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Counting Faces of Polytopes Carl Lee University of Kentucky James Madison University|March 2015 Carl Lee (UK) Counting Faces of Polytopes James Madison University 1 / 36 Convex Polytopes A convex polytope P is the convex hull of a finite set of points in Rd . Example: Cube Carl Lee (UK) Counting Faces of Polytopes James Madison University 2 / 36 4-Cube. f = (16; 32; 24; 8). Question: What are the possible face-vectors of polytopes? Face-Vectors The face-vector of a d-dimensional polytope is f = (f0; f1;:::; fd−1), where fj is the number of faces of dimension j. Define also f−1 = fd = 1. Example: Cube. f = (8; 12; 6). Carl Lee (UK) Counting Faces of Polytopes James Madison University 3 / 36 Question: What are the possible face-vectors of polytopes? Face-Vectors The face-vector of a d-dimensional polytope is f = (f0; f1;:::; fd−1), where fj is the number of faces of dimension j. Define also f−1 = fd = 1. Example: Cube. f = (8; 12; 6). 4-Cube. f = (16; 32; 24; 8). Carl Lee (UK) Counting Faces of Polytopes James Madison University 3 / 36 Face-Vectors The face-vector of a d-dimensional polytope is f = (f0; f1;:::; fd−1), where fj is the number of faces of dimension j. Define also f−1 = fd = 1. Example: Cube. f = (8; 12; 6). 4-Cube. f = (16; 32; 24; 8). Question: What are the possible face-vectors of polytopes? Carl Lee (UK) Counting Faces of Polytopes James Madison University 3 / 36 Three-Dimensional Polytopes Theorem (Euler's Relation) f0 − f1 + f2 = 2 for convex 3-polytopes. Example: Cube. 8 − 12 + 6 = 2. Carl Lee (UK) Counting Faces of Polytopes James Madison University 4 / 36 Initially χ = 0. Bottom vertex. χ changes by 1 − 0 + 0 = 1. Intermediate vertex with k incident lower edges. χ changes by 1 − k + (k − 1) = 0. Top vertex. If its degree is k, then χ changes by 1 − k + k = 1. Total change in χ is therefore 2. Note: This proof technique generalizes to higher dimensions. Three-Dimensional Polytopes Sketch of proof: Sweep the polytope with a plane in general direction. (Think of immersing in water.) Count vertices, edges, and polygons only when fully swept (under water). Watch how χ = f0 − f1 + f2 changes when the plane hits each vertex. Carl Lee (UK) Counting Faces of Polytopes James Madison University 5 / 36 Bottom vertex. χ changes by 1 − 0 + 0 = 1. Intermediate vertex with k incident lower edges. χ changes by 1 − k + (k − 1) = 0. Top vertex. If its degree is k, then χ changes by 1 − k + k = 1. Total change in χ is therefore 2. Note: This proof technique generalizes to higher dimensions. Three-Dimensional Polytopes Sketch of proof: Sweep the polytope with a plane in general direction. (Think of immersing in water.) Count vertices, edges, and polygons only when fully swept (under water). Watch how χ = f0 − f1 + f2 changes when the plane hits each vertex. Initially χ = 0. Carl Lee (UK) Counting Faces of Polytopes James Madison University 5 / 36 Intermediate vertex with k incident lower edges. χ changes by 1 − k + (k − 1) = 0. Top vertex. If its degree is k, then χ changes by 1 − k + k = 1. Total change in χ is therefore 2. Note: This proof technique generalizes to higher dimensions. Three-Dimensional Polytopes Sketch of proof: Sweep the polytope with a plane in general direction. (Think of immersing in water.) Count vertices, edges, and polygons only when fully swept (under water). Watch how χ = f0 − f1 + f2 changes when the plane hits each vertex. Initially χ = 0. Bottom vertex. χ changes by 1 − 0 + 0 = 1. Carl Lee (UK) Counting Faces of Polytopes James Madison University 5 / 36 Top vertex. If its degree is k, then χ changes by 1 − k + k = 1. Total change in χ is therefore 2. Note: This proof technique generalizes to higher dimensions. Three-Dimensional Polytopes Sketch of proof: Sweep the polytope with a plane in general direction. (Think of immersing in water.) Count vertices, edges, and polygons only when fully swept (under water). Watch how χ = f0 − f1 + f2 changes when the plane hits each vertex. Initially χ = 0. Bottom vertex. χ changes by 1 − 0 + 0 = 1. Intermediate vertex with k incident lower edges. χ changes by 1 − k + (k − 1) = 0. Carl Lee (UK) Counting Faces of Polytopes James Madison University 5 / 36 Total change in χ is therefore 2. Note: This proof technique generalizes to higher dimensions. Three-Dimensional Polytopes Sketch of proof: Sweep the polytope with a plane in general direction. (Think of immersing in water.) Count vertices, edges, and polygons only when fully swept (under water). Watch how χ = f0 − f1 + f2 changes when the plane hits each vertex. Initially χ = 0. Bottom vertex. χ changes by 1 − 0 + 0 = 1. Intermediate vertex with k incident lower edges. χ changes by 1 − k + (k − 1) = 0. Top vertex. If its degree is k, then χ changes by 1 − k + k = 1. Carl Lee (UK) Counting Faces of Polytopes James Madison University 5 / 36 Note: This proof technique generalizes to higher dimensions. Three-Dimensional Polytopes Sketch of proof: Sweep the polytope with a plane in general direction. (Think of immersing in water.) Count vertices, edges, and polygons only when fully swept (under water). Watch how χ = f0 − f1 + f2 changes when the plane hits each vertex. Initially χ = 0. Bottom vertex. χ changes by 1 − 0 + 0 = 1. Intermediate vertex with k incident lower edges. χ changes by 1 − k + (k − 1) = 0. Top vertex. If its degree is k, then χ changes by 1 − k + k = 1. Total change in χ is therefore 2. Carl Lee (UK) Counting Faces of Polytopes James Madison University 5 / 36 Three-Dimensional Polytopes Sketch of proof: Sweep the polytope with a plane in general direction. (Think of immersing in water.) Count vertices, edges, and polygons only when fully swept (under water). Watch how χ = f0 − f1 + f2 changes when the plane hits each vertex. Initially χ = 0. Bottom vertex. χ changes by 1 − 0 + 0 = 1. Intermediate vertex with k incident lower edges. χ changes by 1 − k + (k − 1) = 0. Top vertex. If its degree is k, then χ changes by 1 − k + k = 1. Total change in χ is therefore 2. Note: This proof technique generalizes to higher dimensions. Carl Lee (UK) Counting Faces of Polytopes James Madison University 5 / 36 f0; f1; f2 are positive integers What else? Theorem (Steinitz) A positive integer vector (f0; f1; f2) is the face-vector of a 3-polytope if and only if the following conditions hold. f0 − f1 + f2 = 2, f0 ≤ 2f2 − 4, and f2 ≤ 2f0 − 4. Three-Dimensional Polytopes Other necessary conditions: Carl Lee (UK) Counting Faces of Polytopes James Madison University 6 / 36 What else? Theorem (Steinitz) A positive integer vector (f0; f1; f2) is the face-vector of a 3-polytope if and only if the following conditions hold. f0 − f1 + f2 = 2, f0 ≤ 2f2 − 4, and f2 ≤ 2f0 − 4. Three-Dimensional Polytopes Other necessary conditions: f0; f1; f2 are positive integers Carl Lee (UK) Counting Faces of Polytopes James Madison University 6 / 36 Theorem (Steinitz) A positive integer vector (f0; f1; f2) is the face-vector of a 3-polytope if and only if the following conditions hold. f0 − f1 + f2 = 2, f0 ≤ 2f2 − 4, and f2 ≤ 2f0 − 4. Three-Dimensional Polytopes Other necessary conditions: f0; f1; f2 are positive integers What else? Carl Lee (UK) Counting Faces of Polytopes James Madison University 6 / 36 Three-Dimensional Polytopes Other necessary conditions: f0; f1; f2 are positive integers What else? Theorem (Steinitz) A positive integer vector (f0; f1; f2) is the face-vector of a 3-polytope if and only if the following conditions hold. f0 − f1 + f2 = 2, f0 ≤ 2f2 − 4, and f2 ≤ 2f0 − 4. Carl Lee (UK) Counting Faces of Polytopes James Madison University 6 / 36 Three-Dimensional Polytopes Carl Lee (UK) Counting Faces of Polytopes James Madison University 7 / 36 We don't know! But there are some partial results. Four-Dimensional Polytopes What is the characterization of face-vectors of 4-polytopes? Carl Lee (UK) Counting Faces of Polytopes James Madison University 8 / 36 But there are some partial results. Four-Dimensional Polytopes What is the characterization of face-vectors of 4-polytopes? We don't know! Carl Lee (UK) Counting Faces of Polytopes James Madison University 8 / 36 Four-Dimensional Polytopes What is the characterization of face-vectors of 4-polytopes? We don't know! But there are some partial results. Carl Lee (UK) Counting Faces of Polytopes James Madison University 8 / 36 Early proofs (pre-Poincar´e)relied upon the implicit or unproven assumption of \shellability," not established until 1970 by Bruggesser and Mani. Gr¨unbaumdeveloped a \sweeping-like" proof. d-Dimensional Polytopes Theorem (Euler-Poincar´eFormula) For every d-polytope, d−1 X d fj = 1 − (−1) j=0 Carl Lee (UK) Counting Faces of Polytopes James Madison University 9 / 36 Gr¨unbaumdeveloped a \sweeping-like" proof. d-Dimensional Polytopes Theorem (Euler-Poincar´eFormula) For every d-polytope, d−1 X d fj = 1 − (−1) j=0 Early proofs (pre-Poincar´e)relied upon the implicit or unproven assumption of \shellability," not established until 1970 by Bruggesser and Mani. Carl Lee (UK) Counting Faces of Polytopes James Madison University 9 / 36 d-Dimensional Polytopes Theorem (Euler-Poincar´eFormula) For every d-polytope, d−1 X d fj = 1 − (−1) j=0 Early proofs (pre-Poincar´e)relied upon the implicit or unproven assumption of \shellability," not established until 1970 by Bruggesser and Mani.
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