[crdxmcet] angular defect of regular polyhedra

degrees missing from being flat at vertex:

tetrahedron 180
octahedron 120
cube 90
icosahedron 60
dodecahedron 36

(angle defect)*(number of vertices) = 720 degree = 4*pi radian, a theorem of Descartes.

by this metric, dodecahedron is the most flat, the least confusing (at corners) for a map of a sphere.  (does that mean it has a lot of distortion inside each face?)  create a tool to display the earth on a dodecahedral net.  drag any point to anywhere on the net and reproject.  also need rotation.  move pentagons to choose among many possible nets.  what countries fit neatly inside adjacent pentagons?  I don't have a good feel of what is the area and extent of 1/12 or 2/12 of a sphere (1 or 2 pentagons).

move portions of pentagons?  the dissection of a pentagon by a pentagram (star) might be useful, as well as the dissection from the center into pie pieces.

vertex-transitive polyhedra have the same angle defect at every vertex.  the Archimedean solids are another (the other?) family of vertex-transitive polyhedra.  by the theorem of Descartes, the most flat therefore is the polyhedron with the most vertices.  excluding the prisms and antiprisms, the truncated icosidodecahedron has the most vertices with 120.  decagon, hexagon, and square meet at each of its vertices; angle defect is 6 degrees.  how much map distortion is in its 12 relatively large decagons?  (30 squares and 20 hexagons are its other faces.  62 faces total.)

previously, pillars of inaccessibility: the tile diagonally opposite you is blank, perhaps annotated with curved arrows indicating edges connect.  moving your immersed character locally induces a different net.  the faces of a net could discretely change adjacency, or walking around a vertex could cause the net to continuously reproject itself, keeping the inaccessible gap diagonally across from you.  not sure if the latter works.

future work: solid angular defect of 4D regular polytopes.