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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.

##### MSC:
 52B11 $$n$$-dimensional polytopes 05A15 Exact enumeration problems, generating functions
##### Keywords:
3-sphere; $$n$$-polytope; enumeration; algorithm
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##### References:
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