# faces -- Returns the faces of a complex

• Usage:
faces(C)
• Inputs:
• Outputs:

## Description

Compute the faces of a simplicial Complex C. The result is a List L. The j-th entry L#j of L is a list of the faces of C of dimension j.

If the second argument d is specified then a list of the faces of dimension d is returned.

 `i1 : K=QQ;` ```i2 : R=K[x_0..x_4] o2 = R o2 : PolynomialRing``` ```i3 : I=ideal(x_0*x_1,x_1*x_2,x_2*x_3,x_3*x_4,x_4*x_0) o3 = ideal (x x , x x , x x , x x , x x ) 0 1 1 2 2 3 3 4 0 4 o3 : Ideal of R``` ```i4 : C=idealToComplex I o4 = {x x , x x , x x , x x , x x } 2 4 0 3 0 2 1 3 1 4 o4 : complex with 5 facets on the vertices x x x x x 0 1 2 3 4``` ```i5 : faces(C) o5 = {{{}}, {x , x , x , x , x }, {x x , x x , x x , x x , x x 1 0 3 2 4 1 4 1 3 2 4 0 2 0 3 ------------------------------------------------------------------------ }, {}, {}, {}} o5 : List```

 `i6 : K=QQ;` ```i7 : R=K[x_0..x_4] o7 = R o7 : PolynomialRing``` ```i8 : I=ideal(x_0*x_1,x_2*x_3*x_4) o8 = ideal (x x , x x x ) 0 1 2 3 4 o8 : Ideal of R``` ```i9 : C=idealToComplex I o9 = {x x x , x x x , x x x , x x x , x x x , x x x } 0 2 3 1 2 3 1 3 4 1 2 4 0 3 4 0 2 4 o9 : complex with 6 facets on the vertices x x x x x 0 1 2 3 4``` ```i10 : faces(C) o10 = {{{}}, {x , x , x , x , x }, {x x , x x , x x , x x , x 4 1 2 0 3 0 4 2 4 2 3 3 4 1 ----------------------------------------------------------------------- x , x x , x x , x x , x x }, {x x x , x x x , x x x , 2 1 3 0 2 0 3 1 4 0 2 4 0 3 4 1 2 4 ----------------------------------------------------------------------- x x x , x x x , x x x }, {}, {}} 1 2 3 1 3 4 0 2 3 o10 : List``` ```i11 : faces(C,1) o11 = {x x , x x , x x , x x , x x , x x , x x , x x , x x 0 4 2 4 2 3 3 4 1 2 1 3 0 2 0 3 1 4 ----------------------------------------------------------------------- } o11 : List```