by wendy » Sun Dec 30, 2018 7:42 am
Every polytope can be constructed with a directed 'out-vector'. Also, the idea of 'grain' on faces works.
Out-vectors
If you start with a line, you can make it directed as ----> . You can make a polygon of these, with the lines going clockwise, so the outside is on the left, and the inside is on the right. Then this polygon points 'up'. If you take a set of clockwise polygons, you can make a 'clockwise' polyhedron, where all the faces are polyhedral. The arrows of the line go in different directions, and cancel each other out. But the faces would all point outwards, because they represent polygons of clockwise eddies.
This is the basis of the 'vector area' in Ampere's law (m = IA), and that any closed loop encloses a definite area. In effect, any closed volume has a sum of out-vectors of zero. If you remove any number of faces, the area formed by the break will be derived from the sum of the remaining faces.
Grain of faces
Colour-grain is an extension of colour, which is here regarded as zero-dimension. A plain cylinder could be coloured in a single colour. If you make it into a stack of disks, then the disks add a 2D 'grain' to it. (Grain here refers to the direction of growth in wood, which would be a linear colour). If you make the cylinder as a bundle of pencils, then the colour is 1d.
You can make a cube with its faces having a 1d grain, marked by stripes, such that they line up to the next face at right angles. This is the pyritohedral symmetry of a cube. It has mirrors parallel to the faces, and it has a order-3 rotation around its vertices.
Directed edges
If you label opposite vertices of the octahedron as 1, 2, 3, then it is possible to direct the edges with arrows pointing from 1->2, 2->3, 3->1. If these edges are then divided in the ratio of 1.618 to 1, the 12 verticies belong to an icosahedron. Any regular polytope or tiling {p,q,r,...} can be given directed edges, as long as q is even. In 4D, the {3,4,3} gives rise to the snub 24-cell. The {3,6} gives rise to a smaller {3,6}, while the {4,4} gives rise to the snub {4,4} [three triangles and two squares, such that the squares are not touching except at the vertex.]