Wythoff polytope (no ontology)

From Hi.gher. Space

A Wythoff polytope is one that might be derived by a mirror-edge construction on a symmetry group that has a Coxeter-Dynkin Symbol. This is a set of position polytopes for which vector algebra can be consistantly applied.

Some theory

Although Mrs A. B. Stott described the process in expansions and contractions from a unit platonic figure, it is easier to understand it in terms of contracting edges of a fully expanded polyhedron. For the octahedral-mirror polyhedra, the figure in question is the truncated cuboctahedron.

This figure has three kinds of edge, which we might designate as vertex between the octagons and squares, edge between the octagons and hexagons, and face, between the hexagons and squares. Were one to increase the length of the vertex edges, the hexagons would remain the same size, but move out radially from the centre. The octagons and squares would generally become irregular.

The seven octahedral-mirror polyhedra are then derived by setting the vertex, edge or face edges to 0 or 1, as follows. The numbers in brackets represent the edge of the (vertex, edge, face) edges.

• The Cube (1,0,0) derives by keeping the vertex edges, but setting the other two to zero.
• The Cuboctahedron (0,1,0) derives from keeping the edge edges at unity, but reducing the vertex and face edges to zero.
• The Octahedron (0,0,1) comes from keeping the face-edges, but setting the vertex and edge edges to zero.
• The Truncated Octahedron (0,1,1) comes from setting just the vertex-edges to zero.
• The Truncated Cube (1,1,0) comes from sett the face-edges to zero
• The Rhombo-Cuboctahedron (1,0,1) comes from setting the edge-edge to zero.
• The Truncated Cuboctahedron (1,1,1) is the complete figure itself.

One is not restricted to keeping these at 0 or 1. In practice, they can assume any value.

From edge-length to vectors

Although the lengths represent the height of the vertex from the opposite mirror, one can represent it as a vector radiating from the centre of the polytope: the polytope (1,1,1) is the sum of polytopes (1,1,0) and (0,0,1).

The base vectors are as in the ordinary vector analysis, the vectors v = (1,0,0), e = (0,1,0), and f = (0,0,1). The edge is the perpendicular of (1,0,0) to the plane v=0, but the vector (1,0,0) points from the centre of the polyhedron to that point.

In vector analysis, each point on the euclidean space represents a position vector, from (0,0,0) to (x,y,z). Now it represents a position polytope, by dropping perpendiculars to each of the planes v=0, e=0, f=0, and reflecting.

The vectors are generally in an oblique coordinate system. The three vectors of the octahedral-mirrors are

• vertex - (1,1,1) gives the vertices of a cube of edge 2.
• edge - (√2,√2,0) gives the vertices of an cuboctahedron, of edge 2.
• face - (√2,0,0) gives the vertices of an octahedron of edge 2.

Just as the ordinary rectangular coordinates are given by all changes of sign, the resulting points of this system are reflected by all permutations, all change of sign.

One can preform the dot product of vectors of this set, not so much by converting them to an orthogonal set, but by the method of matrix dot. The matrix-dot means that the one vector is first multiplied by a matrix, and then the dot product is taken.

Mathematically, it equates to A_i,j v_i w_j, where v and w are vectors. The matrix A_i,j is made of the dot product of vectors like (1,0,0) and (0,1,0).

This is the Stott matrix, because vectors suggest motion, and the variations along each axis here are exactly the Stott operators.

The Dynkin matrix is the dynkin symbol in matrix form. The matrix represents the dot-product of unit normals to the reflective planes, all vectors set to point inwards. When these vectors are taken at the centre of symmetry, one can see that the angle is the complement of the angles between the planes (ie π-^π∕_n).

One can directly get the dynkin matrix from a_ij = -cos(^π∕_bij), where b_ij is the branch mark between nodes i and j. The diagonal is set as one. I normally double each of the entries, and divide move the value of '1/2' outside, because this makes the numbers easier to remember. √2 occurs more often than its half-value.

Because the dynkin and stott matrices are inverses of each other, one can directly calculate the stott matrix from the dynkin matrix. However, it has an independent meaning. One can calculate for example, the vectors v_i for each of the polytopes with just one node marked (eg x3o3o, o3x3o, o3o3x). The entries are then a_ij = v_i · v_j. Most of the stott matrices were thus calculated.

Applications

Apart from finding the diameter of a wythoff figure, these matrices find use in finding the distance between any two points (such as the top and bottom of a segmentotope), from which the square of the height might be found from subtracting it from the square of the lacing-length.

Segmentotopes occur as sections (or segments), between rows of vertices of uniform polychora.