Wednesday, May 8, 2013

Ko(Tetrahelix) is the 5-4-1 triangulated helix

The Ko() map operation (also known as primal square root of three trisection) converts the tetrahelix (which is a 3-2-1 triangulated helix) into a 5-4-1 triangulated helix, as this paper model shows.

Net for a paper tetrahelix model decorated with Ko(Tetrahelix).

Ko(Tetrahelix) forms a 5-4-1 triangulated helix.
Ko(M) simply superposes the radial, Ra(M), with the edges of the dual, Du(M).

For our purposes, we might just as well superpose the edges of the primal, Id(M), but the same result can be had with Ko(Du(M)). The map operation that superposes the radial, Ra(M), with the edges of the base map is Ki(M), also known as kis, stellation, omnicapping, Psub3, Su2, and 2-dimensional subdivision.

Ki(M) = Ko(Du(M))
Ko(M) = Ki(Du(M)

I was wrong several posts back when I said that a triangulation is mad-weavable if it is the "face-center subdivision" (i.e., kis) of a bipartite map. It suffices to be Ko(M) (or Ki(M)) for any map M.

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