 # S 4 # L 4 . The first inclusion is strict due to the existence of non-rational 4-polytopes (RichterGebert =-=[18]-=-), the second one due to the known examples of non-realizable triangulated 3-spheres, the third since any strongly regular cell decomposition (e. g., a triangulation) of a compact connected 3-manifold... |

25 | Combinatorial aspects of convex polytopes
- Bayer, Lee
- 1993
(Show Context)
Citation Context ...he cones they span! Face Numbers of 4-Polytopes and 3-Spheres 5 4. Fatness and Complexity Instead of linear combinations of face numbers or flag numbers (such as the toric h-vector, the cd-index etc. =-=[4]), in the -=-following we will rely on quotients of such. Thus we obtain homogeneous "density" parameters that characterize extremal polytopes. Such a quotient is the average vertex degree f01 f0 = 2 f1 ... |

16 |
The extended f-vectors of 4-polytopes
- Bayer
- 1987
(Show Context)
Citation Context ...Flag-Vectors(P 4 ).sGunter M. Ziegler The known facts about and partial description of the sets f -Vectors(P 4 ) and Flag-Vectors(P 4 ) have been reviewed in detail in Grunbaum [9, Sect. 10.4], Bayer =-=[2]-=-, Bayer & Lee [4, Sect. 3.8], and Hoppner & Ziegler [11]. Here we will be concerned only with the linear known conditions that are tight at flag(# 4 ), and concentrate of the case of f-vectors rather ... |

15 |
Neighborly cubical polytopes, Discrete Comput. Geometry 24 (2000), 325–344, The Branko Grünbaum birthday issue
- Joswig, Ziegler
(Show Context)
Citation Context ...olytopes (neighborly, stacked, random, etc.). Cubical polytopes (all of whose proper faces are combinatorial cubes) form a natural class of polytopes. A very specific construction by Joswig & Ziegler =-=[12] produced -=-"neighborly cubical" polytopes as special projections of suitably deformed n-cubes to R 4 : These are rational cubical 4-polytopes C n 4 with the graph of the ncube (for n # 4), hence with f... |

14 |
The lower bound conjecture for 3
- Walkup
- 1970
(Show Context)
Citation Context ...cial 4-polytopes and 3-spheres are well-known --- a complete picture is given by the g-Theorem (McMullen [15]), which for 4-polytopes was first established by Barnette [1] and for 3-spheres by Walkup =-=[27]-=-). Face Numbers of 4-Polytopes and 3-Spheres 7 6. Constructions In order to prove completeness for linear descriptions of f - or flag-vector cones, one needs to have at one's disposal enough examples ... |

10 | Polymake: A software package for analyzing convex polytopes, http://www.math.tu-berlin.de/diskregeom/polymake - GAWRILOW, JOSWIG |

8 |
Über die Eulerschen Polyederrelationen, Archivfür Mathematik und
- Steinitz
- 1906
(Show Context)
Citation Context ...combinatorial and geometric properties of 3-dimensional polytopes were isolated a long time ago. Here we mention the three results that will be most relevant to our subsequent discussion: 1. Steinitz =-=[24]-=- characterized the f-vectors (f 0 , f 1 , f 2 ) # Z 3 of the 3-dimensional convex polytopes: They are the integer points in the 2-dimensional convex polyhedral cone that is defined by the three condit... |

4 | A census of flag-vectors of 4-polytopes
- Höppner, Ziegler
- 2000
(Show Context)
Citation Context ...bout and partial description of the sets f -Vectors(P 4 ) and Flag-Vectors(P 4 ) have been reviewed in detail in Grunbaum [9, Sect. 10.4], Bayer [2], Bayer & Lee [4, Sect. 3.8], and Hoppner & Ziegler =-=[11]-=-. Here we will be concerned only with the linear known conditions that are tight at flag(# 4 ), and concentrate of the case of f-vectors rather than flag-vectors. 3. Geometry/Topology/Combinatorics It... |

4 |
Stresses and liftings of cell-complexes, Discrete Comput Geom 21
- Rybnikov
- 1999
(Show Context)
Citation Context ...g the tiles by Schlegel diagrams based on a simplex facet. (The converse direction, from tilings of R 3 to 4-polytopes, is non-trivial: It hinges on non-trivial liftability restrictions; see Rybnikov =-=[19]-=-.) Normal tilings are face-to-face tilings of R d-1 by convex polytopes for which the inradii and circumradii of tiles are bounded from below resp. from above --- see Grunbaum & Shephard [10, Sect. 3.... |

3 |
und Raumeinteilungen, in Encyklopädie der mathematischen Wissenschaften
- Polyeder
- 1922
(Show Context)
Citation Context ...e polytopes with extremal f-vectors: The first inequality is tight if and only if the polytope is simplicial, while the second one is tight if and only if the polytope is simple. 2. In 1922, Steinitz =-=[25]-=- published a characterization of the graphs of 3-polytopes: They are all the planar, three-connected graphs on at least 4 nodes. In modern terms (as reviewed below) and after some additional arguments... |

2 |
Fat 4-polytopes and fatter 3-spheres
- Eppstein, Kuperberg, et al.
(Show Context)
Citation Context ...ace of both of them (which may be empty). These objects appear as "regular CW 3-spheres with the intersection property" as in Bjorner et al. [5, pp. 203, 223]; following Eppstein, Kuperberg =-=& Ziegler [7] we call t-=-hem "strongly regular spheres." The intersection property is equivalent to the fact that the face poset of the cell complex is a lattice. Convex polytopes: P 4 denotes the combinatorial type... |

1 |
Inequalities for f-vectors of 4-polytopes
- Barnette
- 1972
(Show Context)
Citation Context ...nd flag-vectors of simple/simplicial 4-polytopes and 3-spheres are well-known --- a complete picture is given by the g-Theorem (McMullen [15]), which for 4-polytopes was first established by Barnette =-=[1]-=- and for 3-spheres by Walkup [27]). Face Numbers of 4-Polytopes and 3-Spheres 7 6. Constructions In order to prove completeness for linear descriptions of f - or flag-vector cones, one needs to have a... |

1 |
triangulated spheres, Discrete Comput. Geometry 3
- Many
- 1988
(Show Context)
Citation Context ...ertices. Thus one obtains (Pfeifle [17]) that on a large number of vertices there are far more triangulated 3-spheres than there are types of simplicial 4-polytopes, thus resolving a problem of Kalai =-=[14]. 7. Tilin-=-gs There are close connections between d-polytopes ("polyhedral tilings of S d-1 ") and normal polyhedral tilings of R d-1 . In particular, from 4-polytopes one may construct 3-dimensional t... |

1 |
Work in progress
- Paenholz
- 2002
(Show Context)
Citation Context ...ell, with flag-vector flag(E(P 120 )) = (720, 3600, 3600, 720; 5040), fatness F(E(P 120 )) > 5.02, and 720 biyramids over pentagons as facets, could be non-rational, current investigations (Pa#enholz =-=[16]-=-) suggest that E-polytopes are less rigid than one would think at first glance, since in some cases the tangency conditions may be relaxed or dropped.sGunter M. Ziegler Fat 3-spheres: The M g -constru... |

1 |
Work in progress
- Pfeifle
- 2002
(Show Context)
Citation Context ...plied to the fat 3-spheres produced by the M g -construction, the E-construction yields 3-spheres with substantially more nonsimplicial facets than their number of vertices. Thus one obtains (Pfeifle =-=[17]-=-) that on a large number of vertices there are far more triangulated 3-spheres than there are types of simplicial 4-polytopes, thus resolving a problem of Kalai [14]. 7. Tilings There are close connec... |

1 |
Work in progress
- Wamer
- 2002
(Show Context)
Citation Context ...en d-ball into convex cells; via one-point-compactification (e. g. generated by stereographic projection) by one additional vertex we obtain a regular cell-decomposition of a d-sphere; thus we obtain =-=[28]-=- that for all # > 0 f 0 (#) - f 1 (#) + + (-1) d f d-1 (#) = (-1) d-1 . In particular, this implies that if the limits # i := lim ### f i (#) # j f j (#) exist, then they satisfy 0 # # i # 1 2 . Furth... |

1 |
Work in progress
- Paffenholz
- 2002
(Show Context)
Citation Context ...20-cell, with flag-vector flag(E(P120)) = (720, 3600, 3600, 720;5040), fatness F(E(P120)) > 5.02, and 720 biyramids over pentagons as facets, could be non-rational, current investigations (Paffenholz =-=[15]-=-) suggest that E-polytopes are less rigid than one would think at first glance, since in some cases the tangency conditions may be relaxed or dropped.8 Günter M. Ziegler Fat 3-spheres: The Mg-constru... |

1 |
Work in progress
- Waßmer
- 2002
(Show Context)
Citation Context ...en d-ball into convex cells; via one-point-compactification (e. g. generated by stereographic projection) by one additional vertex we obtain a regular cell-decomposition of a d-sphere; thus we obtain =-=[26]-=- that for all ρ > 0 f0(ρ) − f1(ρ) + · · · + (−1) d fd−1(ρ) = (−1) d−1 . In particular, this implies that if the limits φi := lim ρ→∞ fi(ρ) ∑ j fj(ρ)Face Numbers of 4-Polytopes and 3-Spheres 9 exist, ... |