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多胞形 - Study the slicing of a hypercube.
- Read Cube Slices, Pictorial Triangles, and Probability by Don Chakerian and Dave Logothetti.
- How does it relate to the slicing of a cross-polytope?
- Relate it to the other three families of polytopes.
- Infinite Series: Dissecting Hypercubes with Pascal's Triangle
- Relate the Vandermonde determinant to the q-analogue of simplices.
- Relate Schur functions and simplices by way of the Vandermonde determinant.
- Relate polytopes and convexity.
Consider polytopes - does the relations between the components give an algebra of perspectives? - In three dimensions,
- the 3-simplex - a triangle (1) is mapped to a point (0), yielding 1+3+0=4 triangles
- the octahedron - a triangle (1) is mapped to an upside-down triangle (1), yielding 1+3+3+1 triangles, the large, 1 edge from large and 1 vertex from small, 1 vertex from large and 1 edge from small, the small
- the icosahedron - a triangle (1) is related to an upside-down triangle (1) by way of 20=1+3+6+6+3+1 triangles
- In four dimensions, the tetrahedron, as a building block, yields
- the 4-simplex - a tetrahedron (1) is mapped to a point (0), yielding 1+4+0=5 tetrahedra, the original and one for each face
- the 4-orthoplex - a tetrahedron (1) is mapped to an upside-down tetrahedron, yielding 1+4+6+4+1=16 tetrahedra, the large, 1 face from large and 1 vertex from small, 1 edge from large and 1 from small, 1 vertex from large and 1 face from small, and the small tetrahedra.
- the 600-cell - a tetrahedron is mapped to an icosahedron?
- the 24-cell maps an octahedron to an octahedron
- the 8-cell tessarect maps a cube to a cube 8=1+6+1
Readings - The g-conjecture about an h-vectors of a simplicial polytope and the triangulation of a sphere.
Federico Ardila Lecture 1: Course is a combinatorial focus on convex polytopes and hyperplane arrangements. - Euler's theorem: 1 - v + e - f + 1 = 0
- Steinitz's theorem: A 3-polytope exists iff 1 - v + e - f + 1 = 0, v <= 2f - 4, f <= 2v - 4.
- Regular polytopes.
Lecture 2: Definition of polytope P as convex hull of vertices. Sum of lambda x vertex where lambdas are nonnnegative and sum to 1. Lecture 3: Intersections and products of polytopes are also polytopes. - Main theorem of polytopes: Polytopes = convex hulls of finitely many points = bounded intersections of halfspaces.
Lecture 4: V-polyhedron intersected with affine plane is a V-polyhedron. - Dual polytope consists of a where a<=1-x for all x in P.
- Caratheodory's theorem
Lecture 5 and 6: Farkas' lemma versions 1 to 4. Lecture 7: Faces of polytopes. Face of P in direction c is the set of all x in P where c-x is maximal. - Affine spaces.
- Dim(face) = Dim(Aff(face))
- f-vector gives the number of faces in each dimension
- f-poly the generated function where the lowest coefficient gives the number of vertices and the highest coefficient gives the volume
Lecture 8: Construction of faces - Building polytopes: pyramids (adding the center).
- Vertex figures (converting the vertices to faces).
- Face lattice
Lecture 9: Face lattice - P and Q are combinatorially isomorphic if their face lattices are isomorphic.
- Polar (Dual) polytopes. Dual = c in dual space where c x <=1 for all x in P.
- If 0 is in P, then P equals its dual's dual.
Lecture 10: - Dual faces.
- The face lattices of P and its dual are opposites.
- Simple and simplicial polytopes.
- P is simple iff its dual is simplicial.
Lecture 11: The cyclic polytope. Lecture 12: Graphs of polytopes Lecture 13: How good is linear programming? - Hirsch conjecture is false.
Lecture 14: Balinski's theorem: P is a d-polytope implies G(P) is d-connected. Lecture 15: If P is simple, then G(P) determines P combinatorially. Lecture 16: Complexes, subdivisions, triangulations. - Every P has a triangulation.
Lecture 17: Triangulation of d-cross-polytope. Lecture 18: Counting lattice points in polytopes. Lecture 19: Partition functions. Lecture 20: Generating functions for cones. |

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