jmhbrown/sq280-polytope-graphs

Playing around with visible graphs and polytopes.

SQ280-polytope-graphs

Class project for Experimental Mathematics.

How to use this code.

Setup

This code is meant to be run with Sage. It uses the standard packages logging,os,collections, and multiprocessing, and is known to work on Ubuntu 16.04 with the following software versions:

• Sage 7.6
• Python 2.7

Besides having Sage installed, I know of no special setup process. If you have trouble getting this code to work, please let me know!

Project Structure

SQ280-polytope-graphs/
├── output		// For storing Sage objects
│   └── {sphere_N.sobj}			// A bunch of files - including some pre-generated triangulated spheres.
├── src			// Actual Code
│   ├── projection_vectors.sage		// Code for finding some fundamental vectors associated with a polyhedron.
│   ├── sphere.sage			// Code for generating approximations of spheres.
│   ├── utilities.sage			// Helper functions. Used in other parts of the codebase.
│   ├── visible_graphs.sage		// Code for finding visible graphs.
│   └── visualizations.sage		// Code for making (Tachyon) representations of Visible Graph equivalence classes.
└── workspace		// Experiments and junk code.

Common Tasks

Making graphs

Use get_visible_graph in visible_graphs.sage to generate graphs:

sage: get_visible_graph?
Signature:      get_visible_graph(polyhedron, projection_vector)
Docstring:     
   Finds the visible graph associated with a particular vector. INPUT:

      * "polyhedron" -- Polyhedron. Required. Polyhedron whose
        visible graph we want to find.

      * "projection_vector" -- Vector. Required. Vector to project
        along.

   OUTPUT:

      Immutable, canonical version of the visible graph.
Init docstring: x.__init__(...) initializes x; see help(type(x)) for signature
File:           Dynamically generated function. No source code available.
Type:           function

Visualizing Equivalence Classes

Use build_tach_repr in visualizations.sage to make the representations. Use show_tachyon_scene to change camera angle and resolution.

sage: build\_tach\_repr?
Signature:      build\_tach\_repr(polyhedron, **kwargs)
Docstring:     
   Builds a tachyon scene representing the visible graph regions.
   INPUT:

      * "polyhedron" -- Polyhedron. Required. Polyhedron to
        represent.

      * "multithreaded" -- bool. Optional. Whether to use
        multithreading.

      Sphere building options.

      * "seed_sphere" -- A Polyhedron. Optional. A triangulated
        sphere to use for graphics.

      * "build_sphere" -- bool. Optional. Defaults to false. Whether
        to build a sphere for graphics.

      * "n_iterations" -- Integer. Optional. Defaults to 3. Number
        of iterations of sphere subdivisions.

      Tachyon options.

      * "xres" -- Integer. Optional. (default: 800) X-direction
        resolution.

      * "yres" -- Integer. Optional. (default: 800) Y-direction
        resolution.

      * "camera_center" -- List,Tuple. Optional. (default: (2,5,2)
        Camera location.

   OUTPUT:

      Tachyon representation of the sphere, colored by visible graphs.

   EXAMPLES:

      sage: bucky_ball = polytopes.buckyball()
      sage: t = build_tach_repr(bucky_ball, build_sphere=True, sphere_subdivisions=3)
Init docstring: x.__init__(...) initializes x; see help(type(x)) for signature
File:           ~/.sage/temp/wrex/19557/visualizations.sagehA3Wk2.py
Type:           function

Generating Sphere Approximations

Either use the triangulated_sphere function from sphere.sage or pass the parameter build_sphere=True to build_tachyon_repr in visualizations.sage

sage: triangulated_sphere?
Signature:      triangulated_sphere(**kwargs)
Docstring:     
   Makes a triangulation of a unit sphere, by repeatedly dividing the
   faces of an octahedron. The final triangulation will have
   8*4^(subdivisions-1) = 2*4^(subdivision) sides.

   INPUT:

      * "seed_sphere" --  A Polyhedron. Optional. Defaults to an
        octahedron. Base polytope to apply iterations to.

      * "n_iterations" --  An Integer. Optional. Number of times to
        divide faces.

      * "multithreaded" -- Bool. Optional. Whether to run in
        multithreaded mode.

      Save Options:

      * "save_poly" --  Bool. Optional. Defaults to False. Whether
        to save the polytope

      * "save_filepath" --  String. Optional. Defaults to
        cwd+'/output'. Directory to save the polytope in.

      * "save_filename" --  String. Optional. Filename used for save
        file.

   OUTPUT:

      A polyhedron corresponding to a Triangulation of the sphere.
Init docstring: x.__init__(...) initializes x; see help(type(x)) for signature
File:           ~/.sage/temp/wrex/19557/sphere.sageRM9SZz.py
Type:           function

Known Issues.

• triangulated_sphere() throws errors when run in multithreaded mode with polytopes.octahedron() as the seed_sphere.
• Crashes for large triangulated spheres. I've never gotten it to complete a run with a 16385-vertex approximation of a sphere.
• No debug logs in multithreaded mode. This isn't a bug, but it is obnoxious.
• Suspected issues from floating point arithmetic errors when dealing with polytopes with orthogonal sides. (e.g. with cubes.)