added handout

handouts/eg-polyhedra.m2

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+-- 28 Feb 2019, Math 7670 Introduction to Toric Varieties
+-- Mike Stillman
+
+-- Let's compute some with polyhedra in M2:
+restart
+needsPackage "Polyhedra"
+-- Some canned polyhedra:
+P = cyclicPolytope(3,5)
+  vertices P
+  halfspaces P
+  facets P
+  isCompact P
+  isFullDimensional P
+  fVector P
+  Sigma = normalFan P
+  rays Sigma
+  dim Sigma
+  isComplete Sigma
+  isSimplicial Sigma
+  faces(0,Sigma)
+  faces(1,Sigma)
+  faces(2,Sigma)
+  Sigma2 = faceFan P
+
+needsPackage "NormalToricVarieties"  
+  X = normalToricVariety Sigma
+  dim X
+  isSmooth X
+  isSimplicial X
+  rays X
+  max X
+
+needsPackage "ReflexivePolytopesDB"
+  str  = getKreuzerSkarke(5, Limit=>1)
+  tope = first parseKS str
+  A = matrixFromString tope#1
+  P = convexHull A
+  vertices P
+  fVector P
+  isReflexive P
+  isFullDimensional P
+  isSimplicial P  
+
+  -- make a polyhedron from inequalities:
+  (A',b) = halfspaces P
+  P1 = polyhedronFromHData(A', b)
+  P1 == P
+
+  -- polar
+  P' = polar P
+  vertices P'
+  fVector P'
+  isCompact P'
+
+  -- face fan and normal fan
+  Sigma = normalFan P
+  Sigma2 = faceFan polar P
+  Sigma == Sigma2
+  fVector Sigma
+  fVector P
+  fVector polar P
+
+  -- a 3D example
+  topes = parseKS getKreuzerSkarkeDim3();
+  #topes
+  topes#100
+  P = convexHull matrixFromString(last oo)
+  fVector P
+  vertices P
+  Sigma = normalFan P
+  dim Sigma
+  rays Sigma
+  faces(0,Sigma)
+  faces(1,Sigma)
+  faces(2,Sigma)
+  faces(3,Sigma)
+  faceFan P
+