skew polygrams / skew star polygons

[adam ∞]

I am curious, has anyone investigated or seen any information on regular skew polygrams?

When I search for either "skew polygram" or "skew star polygon" I see no results with information about it.

I have in mind a 3-dimensional skew polygon that is a polygram or star polygon.. like a skew polygon, without any area filled in..
It's kind of hard to describe, so here is a quick illustration of something along the lines of what I mean:



^ This one is a skew octagram, (or skew 8-gram), though it's not quite right.. it seems like it almost would share vertices with or fit within a cube or octahedron, but I'm not sure.

something like this maybe:



I am not sure if the angles actually work where you can have them all be the same and meet back up at a starting point, but it seems like it should be possible. I constructed a physical model of what I think would be a regular skew 7-gram recently but I am not sure where it went. I can post an image of the model if I find it or construct another one.

If anyone is good with angles, or vectors or rendering it would interesting to confirm if these are possible as regular polygons.. or if anyone has heard of something like this, maybe by another name, because I am not seeing anything about it when I search for it.

[Klitzing]

The problem you are facing in your example is that you want to do the alternating zigzag of the tetrahedral Petrie polygon with one additional vertex in between. But that one neither can be top nor bottom as either the former or the following one would already be there.

But you could instead use the axial compound of 3 cubes (i.e. the prism of the regular 3 triangle compound) and then considering a {12/5} circuit which alternates between top and bottom. That one then indeed is what you are after.

--- rk

[mr_e_man]

Any regular skew polygon has vertices in two parallel planes, with each set of vertices forming a regular planar n-gon. The skew 2n-gon has each edge connecting a vertex in one plane to a vertex in the other plane.

See the non-convex antiprisms. https://en.wikipedia.org/wiki/Prismatic ... polyhedron