The calling sequences Faces(polyset) and Faces(polyset, dimension = d) return a list of polyhedral sets that are $d$-faces of polyset.  Faces(polyset) uses a default value of dimension = Dimension(polyset) - 1, returning the facets of polyset.  If there are no faces of dimension d (e.g. asking for vertices of a half-space), an empty list is returned.

The PolyhedralSets[Vertices] (or, PolyhedralSets[Vertexes]) command is shorthand for Faces(polyset, dimension = 0).  Similarly, PolyhedralSets[Edges] is shorthand for Faces(polyset, dimension = 1), and PolyhedralSets[Facets] is shorthand for Faces(polyset).

A particular face can be retrieved via Faces(polyset, faceid = id).  The identification number id corresponds to those displayed on the graph returned by PolyhedralSets[Graph].

The ID number of a given set can alternatively by obtained with the ID command.  This returns an integer that identifies a set relative to its faces.  Two unrelated polyhedral sets can have the same ID number, but the faces of a given polyhedral set will always have unique ID numbers.

$\mathrm{with}\left(\mathrm{PolyhedralSets}\right)\:$

$t≔\mathrm{ExampleSets}:-\mathrm{Tetrahedron}\left(\right)\:$$\mathrm{t\_faces}≔\mathrm{Faces}\left(t\right)$

$\mathrm{t\_faces}≔\left[\{\begin{array}{lll}\mathrm{Coordinates}& \:& \left[x_1\,x_2\,x_3\right]\\ \mathrm{Relations}& \:& \left[-x_3\le 1\,-x_2+x_3\le 0\,x_2\le 1\,x_1+x_2-x_3=1\right]\end{array}\,\{\begin{array}{lll}\mathrm{Coordinates}& \:& \left[x_1\,x_2\,x_3\right]\\ \mathrm{Relations}& \:& \left[x_3\le 1\,-x_2\le 1\,x_2-x_3\le 0\,x_1-x_2+x_3=1\right]\end{array}\,\{\begin{array}{lll}\mathrm{Coordinates}& \:& \left[x_1\,x_2\,x_3\right]\\ \mathrm{Relations}& \:& \left[x_3\le 1\,-x_2-x_3\le 0\,x_2\le 1\,-x_2-x_3+x_1=\mathrm{-1}\right]\end{array}\,\{\begin{array}{lll}\mathrm{Coordinates}& \:& \left[x_1\,x_2\,x_3\right]\\ \mathrm{Relations}& \:& \left[-x_3\le 1\,-x_2\le 1\,x_2+x_3\le 0\,x_2+x_3+x_1=\mathrm{-1}\right]\end{array}\right]$

$\mathrm{Plot}\left(\mathrm{t\_faces}\right)$

$\mathrm{s5}≔\mathrm{ExampleSets}:-\mathrm{Simplex}\left(5\right)$

$\mathrm{s5}≔\{\begin{array}{lll}\mathrm{Coordinates}& \:& \left[x_1\,x_2\,x_3\,x_4\,x_5\right]\\ \mathrm{Relations}& \:& \left[-x_5\le 0\,-x_4\le 0\,-x_3\le 0\,-x_2\le 0\,-x_1\le 0\,x_1+x_2+x_3+x_4+x_5\le 1\right]\end{array}$

$\mathrm{s5\_edges}≔\mathrm{Faces}\left(\mathrm{s5}\,\mathrm{dimension}=1\right)$

$\mathrm{s5\_edges}≔\left[\{\begin{array}{lll}\mathrm{Coordinates}& \:& \left[x_1\,x_2\,x_3\,x_4\,x_5\right]\\ \mathrm{Relations}& \:& \left[-x_5\le 0\,x_5\le 1\,x_4+x_5=1\,x_3=0\,x_2=0\,x_1=0\right]\end{array}\,\{\begin{array}{lll}\mathrm{Coordinates}& \:& \left[x_1\,x_2\,x_3\,x_4\,x_5\right]\\ \mathrm{Relations}& \:& \left[-x_5\le 0\,x_5\le 1\,x_4=0\,x_3+x_5=1\,x_2=0\,x_1=0\right]\end{array}\,\{\begin{array}{lll}\mathrm{Coordinates}& \:& \left[x_1\,x_2\,x_3\,x_4\,x_5\right]\\ \mathrm{Relations}& \:& \left[x_5=0\,-x_4\le 0\,x_4\le 1\,x_3+x_4=1\,x_2=0\,x_1=0\right]\end{array}\,\{\begin{array}{lll}\mathrm{Coordinates}& \:& \left[x_1\,x_2\,x_3\,x_4\,x_5\right]\\ \mathrm{Relations}& \:& \left[-x_5\le 0\,x_5\le 1\,x_4=0\,x_3=0\,x_2+x_5=1\,x_1=0\right]\end{array}\,\{\begin{array}{lll}\mathrm{Coordinates}& \:& \left[x_1\,x_2\,x_3\,x_4\,x_5\right]\\ \mathrm{Relations}& \:& \left[x_5=0\,-x_4\le 0\,x_4\le 1\,x_3=0\,x_2+x_4=1\,x_1=0\right]\end{array}\,\{\begin{array}{lll}\mathrm{Coordinates}& \:& \left[x_1\,x_2\,x_3\,x_4\,x_5\right]\\ \mathrm{Relations}& \:& \left[x_5=0\,x_4=0\,-x_3\le 0\,x_3\le 1\,x_2+x_3=1\,x_1=0\right]\end{array}\,\{\begin{array}{lll}\mathrm{Coordinates}& \:& \left[x_1\,x_2\,x_3\,x_4\,x_5\right]\\ \mathrm{Relations}& \:& \left[-x_5\le 0\,x_5\le 1\,x_4=0\,x_3=0\,x_2=0\,x_1+x_5=1\right]\end{array}\,\{\begin{array}{lll}\mathrm{Coordinates}& \:& \left[x_1\,x_2\,x_3\,x_4\,x_5\right]\\ \mathrm{Relations}& \:& \left[x_5=0\,-x_4\le 0\,x_4\le 1\,x_3=0\,x_2=0\,x_1+x_4=1\right]\end{array}\,\{\begin{array}{lll}\mathrm{Coordinates}& \:& \left[x_1\,x_2\,x_3\,x_4\,x_5\right]\\ \mathrm{Relations}& \:& \left[x_5=0\,x_4=0\,-x_3\le 0\,x_3\le 1\,x_2=0\,x_1+x_3=1\right]\end{array}\,\{\begin{array}{lll}\mathrm{Coordinates}& \:& \left[x_1\,x_2\,x_3\,x_4\,x_5\right]\\ \mathrm{Relations}& \:& \left[x_5=0\,x_4=0\,x_3=0\,-x_2\le 0\,x_2\le 1\,x_1+x_2=1\right]\end{array}\,\{\begin{array}{lll}\mathrm{Coordinates}& \:& \left[x_1\,x_2\,x_3\,x_4\,x_5\right]\\ \mathrm{Relations}& \:& \left[-x_5\le 0\,x_5\le 1\,x_4=0\,x_3=0\,x_2=0\,x_1=0\right]\end{array}\,\{\begin{array}{lll}\mathrm{Coordinates}& \:& \left[x_1\,x_2\,x_3\,x_4\,x_5\right]\\ \mathrm{Relations}& \:& \left[x_5=0\,-x_4\le 0\,x_4\le 1\,x_3=0\,x_2=0\,x_1=0\right]\end{array}\,\{\begin{array}{lll}\mathrm{Coordinates}& \:& \left[x_1\,x_2\,x_3\,x_4\,x_5\right]\\ \mathrm{Relations}& \:& \left[x_5=0\,x_4=0\,-x_3\le 0\,x_3\le 1\,x_2=0\,x_1=0\right]\end{array}\,\{\begin{array}{lll}\mathrm{Coordinates}& \:& \left[x_1\,x_2\,x_3\,x_4\,x_5\right]\\ \mathrm{Relations}& \:& \left[x_5=0\,x_4=0\,x_3=0\,-x_2\le 0\,x_2\le 1\,x_1=0\right]\end{array}\,\{\begin{array}{lll}\mathrm{Coordinates}& \:& \left[x_1\,x_2\,x_3\,x_4\,x_5\right]\\ \mathrm{Relations}& \:& \left[x_5=0\,x_4=0\,x_3=0\,x_2=0\,-x_1\le 0\,x_1\le 1\right]\end{array}\right]$

$\mathrm{p1}≔\mathrm{PolyhedralSet}\left(\left[3\le x\,10\le y+x\,x\le 10\right]\,\left[x\,y\,z\right]\right)\;$$\mathrm{p2}≔\mathrm{PolyhedralSet}\left(\left[y\le 5\,3\le y+x\,x\le 7\right]\,\left[x\,y\,z\right]\right)\;$$\mathrm{ID}\left(\mathrm{p1}\right)\;$$\mathrm{ID}\left(\mathrm{p2}\right)$

$\mathrm{p1}≔\{\begin{array}{lll}\mathrm{Coordinates}& \:& \left[x\,y\,z\right]\\ \mathrm{Relations}& \:& \left[-y-x\le \mathrm{-10}\,-x\le \mathrm{-3}\,x\le 10\right]\end{array}$

$\mathrm{p2}≔\{\begin{array}{lll}\mathrm{Coordinates}& \:& \left[x\,y\,z\right]\\ \mathrm{Relations}& \:& \left[y\le 5\,-y-x\le \mathrm{-3}\,x\le 7\right]\end{array}$

$26$

$26$

$\mathrm{map}\left(\mathrm{ID}\,\mathrm{Faces}\left(\mathrm{p1}\right)\right)$

$\left[5\,11\,25\right]$

$\mathrm{map}\left(\mathrm{ID}\,\mathrm{Faces}\left(\mathrm{p2}\right)\right)$

$\left[17\,23\,25\right]$

The PolyhedralSets[Faces] and PolyhedralSets[ID] commands were introduced in Maple 2015.

For more information on Maple 2015 changes, see Updates in Maple 2015.