Fat 4-polytopes and fatter 3-spheres

We introduce the fatness parameter of a 4-dimensional polytope P, defined as \phi(P)=(f_1+f_2)/(f_0+f_3). It arises in an important open problem in 4-dimensional combinatorial geometry: Is the fatness of convex 4-polytopes bounded? We describe and analyze a hyperbolic geometry construction that produces 4-polytopes with fatness \phi(P)>5.048, as well as the first infinite family of 2-simple, 2-simplicial 4-polytopes. Moreover, using a construction via finite covering spaces of surfaces, we show that fatness is not bounded for the more general class of strongly regular CW decompositions of the 3-sphere.

[1]  Paul Erdös.  Über die Primzahlen gewisser arithmetischer Reihen , 1935 .

[2]  P. McMullen.  The numbers of faces of simplicial polytopes , 1971 .

[3]  B. Bollobás,et al.  Extremal Graph Theory , 2013 .

[4]  R. Stanley.  The number of faces of a simplicial convex polytope , 1980 .

[5]  Carl W. Lee,et al.  A Proof of the Sufficiency of McMullen's Conditions for f-Vectors of Simplicial Convex Polytopes , 1981, J. Comb. Theory, Ser. A.

[6]  James R. Munkres,et al.  Elements of algebraic topology , 1984 .

[7]  È. Vinberg,et al.  Hyperbolic reflection groups , 1985 .

[8]  Joseph Malkevitch.  TILING CONVEX POLYGONS WITH EQUILATERAL TRIANGLES AND SQUARES , 1985 .

[9]  Louis J. Billera,et al.  Generalized Dehn-Sommerville relations for polytopes, spheres and Eulerian partially ordered sets , 1985 .

[10]  Joseph H. Silverman,et al.  The arithmetic of elliptic curves , 1986, Graduate texts in mathematics.

[11]  Margaret Bayer,et al.  The extended f-vectors of 4-polytopes , 1987, J. Comb. Theory, Ser. A.

[12]  N. J. A. Sloane,et al.  Sphere Packings, Lattices and Groups , 1987, Grundlehren der mathematischen Wissenschaften.

[13]  Chee-Keng Yap,et al.  Computational complexity of combinatorial surfaces , 1990, SCG '90.

[14]  Average kissing numbers for non-congruent sphere packings , 1994, math/9405218.

[15]  G. Ziegler.  Lectures on Polytopes , 1994 .

[16]  William Fulton.  Algebraic Topology: A First Course , 1995 .

[17]  János Pach,et al.  Combinatorial Geometry , 2012 .

[18]  G. Kuperberg,et al.  Highly saturated packings and reduced coverings , 1995, math/9511225.

[19]  W. Thurston,et al.  Three-Dimensional Geometry and Topology, Volume 1 , 1997, The Mathematical Gazette.

[20]  Günter M. Ziegler,et al.  Neighborly Cubical Polytopes , 2000, Discret. Comput. Geom..