# 1-skeletons of Platonic solids¶

The methods defined here appear in `sage.graphs.graph_generators`.

sage.graphs.generators.platonic_solids.DodecahedralGraph()

Return a Dodecahedral graph (with 20 nodes)

The dodecahedral graph is cubic symmetric, so the spring-layout algorithm will be very effective for display. It is dual to the icosahedral graph.

PLOTTING: The Dodecahedral graph should be viewed in 3 dimensions. We choose to use a planar embedding of the graph. We hope to add rotatable, 3-dimensional viewing in the future. In such a case, a argument will be added to select the desired layout.

EXAMPLES:

Construct and show a Dodecahedral graph:

```sage: g = graphs.DodecahedralGraph()
sage: g.show()  # long time
```

Create several dodecahedral graphs in a Sage graphics array They will be drawn differently due to the use of the spring-layout algorithm:

```sage: g = []
sage: j = []
sage: for i in range(9):
....:     k = graphs.DodecahedralGraph()
....:     g.append(k)
sage: for i in range(3):
....:     n = []
....:     for m in range(3):
....:         n.append(g[3*i + m].plot(vertex_size=50, vertex_labels=False))
....:     j.append(n)
sage: G = graphics_array(j)
sage: G.show()  # long time
```
sage.graphs.generators.platonic_solids.HexahedralGraph()

Return a hexahedral graph (with 8 nodes).

A regular hexahedron is a 6-sided cube. The hexahedral graph corresponds to the connectivity of the vertices of the hexahedron. This graph is equivalent to a 3-cube.

PLOTTING: The Hexahedral graph should be viewed in 3 dimensions. We choose to use a planar embedding of the graph. We hope to add rotatable, 3-dimensional viewing in the future. In such a case, a argument will be added to select the desired layout.

EXAMPLES:

Construct and show a Hexahedral graph:

```sage: g = graphs.HexahedralGraph()
sage: g.show()  # long time
```

Create several hexahedral graphs in a Sage graphics array. They will be drawn differently due to the use of the spring-layout algorithm:

```sage: g = []
sage: j = []
sage: for i in range(9):
....:     k = graphs.HexahedralGraph()
....:     g.append(k)
sage: for i in range(3):
....:     n = []
....:     for m in range(3):
....:         n.append(g[3*i + m].plot(vertex_size=50, vertex_labels=False))
....:     j.append(n)
sage: G = graphics_array(j)
sage: G.show()  # long time
```
sage.graphs.generators.platonic_solids.IcosahedralGraph()

Return an Icosahedral graph (with 12 nodes).

The regular icosahedron is a 20-sided triangular polyhedron. The icosahedral graph corresponds to the connectivity of the vertices of the icosahedron. It is dual to the dodecahedral graph. The icosahedron is symmetric, so the spring-layout algorithm will be very effective for display.

PLOTTING: The Icosahedral graph should be viewed in 3 dimensions. We choose to use a planar embedding of the graph. We hope to add rotatable, 3-dimensional viewing in the future. In such a case, a argument will be added to select the desired layout.

EXAMPLES:

Construct and show an Octahedral graph:

```sage: g = graphs.IcosahedralGraph()
sage: g.show()  # long time
```

Create several icosahedral graphs in a Sage graphics array. They will be drawn differently due to the use of the spring-layout algorithm:

```sage: g = []
sage: j = []
sage: for i in range(9):
....:     k = graphs.IcosahedralGraph()
....:     g.append(k)
sage: for i in range(3):
....:     n = []
....:     for m in range(3):
....:         n.append(g[3*i + m].plot(vertex_size=50, vertex_labels=False))
....:     j.append(n)
sage: G = graphics_array(j)
sage: G.show()  # long time
```
sage.graphs.generators.platonic_solids.OctahedralGraph()

Return an Octahedral graph (with 6 nodes).

The regular octahedron is an 8-sided polyhedron with triangular faces. The octahedral graph corresponds to the connectivity of the vertices of the octahedron. It is the line graph of the tetrahedral graph. The octahedral is symmetric, so the spring-layout algorithm will be very effective for display.

PLOTTING: The Octahedral graph should be viewed in 3 dimensions. We choose to use a planar embedding of the graph. We hope to add rotatable, 3-dimensional viewing in the future. In such a case, a argument will be added to select the desired layout.

EXAMPLES:

Construct and show an Octahedral graph:

```sage: g = graphs.OctahedralGraph()
sage: g.show()  # long time
```

Create several octahedral graphs in a Sage graphics array They will be drawn differently due to the use of the spring-layout algorithm:

```sage: g = []
sage: j = []
sage: for i in range(9):
....:     k = graphs.OctahedralGraph()
....:     g.append(k)
sage: for i in range(3):
....:     n = []
....:     for m in range(3):
....:         n.append(g[3*i + m].plot(vertex_size=50, vertex_labels=False))
....:     j.append(n)
sage: G = graphics_array(j)
sage: G.show()  # long time
```
sage.graphs.generators.platonic_solids.TetrahedralGraph()

Return a tetrahedral graph (with 4 nodes).

A tetrahedron is a 4-sided triangular pyramid. The tetrahedral graph corresponds to the connectivity of the vertices of the tetrahedron. This graph is equivalent to a wheel graph with 4 nodes and also a complete graph on four nodes. (See examples below).

PLOTTING: The Tetrahedral graph should be viewed in 3 dimensions. We choose to use a planar embedding of the graph. We hope to add rotatable, 3-dimensional viewing in the future. In such a case, a argument will be added to select the desired layout.

EXAMPLES:

Construct and show a Tetrahedral graph:

```sage: g = graphs.TetrahedralGraph()
sage: g.show()  # long time
```

The following example requires networkx:

```sage: import networkx as NX
```

Compare this Tetrahedral, Wheel(4), Complete(4), and the Tetrahedral plotted with the spring-layout algorithm below in a Sage graphics array:

```sage: tetra_pos = graphs.TetrahedralGraph()
sage: tetra_spring = Graph(NX.tetrahedral_graph())
sage: wheel = graphs.WheelGraph(4)
sage: complete = graphs.CompleteGraph(4)
sage: g = [tetra_pos, tetra_spring, wheel, complete]
sage: j = []
sage: for i in range(2):
....:     n = []
....:     for m in range(2):
....:         n.append(g[i + m].plot(vertex_size=50, vertex_labels=False))
....:     j.append(n)
sage: G = graphics_array(j)
sage: G.show()  # long time
```