Polytopes with directed elements.

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

Polytopes with directed elements.

Postby ubersketch » Tue Dec 25, 2018 9:28 pm

I’ve been thinking about polytopes with directed elements (non-directed elements are allowed,) and I’ve been asking this question. Do only edges have to be directed or do higher dimensional elements? I think only edges should be, since a directed square can easily be described by arrows.
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Re: Polytopes with directed elements.

Postby mr_e_man » Thu Dec 27, 2018 5:35 am

A directed segment could be described as a pair of directed vertices (one positive and one negative). But this doesn't extend to whole polygons, because one vertex is shared by two edges, which may have conflicting directions to give the vertex. So the edges must support themselves; they can't unload their directions on the vertices.

Likewise, a directed polygon could be described as a cycle of directed edges. But this doesn't extend to whole polyhedra, because one edge is shared by two faces, which may have conflicting directions to give the edge. And one face is surrounded by several edges, which may have conflicting directions to give the face. So the faces must support themselves; they can't unload their directions on the edges.
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Re: Polytopes with directed elements.

Postby ubersketch » Sat Dec 29, 2018 10:52 pm

I see. Thanks for your input.
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Re: Polytopes with directed elements.

Postby 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.]
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