# Visualization of polyhedron objects in Sage¶

Author: sarah-marie belcastro <smbelcas@toroidalsnark.net>, Jean-Philippe Labbé <labbe@math.fu-berlin.de>

There are different ways to visualize polyhedron object of dimension at most 4.

## render_solid¶

This plots the polyhedron as a solid. You can also adjust the opacity parameter.

sage: Cube = polytopes.cube()
sage: Cube.render_solid(opacity=0.7)
Graphics3d Object


## render_wireframe¶

This plots the graph (with unbounded edges) of the polyhedron

sage: Cube.render_wireframe()
Graphics3d Object


## plot¶

The plot method draws the graph, the polygons and vertices of the polyhedron all together.

sage: Cube.plot()
Graphics3d Object


## show¶

This is similar to plot but does not return an object that you can manipulate.

## schlegel_projection¶

It is possible to visualize 4-dimensional polytopes using a schlegel diagram.

sage: HC = polytopes.hypercube(4)
sage: HC.schlegel_projection()
The projection of a polyhedron into 3 dimensions
sage: HC.schlegel_projection().plot()
Graphics3d Object


We can see it from a different perspective by choosing point at a different distance:

sage: HC.schlegel_projection(position=1/4).plot()
Graphics3d Object


It is possible to choose from which facet one sees the projection:

sage: tHC = HC.face_truncation(HC.faces(0))
sage: tHC.facets()
(A 3-dimensional face of a Polyhedron in QQ^4 defined as the convex hull of 10 vertices,
A 3-dimensional face of a Polyhedron in QQ^4 defined as the convex hull of 10 vertices,
A 3-dimensional face of a Polyhedron in QQ^4 defined as the convex hull of 10 vertices,
A 3-dimensional face of a Polyhedron in QQ^4 defined as the convex hull of 10 vertices,
A 3-dimensional face of a Polyhedron in QQ^4 defined as the convex hull of 4 vertices,
A 3-dimensional face of a Polyhedron in QQ^4 defined as the convex hull of 8 vertices,
A 3-dimensional face of a Polyhedron in QQ^4 defined as the convex hull of 8 vertices,
A 3-dimensional face of a Polyhedron in QQ^4 defined as the convex hull of 8 vertices,
A 3-dimensional face of a Polyhedron in QQ^4 defined as the convex hull of 8 vertices)
sage: tHC.schlegel_projection(tHC.facets()).plot()
Graphics3d Object


## tikz¶

This method returns a tikz picture of the polytope (must be 2 or 3-dimensional). For more detail see the tutorial Draw polytopes in LaTeX using TikZ.

sage: c = polytopes.cube()
sage: c.tikz().splitlines()[:5]
['\\begin{tikzpicture}%',
'\t[x={(1.000000cm, 0.000000cm)},',
'\ty={(-0.000000cm, 1.000000cm)},',
'\tz={(0.000000cm, -0.000000cm)},',
'\tscale=1.000000,']