shape rotator

Generate polyhedra using Conway polyhedron notation. Implemented in Prolog.

(Each row in the picture was produced by applying the same operation(s) to all five platonic solids. You can see some relationships between the platonic solids, e.g. aaT = aC, dI = D.)

  T   C   O   I   D
 tT  tC  tO  tI  tD
 aT  aC  aO  aI  aD
aaT aaC aaO aaI aaD
 dT  dC  dO  dI  dD

Dependencies

The program requires SWI Prolog, which available here (or with nix, do nix develop).

Usage

swipl main.pl

gen_shape("your string here").

Some examples:

• gen_shape("eD").
• gen_shape("dkC").
• gen_shape("aI").

Your string should end with one of the supported seed shapes (see below), and can have operators to the left of the seed which are applied from right to left.

Seed shapes (uppercase):

• T (tetrahedron)
• C (cube)
• O (octahedron)
• D (dodecahedron)
• I (icosahedron)

Operations:

• a (ambo)
• d (dual)
• g (gyro) [not yet implemented]
• k (kis)
• j (join)
• s (snub) [not yet implemented]
• t (truncate)
• b (bevel)
• e (expand)
• m (meta)
• o (ortho)
• n (needle)
• z (zip)

Viewing STL output

Find output files in ./out. If you have macOS, you can rotate the generated 3D shapes in Preview.app! Viewers are available online, too.

Internals

• Until it’s time to render them, polyhedra are represented as follows:

  ap([face([edge(point(1,2,3), point(4,5,6))
            edge(point(1,2,3), point(9,8,7))
            edge(point(4,5,6), point(9,8,7)) ...])
      face([...])
      face([...])
      face([...]) ...])
      

  The order of face~s within the ~ap does not matter, nor does the order of the edges within the face.

  The order of point~s within the ~edge(A, B) does matter (A @=< B) for easier edge comparison.

• For STL output, each face is broken into triangles.
• Several assumptions are made about the properties of the polyhedron. If you manage to violate them (which is not difficult at the moment), you’ll probably get some fun visual bugs (or a div/0 error.)
  • Every face(edge1, edge2, edge3, ...) lies flat in some plane.
  • Every 2d and 3d shape is always convex (and consequently, the midpoint of any face is inside the face)
  • No edge or face would intersect the origin when extended. I think we even assume that the origin always lies inside the 3d shape.

Features to add

• gyro operator and friends
• Improve kis, which naively puts a constant-height pyramid on each face
• reflect (easy)
• number of vertices, edges, faces
• symmetry stats, groups
• SVG output (project to 2d)
• Comparison with known polyhedra (graph isomorphism)
• Better normalization

Resources

• wikipedia: https://en.wikipedia.org/wiki/Conway_polyhedron_notation
• operators in simple english: https://www.georgehart.com/virtual-polyhedra/conway_notation.html
• 3d views of a bunch of named polyhedra: http://dmccooey.com/polyhedra/index.html
• online visualizer that already does a pretty excellent job: https://levskaya.github.io/polyhedronisme/
• Projection onto the plane: https://en.wikipedia.org/wiki/Orthographic_projection
• Polyhedral graph: https://en.wikipedia.org/wiki/Polyhedral_graph
  • Do the numbers of faces, edges, and vertices of a convex polyhedron uniquely specify its graph? I would think no, but I can’t find the answer. So probably no.
• List of all Johnson solids (for later later): https://en.wikipedia.org/wiki/Johnson_solid