trac #18779: polytopes.gosset_3_21 and graphs.GossetGraph

src/sage/geometry/polyhedron/library.py

@@ -808,6 +808,31 @@ def six_hundred_cell(self, exact=False):
             verts.extend(p(x) for x in pts)
         return Polyhedron(vertices=verts, base_ring=base_ring)
 
+    def gosset_3_21(self):
+        r"""
+        Return the Gosset `3_{21}` polytope.
+
+        The Gosset `3_{21}` polytope is a uniform 7-polytope. It has 56
+        vertices, and 702 facets: `126` `3_{11}` and `576` `6`-simplex. For more
+        information, see the :wikipedia:`3_21_polytope`.
+
+        EXAMPLES::
+
+            sage: g = polytopes.gosset_3_21(); g
+            A 7-dimensional polyhedron in ZZ^8 defined as the convex hull of 56 vertices
+            sage: g.f_vector() # not tested (~16s)
+            (1, 120, 720, 1200, 600, 1)
+        """
+        from itertools import combinations
+        verts = []
+        for i,j in combinations(range(8),2):
+            x = [1]*8
+            x[i] = x[j] = -3
+            verts.append(x)
+            verts.append([-xx for xx in x])
+
+        return Polyhedron(vertices=verts, base_ring=ZZ)
+
     @rename_keyword(deprecation=18213, points_n='n', dim_n='dim')
     def cyclic_polytope(self, dim, n, base_ring=QQ):
         """

src/sage/graphs/generators/smallgraphs.py

@@ -1488,6 +1488,45 @@ def BuckyBall():
 
     return g
 
+def GossetGraph():
+    r"""
+    Return the Gosset graph.
+
+    The Gosset graph is the skeleton of the
+    :meth:`~sage.geometry.polyhedron.library.polytopes.gosset_3_21` polytope. It
+    has with 56 vertices and degree 27. For more information, see the
+    :wikipedia:`Gosset_graph`.
+
+    EXAMPLE::
+
+        sage: g = graphs.GossetGraph(); g
+        Gosset Graph: Graph on 56 vertices
+
+        sage: g.order(), g.size()
+        (56, 756)
+
+    TESTS::
+
+        sage: g.is_isomorphic(polytopes.gosset_3_21().graph()) # not tested (~16s)
+        True
+    """
+    string = ('w~~~~rt{~Z\\ZxnvYZYmlfrb}|hDuhLlcmmMNf_^zzQGNYcP\\kcRZbaJjoNBx{'+
+              '?N~o^}?A`}F_Kbbm_[QZ\\_]Cj\\oN_dm{BzB{?]WIMM@tPQRYBYRPIuAyJgQv?'+
+              '|Bxb_M[kWIR@jTQcciDjShXCkFMgpwqBKxeKoS`TYqdTCcKtkdKwWQXrbEZ@OdU'+
+              'mITZ@_e[{KXn?YPABzvY?IcO`zvYg@caC\\zlf?BaGR]zb{?@wOjv`~w??N_n_~'+
+              '~w???^_^~~{')
+
+    G = Graph(string,name="Gosset Graph")
+
+    ordering = [0, 2, 4, 6, 43, 23, 50, 18, 28, 9, 8, 7, 44, 3, 26, 35, 16, 14,
+                33, 15, 54, 30, 17, 21, 10, 13, 36, 31, 55, 53, 51, 49, 12, 32,
+                5, 37, 27, 46, 47, 48, 11, 52, 29, 20, 39, 41, 22, 40, 1, 25, 38,
+                34, 45, 42, 19, 24]
+
+    _circle_embedding(G,ordering)
+
+    return G
+
 def DoubleStarSnark():
     r"""
     Returns the double star snark.

src/sage/graphs/graph_generators.py

@@ -111,6 +111,7 @@ def __append_to_doc(methods):
      "FranklinGraph",
      "FruchtGraph",
      "GoldnerHararyGraph",
+     "GossetGraph",
      "GrayGraph",
      "GrotzschGraph",
      "HallJankoGraph",
@@ -1859,6 +1860,7 @@ def quadrangulations(self, order, minimum_degree=None, minimum_connectivity=None
     FranklinGraph            = staticmethod(sage.graphs.generators.smallgraphs.FranklinGraph)
     FruchtGraph              = staticmethod(sage.graphs.generators.smallgraphs.FruchtGraph)
     GoldnerHararyGraph       = staticmethod(sage.graphs.generators.smallgraphs.GoldnerHararyGraph)
+    GossetGraph              = staticmethod(sage.graphs.generators.smallgraphs.GossetGraph)
     GrayGraph                = staticmethod(sage.graphs.generators.smallgraphs.GrayGraph)
     GrotzschGraph            = staticmethod(sage.graphs.generators.smallgraphs.GrotzschGraph)
     HallJankoGraph           = staticmethod(sage.graphs.generators.smallgraphs.HallJankoGraph)