add doc for #26077

src/sage/geometry/polyhedron/constructor.py

@@ -433,6 +433,19 @@ def Polyhedron(vertices=None, rays=None, lines=None,
          A vertex at (0, 31/2, 31/2, 0, 0, 0), A vertex at (0, 31/2, 0, 0, 31/2, 0),
          A vertex at (0, 0, 0, 31/2, 31/2, 0))
 
+    Regular icosahedron, centered at `0` with edge length `2`, with vertices given
+    by the cyclic shifts of `(0, \pm 1, \pm (1+\sqrt(5))/2)`, cf.
+    :wikipedia:`Regular_icosahedron`. It needs a number field::
+
+        sage: R0.<r0>=QQ[]
+        sage: R1.<r1>=NumberField(r0^2-5, embedding=AA(5)**(1/2))
+        sage: grat=(1+r1)/2
+        sage: v=[[0,1,grat],[0,1,-grat],[0,-1,grat],[0,-1,-grat]]
+        sage: pp=Permutation((1,2,3))
+        sage: icosah = Polyhedron(map((pp^2).action,v)+map(pp.action,v)+v,  base_ring=R1)
+        sage: len(icosah.faces(2))
+        20
+
     When the input contains elements of a Number Field, they require an
     embedding::
 
@@ -493,6 +506,10 @@ def Polyhedron(vertices=None, rays=None, lines=None,
         Traceback (most recent call last):
         ...
         ValueError: no appropriate backend for computations with Real Field with 53 bits of precision
+
+    .. SEEALSO::
+
+        :mod:`Library of polytopes <sage.geometry.polyhedron.library>`
     """
     # Clean up the arguments
     vertices = _make_listlist(vertices)