Chiral extensions of chiral polytopes

polytope Abstract polytope November, 2013 – p. 2 Abstract polytope Abstract polytope −→ combinatorial generalization of convex polytopepolytope Abstract polytope −→ combinatorial generalization of convex polytope November, 2013 – p. 2 Abstract polytope Abstract polytope −→ combinatorial generalization of convex polytopepolytope Abstract polytope −→ combinatorial generalization of convex polytope November, 2013 – p. 2 Abstract polytope Abstract polytope −→ combinatorial generalization of convex polytopepolytope Abstract polytope −→ combinatorial generalization of convex polytope November, 2013 – p. 2 Abstract polytope Abstract polytope −→ combinatorial generalization of convex polytopepolytope Abstract polytope −→ combinatorial generalization of convex polytope November, 2013 – p. 2 Abstract polytope Abstract polytope −→ combinatorial generalization of convex polytopepolytope Abstract polytope −→ combinatorial generalization of convex polytope November, 2013 – p. 2 Abstract polytope Abstract polytope −→ combinatorial generalization of convex polytopepolytope Abstract polytope −→ combinatorial generalization of convex polytope November, 2013 – p. 2 Abstract polytope Abstract polytopepolytope Abstract polytope November, 2013 – p. 3 Abstract polytope Abstract polytope POSETpolytope Abstract polytope POSET November, 2013 – p. 3 Abstract polytope Abstract polytope POSET Unique maximal and minimal elementspolytope Abstract polytope POSET Unique maximal and minimal elements November, 2013 – p. 3 Abstract polytope Abstract polytope POSET Unique maximal and minimal elements Rank functionpolytope Abstract polytope POSET Unique maximal and minimal elements Rank function November, 2013 – p. 3 Abstract polytope Abstract polytope POSET Unique maximal and minimal elements Rank function Strongly flag-connected November, 2013 – p. 3polytope Abstract polytope POSET Unique maximal and minimal elements Rank function Strongly flag-connected November, 2013 – p. 3 Abstract polytope Abstract polytope POSET Unique maximal and minimal elements Rank function Strongly flag-connected Diamond condition November, 2013 – p. 3polytope Abstract polytope POSET Unique maximal and minimal elements Rank function Strongly flag-connected Diamond condition November, 2013 – p. 3 Abstract polytope Between an (i+ 1)-face and an (i− 1)-face there are precisely two i-facespolytope Between an (i+ 1)-face and an (i− 1)-face there are precisely two i-faces November, 2013 – p. 4 Abstract polytope Between an (i+ 1)-face and an (i− 1)-face there are precisely two i-faces Every edge (1-face) contains precisely two vertices (0-faces)polytope Between an (i+ 1)-face and an (i− 1)-face there are precisely two i-faces Every edge (1-face) contains precisely two vertices (0-faces) November, 2013 – p. 4 Abstract polytope Between an (i+ 1)-face and an (i− 1)-face there are precisely two i-faces Every edge (1-face) contains precisely two vertices (0-faces) In a polygon (2-face), every vertex (0-face) belongs precisely to two edges (1-faces) November, 2013 – p. 4polytope Between an (i+ 1)-face and an (i− 1)-face there are precisely two i-faces Every edge (1-face) contains precisely two vertices (0-faces) In a polygon (2-face), every vertex (0-face) belongs precisely to two edges (1-faces) November, 2013 – p. 4 Abstract polytope Between an (i+ 1)-face and an (i− 1)-face there are precisely two i-faces Every edge (1-face) contains precisely two vertices (0-faces) In a polygon (2-face), every vertex (0-face) belongs precisely to two edges (1-faces) In a polyhedron (3-face), every edge (1-face) belongs precisely to two polygons (2-faces) November, 2013 – p. 4polytope Between an (i+ 1)-face and an (i− 1)-face there are precisely two i-faces Every edge (1-face) contains precisely two vertices (0-faces) In a polygon (2-face), every vertex (0-face) belongs precisely to two edges (1-faces) In a polyhedron (3-face), every edge (1-face) belongs precisely to two polygons (2-faces) November, 2013 – p. 4 Abstract polytope Between an (i+ 1)-face and an (i− 1)-face there are precisely two i-faces Every edge (1-face) contains precisely two vertices (0-faces) In a polygon (2-face), every vertex (0-face) belongs precisely to two edges (1-faces) In a polyhedron (3-face), every edge (1-face) belongs precisely to two polygons (2-faces) etc. November, 2013 – p. 4polytope Between an (i+ 1)-face and an (i− 1)-face there are precisely two i-faces Every edge (1-face) contains precisely two vertices (0-faces) In a polygon (2-face), every vertex (0-face) belongs precisely to two edges (1-faces) In a polyhedron (3-face), every edge (1-face) belongs precisely to two polygons (2-faces) etc. November, 2013 – p. 4