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# dim(Face) -- Compute the dimension of a face.

- Function: dim
- Usage:
    `dim(F)`
- Inputs:
  - F, a face,
- Outputs:
  - an integer, bigger or equal to -1

## Description

Computes the dimension of a face. If F.indices is present (usually the case by construction) this requires no computations.

If F.indices is not present but a polynomial ring R can be associated to F (which is the case if F.ofComplex is present (or given as a second argument) or F is non-empty) then R.grading (which can be installed by addCokerGrading) is used to compute the dimension of the plane spanned by F.

    i1 : R=QQ[x_0..x_4]

    o1 = R

    o1 : PolynomialRing

    i2 : addCokerGrading R

    o2 = | -1 -1 -1 -1 |
         | 1  0  0  0  |
         | 0  1  0  0  |
         | 0  0  1  0  |
         | 0  0  0  1  |

                  5       4
    o2 : Matrix ZZ  <-- ZZ

    i3 : C=simplex R

    o3 = 4: x x x x x
             0 1 2 3 4

    o3 : complex of dim 4 embedded in dim 4 (printing facets)
         equidimensional, simplicial, F-vector {1, 5, 10, 10, 5, 1}, Euler = 0

    i4 : bC=boundaryOfPolytope C

    o4 = 3: x x x x  x x x x  x x x x  x x x x  x x x x
             0 1 2 3  0 1 2 4  0 1 3 4  0 2 3 4  1 2 3 4

    o4 : complex of dim 3 embedded in dim 4 (printing facets)
         equidimensional, simplicial, F-vector {1, 5, 10, 10, 5, 0}, Euler = -1

    i5 : F=bC.fc_2_0

    o5 = x x x
          0 1 2

    o5 : face with 3 vertices

    i6 : dim(face vert F,R)

    o6 = 2

## Caveat

If F.indices is not present this returns a dimension as explained above but note that this does not check whether F is a face of the convex hull of the rows of R.grading.

## See also

- Complex -- The class of all embedded complexes.
- CoComplex -- The class of all embedded co-complexes.

## Ways to use this method:

- dim(Face) -- Compute the dimension of a face.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/SRdeformations.m2:5308:0.