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The enumeration of four-dimensional polytopes. (English) Zbl 0744.52006
A 3-sphere is a (face-to-face) cell complex on a sphere in 4-dimensional space. Radial projection gives, for each convex polytope in 4-space, a combinatorially equivalent 3-sphere whereas it is known that not all 3- spheres have their polytope counterpart. The paper gives an algorithm for enumerating all possible combinatorial types of 3-spheres. Using a cutting operation (similar to cutting a polytope with a hyperplane) it is possible to derive all 3-spheres with \(n\) cells from the one with \(n-1\) cells. The paper gives implementation details and computational results, including complete listings of 3-spheres with few vertices. In most cases, the produced 3-spheres could be classified into polytopical or not.

52B11 \(n\)-dimensional polytopes
05A15 Exact enumeration problems, generating functions
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[1] Altshuler, A.; Steinberg, L., The complete enumeration of the 4-polytopes and 3-spheres with eight vertices, Pacific J. of math., 117, 1-16, (1985) · Zbl 0512.52003
[2] Altshuler, A.; Steinberg, L., Enumeration of the quasi simplicial 3-spheres and 4-polytopes with eight vertices, Pacific J. of math., 113, 269-288, (1984) · Zbl 0512.52004
[3] Altshuler, A.; Bokowski, J.; Steinberg, L., The classification of simplicial 3-spheres with nine vertices into polytopes and non-polytopes, Discrete math., 31, 115-124, (1980) · Zbl 0468.52008
[4] Altshuler, A.; Steinberg, L., Neighborly combinatorial 3-manifolds with 9 vertices, Discrete math., 8, 113-137, (1974) · Zbl 0292.57011
[5] Barnette, D., Diagrams and schlegel diagrams, () · Zbl 0245.52005
[6] Barnette, D., The triangulations of the 3-sphere with up to 8 vertices, J. combin. theory ser. A, 14, 37-52, (1975) · Zbl 0251.52012
[7] Bokowski, J.; Sturmfels, B., Polytopal and nonpolytopal spheres an algorithmic approach, Israel J. of math., 57, 257-271, (1987) · Zbl 0639.52004
[8] Brückner, M., Über die ableitung der allgemeinen polytope und die nach isomorphismus verschiedenen typen der allgemeinen achtzelle (oktatope), Verh. kon. akad. v. wetensch., 1, 1-27, (1905), Sec. 10 · JFM 39.0629.01
[9] Engel, P., On the enumeration of polyhedra, Discrete math., 41, 215-218, (1982) · Zbl 0491.52004
[10] Engel, P., Geometric crystallography, (1986), Reidel Dordrecht · Zbl 0659.51001
[11] Grünbaum, B., Convex polytopes, (1967), Wiley New York · Zbl 0163.16603
[12] Grünbaum, B.; Sreedharan, V.P., An enumeration of simplicial 4-polytopes with 8 vertices, J. combin. theory, 2, 437-465, (1967) · Zbl 0156.43304
[13] Kleinschmidt, P., Sphären mit wenigen ecken, Geom. dedicata, 5, 307-320, (1976) · Zbl 0347.52001
[14] Mani, P., Spheres with few vertices, J. combin. theory ser. A, 13, 346-352, (1972) · Zbl 0248.52006
[15] M.A. Perles (results stated in Grünbaum [8, p. 424]).
[16] Schulz, C., Nicht-polytopale 3-sphären mit 8 ecken, Geom. dedicata, 13, 325-329, (1982) · Zbl 0516.52004
[17] Steinitz, E.; Rademacher, H., Vorlesung über die theorie der polyeder, (1934), Springer Berlin · Zbl 0325.52001
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