by **wendy** » Sun Dec 31, 2017 3:46 pm

The various operators do not need to be connected with the CD diagram, but rather one can think in terms of polytopes interacting etc.

Truncation + Rectification.

The process here is to imagine a polytope getting bigger, and the dual getting smaller. This is the 'decent of the dual' The particular truncates + rectates are then points in this process.

0 truncate, 0 rectate. The figure is entirely inside the dual, and at the rectate, the vertices of the figure touch the faces of the dual.

1 truncate, 1 rectate. As the figure rises above the dual, the overtices of the truncate now travel down the edges of the original figure. The rectate happens when the edges are just consumed.

2 truncate, 2 rectate. The vertices of the 2truncate head towards the centre of the hedra (2faces), eventually meeting in the centre of the 2faces (the 2rectate).

&c.

Apiculate + Surtegmate.

This is the dual of the previous process. It produces a convex hull of duals expanding around a figure.

0-apiculate, 0-surtegmate. The dual is expanding inside the figure, (apiculate), reaching the the surtegmate when the vertices of the dual touch the face-centres of the polytope.

1-apiculate, 1-surtegmetate. As the dual expands, it raises peaks on the faces of the figure, these peaks end in this process, when the full edge of the dual is on the surface, the peaks join by pairs in the 1-surtegmate.

2-apiculate, 2-surtegmate. As the dual continues to expand, the edges of the dual are now exposed as the tops of the peaks. This ends when the polygons of the dual are now fully on the surface.

&c.

apiculate = to raise peaks on the

surtegmate = surface tegums (a convex cover of a two orthogonal polytopes, like a rhombus is a cover of its two diagonals).

runcinate, strombiate

The runcinate process, is to push each face outwards, without making them any larger. For example, if you imagine a cube inside a skin, and then pump the cube up, the faces would remain rigid, but move outwards. The skin would expand over the gaps, so you get a rhombo-cuboctahedron. The edges have become squares, and the vertices have stretched into triangles.

In a construction, you take a figure and its dual, and construct it so that each vertices of the figure are replaced with the faces of the dual. The gaps in between are filled with prisms of the surtope of the figure, and the matching surtope of the dual. eg in the runcinated {5,3,3}, the dodecahedra and tetrahedra meet at corners, the pentagons of the dodecahedra form prisms, with the height as the edges of the tetrahedra, the faces of the tetrahedra form prisms with the edges of the dodecahedron.

The strombiate is formed by putting the figure and its dual on the same sphere. The faces of the strombiate are then the the intersections of the figure's faces and its dual. These are generalised rhombuses, or by John Conway, 'strombuses'. The faces are actually the dual of the antiprism of a figure and its dual, or the anti-tegums.

omnitruncate, vaniated.

A pair of duals here. A 'flag' is a triangle or simplex, whose vertices are the centres of the vertex, edge, 2face, 3face, ... Vane = flag (eg a weather-vane is a metal flag that rotates to point to the weather).

vaniated X is to replace the surface of X, with the flags of the surface.

omnitruncate, in the sense of Stott, is to mark every node of the dynkin graph, but here, is to 'expand' a polytope, by pushing the vertex against every surtope.