Is there a way to flip a convex group of faces to concave?

[Polytopos]

I'm trying to make a shape in Stell4D that I call the concave triakis octahedron. It has the same face dimensions as the triakis octrahedron but instead of the vertex in the center of the octahedral sides being raised outwards to form the three extra isosceles triangle faces, the vertex is pushed inwards by the same amount.

I've made a paper model attached.

Is there a way to achieve this in Stella?




The isosceles triangle faces have two short edges and one long edge. If the short edge is length 1, the long edge is length 1 + 1/sqroot(2) which is about 1.7071.

[robertw]

It can be done, but there's not a specific operation for it. Here's what I'd do...

Cut off one of the convex peaks using faceting mode. First, drop to the "3-fold pyramidal" subsymmetry group, so we can concentrate on just one peak. Show the original shape in one view, and the faceting preview in another. Enter faceting mode.



In faceting mode, Shift-Left-click on the 3 vertices around the base of the peak, as shown above. Then hit Enter (or Shift-Right-click anywhere) to accept that as a new facet.

You can do the same to trace the tree vertices of one of the side triangles too, or just Ctrl+Right-click on the original triangle to copy it as a new facet.

Now you have all the faces required for just the peak (including its base). Click the left-and-down arrow button to make this the new base model (or use "Faceting->Create Faceted Polyhedron" from the menu).



You probably want to leave faceting mode now, can just hit Esc for that, or select teh normal mode again.

Put this in a memory slot by typing "m1" (or the "Edit->Put Model in Memory" submenu).

You could use augmentation to remove the peaks from the original model at this point, but quicker to just load an octahedron.

Select a face of the octahedron, and hit "a" (or "Poly->Augment Polyhedron"). In the panel that appears, select "Augment using... Memory" in the top section, and "Excavate" lower down. Hit OK. You should see the result you want now. If all looks good, you can confirm the augmentation by hitting Enter or the small green tick button at the top right of the view.

[Polytopos]

Thank you so much Robert for your very thoughtful answer. After a few failed attempts I managed to follow your detailed instructions and produce the concave triakis octahedron! I'm really so so happy about this. I've dreamed of this shape for so many years, and now it's finally before me.

Here is a cross-eyed stereogram of it:

[robertw]

Glad to help. Now you can do similar for various other Archimedean duals too.