Sunday, February 18, 2018

[pkobcsan] Polyhedron facts

Cube: 6 faces, 8 vertices, 12 edges, 24 face-vertex tuples, 24 face-edge tuples, 24 edge-vertex tuples.  98 items total, maybe good for a deck of cards for a game.

Same for octahedron.

Tetrahedron: 4 4 6, 12 12 12 = 50 items.

Dodecahedron / icosahedron: 12 20 30 60 60 60 = 242 items.

Vaguely an extension of half-edge data structures for computer graphics.

For a general polyhedron, the cards could include additional information in the style of Pokemon cards.  Faces, edges, vertices get names.  Face: fraction of total surface area, number of sides, number of holes, fraction of the interior volume closer to this face than to any other face.  Edge: length (in what units?), dihedral angle, adjacent faces, adjacent vertices, fraction of close interior volume.  Vertex: degree, spherical angle of interior, fraction of close interior volume.  Face-Edge: which vertex is to the left and which is to the right when standing on the edge facing into the face, fraction of the area of the face closer to this edge than any other edge, fraction of the perimeter of the face.  Face-Vertex: angle, which edges are to the left and to the right when standing on the vertex facing into the face, fraction of the area of the face closer to this vertex than any other vertex.  Edge-Vertex: which face is to the left and to the right when traveling along that edge to the vertex.  Edge-edge: angle at the shared vertex (this can be an internal angle for octahedron and icosahedron, generally any vertex with more than 3 edges).

Edge length could be scaled relative to the longest edge, total length of all edges, maximum distance between two vertices (diameter).

In general, a card contains a fixed amount of information, so we do not for example list all the edges of a face because that is unbounded in number.

We could also define the "angle" of an edge on a polygonal face by imagining the face at the center of a very large coplanar circle, and computing what part of that circle is closer to some point on that edge than to any other edge on the face.  For nonconvex polygons, this angle could be zero.  This probably already has a name.  It seems related to Voronoi.  Similarly, spherical angle of a face or vertex.  Maybe also angle of an edge visible from an infinite sphere.

Or, again consider the plane of a face.  Bisect the angles of the vertices adjacent to an edge of a face: give the angle between the two lines, negative if they intersect on the exterior of the polygon due to concavity.  Call this local convexity.

We could also concretely define a center of a face or polyhedron as the centroid and give angles, areas, volumes, distances.  These values might be negative for faces with holes, concave shapes.

Previously, on adding edge cards to a deck representing the vertices of the Hoffman-Singleton graph.

In 4D each face touches 2 cells.  Edges bound an arbitrary number of faces so we would no longer list all of them.  We probably need to consider 3-tuples instead of (in addition to?) 2.  Use ideas from abstract polytopes to organize things.

No comments :