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Two “simple” 3-spheres
Discrete Mathematics 67 (1987) 97-99 North-Holland
97
COMMUNICATION
T W O "SIMPLE" 3 - S P H E R E S D.W. BARNETTE Department of Mathematics, University of California, Davis, CA 95616, U.S.A. Received 31 March 1987 Revised 3 April 1987
Communicated by Branko Griinbaum
1. Introduction By a theorem of Steinitz, every 2-cell complex that is a 2-sphere is isomorphic to the boundary of a 3-dimensional convex polytope [5]. By modifying a proof of Steinitz's theorem, one can prove that given any combinatorial type of 3-polytope there is a representative of that type with one facet having a preassigned shape
[3]. The analogous theorems fail for 3-spheres [1, 2, 4]. In this paper we present two examples which show that these properties do not hold for 3-spheres. The significance of these examples is that they are very easy to visualize and the proofs of their properties are extremely simple.
2. Preassigning the shape of a facet Consider the 4-dimensional prism P whose bases are pyramids over quadrilaterals. Figure 1 shows a Schlegel diagram of P. Four of the five lateral facets of P are triangular prisms. In any such triangular prism three of the edges, such as el, e2 and e3 in Fig. 1, must belong to a pencil of lines (i.e., their attine hulls form a pencil of lines). Similarly el, e3, e4; el, e4, es; and el, e5, e2 must belong to pencils of lines. It follows that the four edges e2, e3, e4 and e5 of the cubical facet belong to a pencil of lines. Since there exist many combinatorial cubes in which there are no quadruples of edges belonging to pencils (e.g. a regular cube with 3 facets meeting at a vertex moved a small amount), the shape of the cubical facet cannot be preassigned arbitrarily. 0012-365X/87/$3.50 ~) 1987, Elsevier Science Publishers B.V. (North-Holland)
98
D. W. Barnette
e, I
ez
Fig. 1.
3. A nonpolyhedral sphere We begin with a simplex S, and to the edges el, e2, e3 and e4 of S (see Fig. 2) we glue the 2-complex C in Fig. 3. We do the gluing so that Int C c Int S,
X,
e,
e~
X3
Xz
Fig. 2.
bending C as necessary. (It is easily seen that there is no embedding of this kind if the 2-faces of S and C are planar polygons.) The complex C breaks up Int S into two 3-cells $1 and $2. We place a new vertex in $1 and take the join of the vertex
-
e,
Oi
Xl
X2.
Fig. 3.
t
Two "simple" 3-spheres
99
with each 2-cell on the boundary of $1. We place a vertex inside $2 and do the same. This produces our nonpolyhedral 3-sphere ~. Suppose the boundary complex of a convex 4-polytope P was isomorphic to ~. The affine hull of F1 O F2 is a 3-flat in E 4. This 3-flat is also the atiine hull of S because Aff S = .Aft {xl, x2, x3, x4}. The faces F1, F2 and S, however, cannot lie in the same 3-flat in E 4 since F1 and F2 lie on facets that are distinct from S.
References [1] D. Barnette, The triangulations of the 3-sphere with up to 8 vertices, J. Combin. Theory 14(1973) 37-52. [2] D. Barnette, Diagrams and Schlegel diagrams, in: R.K. Guy et al., eds., Combinatorial Structures and their Applications (Gordon and Breach, New York, 1970) 1-4. [3] D. Barnette and B. Griinbaum, On Steinitz's theorem concerning convex 3-polytopes and some properties of planar graphs, Lecture Notes in Math., Vol. 110 (Springer, Berlin, 1969) 27-40. [4] P. Kleinschmidt, On facets with non-arbitrary shapes, Pacific J. Math. 65 (1976) 97-101. [5] E. Steinitz and H. Rademacher, Vorlesungen fiber die Theorie der Polyeder (Springer, Berlin, 1934).
97
COMMUNICATION
T W O "SIMPLE" 3 - S P H E R E S D.W. BARNETTE Department of Mathematics, University of California, Davis, CA 95616, U.S.A. Received 31 March 1987 Revised 3 April 1987
Communicated by Branko Griinbaum
1. Introduction By a theorem of Steinitz, every 2-cell complex that is a 2-sphere is isomorphic to the boundary of a 3-dimensional convex polytope [5]. By modifying a proof of Steinitz's theorem, one can prove that given any combinatorial type of 3-polytope there is a representative of that type with one facet having a preassigned shape
[3]. The analogous theorems fail for 3-spheres [1, 2, 4]. In this paper we present two examples which show that these properties do not hold for 3-spheres. The significance of these examples is that they are very easy to visualize and the proofs of their properties are extremely simple.
2. Preassigning the shape of a facet Consider the 4-dimensional prism P whose bases are pyramids over quadrilaterals. Figure 1 shows a Schlegel diagram of P. Four of the five lateral facets of P are triangular prisms. In any such triangular prism three of the edges, such as el, e2 and e3 in Fig. 1, must belong to a pencil of lines (i.e., their attine hulls form a pencil of lines). Similarly el, e3, e4; el, e4, es; and el, e5, e2 must belong to pencils of lines. It follows that the four edges e2, e3, e4 and e5 of the cubical facet belong to a pencil of lines. Since there exist many combinatorial cubes in which there are no quadruples of edges belonging to pencils (e.g. a regular cube with 3 facets meeting at a vertex moved a small amount), the shape of the cubical facet cannot be preassigned arbitrarily. 0012-365X/87/$3.50 ~) 1987, Elsevier Science Publishers B.V. (North-Holland)
98
D. W. Barnette
e, I
ez
Fig. 1.
3. A nonpolyhedral sphere We begin with a simplex S, and to the edges el, e2, e3 and e4 of S (see Fig. 2) we glue the 2-complex C in Fig. 3. We do the gluing so that Int C c Int S,
X,
e,
e~
X3
Xz
Fig. 2.
bending C as necessary. (It is easily seen that there is no embedding of this kind if the 2-faces of S and C are planar polygons.) The complex C breaks up Int S into two 3-cells $1 and $2. We place a new vertex in $1 and take the join of the vertex
-
e,
Oi
Xl
X2.
Fig. 3.
t
Two "simple" 3-spheres
99
with each 2-cell on the boundary of $1. We place a vertex inside $2 and do the same. This produces our nonpolyhedral 3-sphere ~. Suppose the boundary complex of a convex 4-polytope P was isomorphic to ~. The affine hull of F1 O F2 is a 3-flat in E 4. This 3-flat is also the atiine hull of S because Aff S = .Aft {xl, x2, x3, x4}. The faces F1, F2 and S, however, cannot lie in the same 3-flat in E 4 since F1 and F2 lie on facets that are distinct from S.
References [1] D. Barnette, The triangulations of the 3-sphere with up to 8 vertices, J. Combin. Theory 14(1973) 37-52. [2] D. Barnette, Diagrams and Schlegel diagrams, in: R.K. Guy et al., eds., Combinatorial Structures and their Applications (Gordon and Breach, New York, 1970) 1-4. [3] D. Barnette and B. Griinbaum, On Steinitz's theorem concerning convex 3-polytopes and some properties of planar graphs, Lecture Notes in Math., Vol. 110 (Springer, Berlin, 1969) 27-40. [4] P. Kleinschmidt, On facets with non-arbitrary shapes, Pacific J. Math. 65 (1976) 97-101. [5] E. Steinitz and H. Rademacher, Vorlesungen fiber die Theorie der Polyeder (Springer, Berlin, 1934).