Thursday, May 23, 2013

Even-frequency PUCK weaving patterns via the map operation Meta

For orientable surfaces the map operation Bevel is a reliable source of bipartite cubic maps. The duals of these maps are chess-colorable triangulations, and therefore, the chess-colorable triangulations needed for even-frequency PUCK weaving can be generated directly through the map operation Meta ( since Mt(M) = Du(Be(M)). )

Here are some examples of Meta applied to maps that were already triangulations in the first place.

A chess-coloring of Mt(Octahedron)

A chess-coloring of Mt(Icosahedron).

A chess-coloring of Mt(Tetrahelix).

A chess-coloring of Mt(a high frequency triangulation of the Sphere).

A chess-coloring of Mt(a multi-resolution surface mesh.)

Using the map operation Meta to create a PUCK weaving pattern effectively replaces each edge rhomb with a 3-D module like this one, composed of four equilateral triangles: two "outtie" triangles and two "innie" triangles.

This tessellation is 3-regular and even-faced, but it is not the Bevel of anything.

This tessellation is 3-regular and even faced; it is the Bevel of the square grid.

This tessellation is 3-regular and even-faced; it is the Bevel of both the triangle and the hexagon grids.

No comments: